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Номер 24.9, страница 174 - гдз по физике 10-11 класс задачник Гельфгат, Генденштейн
Авторы: Гельфгат И. М., Генденштейн Л. Э., Кирик Л. А.
Тип: Задачник
Издательство: Илекса
Год издания: 2008 - 2026
Уровень обучения: профильный
Цвет обложки: красный лупа, парень едет на велосипеде
ISBN: 978-5-89237-252-7
Популярные ГДЗ в 10 классе
Электродинамика. Электрическое поле. 24. Потенциал электрического поля. Проводники и диэлектрики в электрическом поле. Упражнения - номер 24.9, страница 174.
App\Models\Task {#1030 // resources/views/models/task/default.blade.php #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1319671 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "174" "field_page_end" => "175" "field_url" => "/10-klass/fizika/gelfgat-zadachnik/24-9" "field_display_title" => "24.9" "field_outside_task" => "0" "field_task_type" => Illuminate\Database\Eloquent\Collection {#1037 #items: array:1 [ 0 => App\Models\Term {#1036 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 26 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "номер" "field_cases" => array:6 [ "field_accusative_case" => "номер" "field_creative_case" => "номером" "field_dative_case" => "номеру" "field_genitive_case" => "номера" "field_nominative_case" => "номер" "field_prepositional_case" => "номере" ] "field_short_name" => "№" ] #original: array:6 [ "id" => 26 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "номер" "field_cases" => array:6 [ "field_accusative_case" => "номер" "field_creative_case" => "номером" "field_dative_case" => "номеру" "field_genitive_case" => "номера" "field_nominative_case" => "номер" "field_prepositional_case" => "номере" ] "field_short_name" => "№" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1035 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1074 #items: array:1 [ 0 => App\Models\Branch {#1034 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1319339 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Электродинамика" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1039 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "163" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1038 #items: array:1 [ 0 => App\Models\Book {#1040 #connection: "mysql" #table: "books" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:50 [ "id" => 5400 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1041 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1043 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1046 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1048 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1052 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1053 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1055 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1057 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1059 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1060 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1061 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1063 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1064 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1065 …2} "field_under_the_edition_degree" => null "field_cover_description" => "лупа, парень едет на велосипеде" "field_publication_year" => "2008" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебное издание" "field_allowed" => null "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "459" "field_priority" => null "field_default_folder" => "/fizika_10/gelfgat-z/" "field_isbn" => "978-5-89237-252-7" "field_cover" => array:1 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => "0" "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1066 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1067 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1068 …2} "field_url" => "/10-klass/fizika/gelfgat-zadachnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1069 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #original: array:50 [ "id" => 5400 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1041 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1043 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1046 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1048 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1052 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1053 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1055 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1057 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1059 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1060 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1061 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1063 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1064 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1065 …2} "field_under_the_edition_degree" => null "field_cover_description" => "лупа, парень едет на велосипеде" "field_publication_year" => "2008" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебное издание" "field_allowed" => null "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "459" "field_priority" => null "field_default_folder" => "/fizika_10/gelfgat-z/" "field_isbn" => "978-5-89237-252-7" "field_cover" => array:1 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => "0" "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1066 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1067 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1068 …2} "field_url" => "/10-klass/fizika/gelfgat-zadachnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1069 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1071 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1319339 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Электродинамика" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1039} "field_page_start" => "163" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1038} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1071} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1268 #items: array:4 [ 0 => App\Models\Branch {#1072 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1319339 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Электродинамика" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1075 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "163" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1073 #items: array:1 [ 0 => App\Models\Book {#1076 #connection: "mysql" #table: "books" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:50 [ "id" => 5400 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1077 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1079 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1082 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1084 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1088 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1089 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1091 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1093 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1095 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1096 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1097 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1099 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1100 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1101 …2} "field_under_the_edition_degree" => null "field_cover_description" => "лупа, парень едет на велосипеде" "field_publication_year" => "2008" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебное издание" "field_allowed" => null "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "459" "field_priority" => null "field_default_folder" => "/fizika_10/gelfgat-z/" "field_isbn" => "978-5-89237-252-7" "field_cover" => array:1 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => "0" "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1102 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1103 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1104 …2} "field_url" => "/10-klass/fizika/gelfgat-zadachnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1105 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #original: array:50 [ "id" => 5400 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1077 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1079 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1082 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1084 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1088 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1089 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1091 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1093 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1095 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1096 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1097 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1099 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1100 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1101 …2} "field_under_the_edition_degree" => null "field_cover_description" => "лупа, парень едет на велосипеде" "field_publication_year" => "2008" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебное издание" "field_allowed" => null "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "459" "field_priority" => null "field_default_folder" => "/fizika_10/gelfgat-z/" "field_isbn" => "978-5-89237-252-7" "field_cover" => array:1 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => "0" "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1102 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1103 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1104 …2} "field_url" => "/10-klass/fizika/gelfgat-zadachnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1105 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1107 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1319339 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Электродинамика" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1075} "field_page_start" => "163" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1073} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1107} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Branch {#1108 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1319340 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Электрическое поле" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1109 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "163" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1110 #items: array:1 [ 0 => App\Models\Book {#1111 #connection: "mysql" #table: "books" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:50 [ "id" => 5400 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1112 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1114 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1117 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1119 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1123 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1124 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1126 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1128 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1130 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1131 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1132 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1134 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1135 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1136 …2} "field_under_the_edition_degree" => null "field_cover_description" => "лупа, парень едет на велосипеде" "field_publication_year" => "2008" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебное издание" "field_allowed" => null "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "459" "field_priority" => null "field_default_folder" => "/fizika_10/gelfgat-z/" "field_isbn" => "978-5-89237-252-7" "field_cover" => array:1 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => "0" "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1137 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1138 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1139 …2} "field_url" => "/10-klass/fizika/gelfgat-zadachnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1140 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #original: array:50 [ "id" => 5400 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1112 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1114 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1117 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1119 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1123 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1124 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1126 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1128 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1130 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1131 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1132 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1134 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1135 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1136 …2} "field_under_the_edition_degree" => null "field_cover_description" => "лупа, парень едет на велосипеде" "field_publication_year" => "2008" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебное издание" "field_allowed" => null "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "459" "field_priority" => null "field_default_folder" => "/fizika_10/gelfgat-z/" "field_isbn" => "978-5-89237-252-7" "field_cover" => array:1 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => "0" "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1137 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1138 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1139 …2} "field_url" => "/10-klass/fizika/gelfgat-zadachnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1140 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1142 #items: array:1 [ 0 => App\Models\Branch {#1143 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1319339 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Электродинамика" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1144 …2} "field_page_start" => "163" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1145 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1146 …2} ] #original: array:24 [ "id" => 1319339 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Электродинамика" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1144 …2} "field_page_start" => "163" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1145 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1146 …2} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1319340 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Электрическое поле" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1109} "field_page_start" => "163" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1110} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1142} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 2 => App\Models\Branch {#1147 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1319344 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "24. Потенциал электрического поля. Проводники и диэлектрики в электрическом поле" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1148 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "170" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1149 #items: array:1 [ 0 => App\Models\Book {#1150 #connection: "mysql" #table: "books" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:50 [ "id" => 5400 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1151 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1153 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1156 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1158 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1162 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1163 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1165 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1167 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1169 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1170 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1171 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1173 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1174 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1175 …2} "field_under_the_edition_degree" => null "field_cover_description" => "лупа, парень едет на велосипеде" "field_publication_year" => "2008" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебное издание" "field_allowed" => null "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "459" "field_priority" => null "field_default_folder" => "/fizika_10/gelfgat-z/" "field_isbn" => "978-5-89237-252-7" "field_cover" => array:1 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => "0" "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1176 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1177 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1178 …2} "field_url" => "/10-klass/fizika/gelfgat-zadachnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1179 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #original: array:50 [ "id" => 5400 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1151 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1153 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1156 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1158 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1162 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1163 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1165 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1167 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1169 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1170 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1171 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1173 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1174 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1175 …2} "field_under_the_edition_degree" => null "field_cover_description" => "лупа, парень едет на велосипеде" "field_publication_year" => "2008" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебное издание" "field_allowed" => null "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "459" "field_priority" => null "field_default_folder" => "/fizika_10/gelfgat-z/" "field_isbn" => "978-5-89237-252-7" "field_cover" => array:1 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => "0" "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1176 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1177 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1178 …2} "field_url" => "/10-klass/fizika/gelfgat-zadachnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1179 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1181 #items: array:1 [ 0 => App\Models\Branch {#1182 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1319340 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Электрическое поле" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1183 …2} "field_page_start" => "163" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1184 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1185 …2} ] #original: array:24 [ "id" => 1319340 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Электрическое поле" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1183 …2} "field_page_start" => "163" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1184 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1185 …2} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1319344 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "24. Потенциал электрического поля. Проводники и диэлектрики в электрическом поле" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1148} "field_page_start" => "170" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1149} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1181} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 3 => App\Models\Branch {#1190 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1319345 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Упражнения" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1191 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "170" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1192 #items: array:1 [ 0 => App\Models\Book {#1193 #connection: "mysql" #table: "books" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:50 [ "id" => 5400 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1194 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1196 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1199 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1201 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1205 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1206 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1208 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1210 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1212 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1213 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1214 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1216 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1217 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1218 …2} "field_under_the_edition_degree" => null "field_cover_description" => "лупа, парень едет на велосипеде" "field_publication_year" => "2008" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебное издание" "field_allowed" => null "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "459" "field_priority" => null "field_default_folder" => "/fizika_10/gelfgat-z/" "field_isbn" => "978-5-89237-252-7" "field_cover" => array:1 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => "0" "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1219 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1220 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1221 …2} "field_url" => "/10-klass/fizika/gelfgat-zadachnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1222 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #original: array:50 [ "id" => 5400 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => 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Illuminate\Database\Eloquent\Collection {#1217 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1218 …2} "field_under_the_edition_degree" => null "field_cover_description" => "лупа, парень едет на велосипеде" "field_publication_year" => "2008" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебное издание" "field_allowed" => null "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "459" "field_priority" => null "field_default_folder" => "/fizika_10/gelfgat-z/" "field_isbn" => "978-5-89237-252-7" "field_cover" => array:1 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => "0" "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1219 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1220 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1221 …2} "field_url" => "/10-klass/fizika/gelfgat-zadachnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1222 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1224 #items: array:1 [ 0 => App\Models\Branch {#1225 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1319344 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "24. 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Проводники и диэлектрики в электрическом поле" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1226 …2} "field_page_start" => "170" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1227 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1228 …2} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1319345 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Упражнения" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1191} "field_page_start" => "170" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1192} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1224} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1279 #items: array:2 [ 0 => App\Models\Element {#1287 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ "id" => 1493862 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1296 #items: array:1 [ 0 => App\Models\Edition {#1288 #connection: "mysql" #table: "editions" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 5430 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1290 …2} "field_content_type" => "free" "field_content_mode" => "text, image" "field_page_content_mode" => "image" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Автор" "field_moderator" => "stas" "field_edition_type" => "statement" "field_root_dir" => "0-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1291 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1292 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1293 …2} "field_content_source" => null ] #original: array:21 [ "id" => 5430 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1290 …2} "field_content_type" => "free" "field_content_mode" => "text, image" "field_page_content_mode" => "image" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Автор" "field_moderator" => "stas" "field_edition_type" => "statement" "field_root_dir" => "0-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1291 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1292 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1293 …2} "field_content_source" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "1319671" "type" => "task" ] "text" => "<p><strong>24.9.</strong> Вернитесь к условию задачи 24.8 и определите потенциал поля $\varphi$ на расстоянии $\text{r}$ от центра шара. Постройте график зависимости $\varphi(r)$.</p><p>☑ $\varphi = \frac{Q(3R^2 - r^2)}{8\pi\varepsilon_0 R^3}$ при $r < R$; $\varphi = \frac{Q}{4\pi\varepsilon_0 r}$ при $r > R$. График $\varphi(r)$ см. на рис. б.</p><p><strong>Решение.</strong> При $r > R$ поле заряженного шара совпадает с полем точечного заряда, поэтому $\varphi = \frac{Q}{4\pi\varepsilon_0 r}$. При $r = R$ потенциал $\varphi_0 = \frac{Q}{4\pi\varepsilon_0 R}$. Чтобы найти потенциал при $r < R$, вычислим работу поля по перемещению заряда $\text{q}$ из точки, находящейся на расстоянии $\text{r}$ от центра шара, к поверхности шара. При малом перемещении $\Delta A = qE\Delta r$. Значит, полная работа составит $A = qS_E$ (см. рис. а). Площадь под графиком</p><p>$S_E = \frac{E(r)+E(R)}{2}(R-r) = \frac{Q(R^2 - r^2)}{8\pi\varepsilon_0 R^3}$.</p><p>С другой стороны, $A = q(\varphi(r) - \varphi_0)$.</p><p>Отсюда $\varphi = \frac{Q(3R^2 - r^2)}{8\pi\varepsilon_0 R^3}$. График $\varphi(r)$ приведен на рисунке б. При $r < R$ это парабола, при $r > R$ — гипербола.</p>" "img" => array:2 [ 0 => array:5 [ "name" => "9-1.jpg" "alt" => null "width" => "1426" "height" => 960 "path" => "/media/fizika_10/gelfgat-z/0-24/9-1.webp?ts=1756251100" ] 1 => array:5 [ "name" => "9-2.jpg" "alt" => null "width" => "1336" "height" => 887 "path" => "/media/fizika_10/gelfgat-z/0-24/9-2.webp?ts=1756251100" ] ] ] #original: array:7 [ "id" => 1493862 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1296} "task" => array:2 [ "refs" => "1319671" "type" => "task" ] "text" => "<p><strong>24.9.</strong> Вернитесь к условию задачи 24.8 и определите потенциал поля $\varphi$ на расстоянии $\text{r}$ от центра шара. Постройте график зависимости $\varphi(r)$.</p><p>☑ $\varphi = \frac{Q(3R^2 - r^2)}{8\pi\varepsilon_0 R^3}$ при $r < R$; $\varphi = \frac{Q}{4\pi\varepsilon_0 r}$ при $r > R$. График $\varphi(r)$ см. на рис. б.</p><p><strong>Решение.</strong> При $r > R$ поле заряженного шара совпадает с полем точечного заряда, поэтому $\varphi = \frac{Q}{4\pi\varepsilon_0 r}$. При $r = R$ потенциал $\varphi_0 = \frac{Q}{4\pi\varepsilon_0 R}$. Чтобы найти потенциал при $r < R$, вычислим работу поля по перемещению заряда $\text{q}$ из точки, находящейся на расстоянии $\text{r}$ от центра шара, к поверхности шара. При малом перемещении $\Delta A = qE\Delta r$. Значит, полная работа составит $A = qS_E$ (см. рис. а). Площадь под графиком</p><p>$S_E = \frac{E(r)+E(R)}{2}(R-r) = \frac{Q(R^2 - r^2)}{8\pi\varepsilon_0 R^3}$.</p><p>С другой стороны, $A = q(\varphi(r) - \varphi_0)$.</p><p>Отсюда $\varphi = \frac{Q(3R^2 - r^2)}{8\pi\varepsilon_0 R^3}$. График $\varphi(r)$ приведен на рисунке б. При $r < R$ это парабола, при $r > R$ — гипербола.</p>" "img" => array:2 [ 0 => array:5 [ "name" => "9-1.jpg" "alt" => null "width" => "1426" "height" => 960 "path" => "/media/fizika_10/gelfgat-z/0-24/9-1.webp?ts=1756251100" ] 1 => array:5 [ "name" => "9-2.jpg" "alt" => null "width" => "1336" "height" => 887 "path" => "/media/fizika_10/gelfgat-z/0-24/9-2.webp?ts=1756251100" ] ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Element {#1294 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 1691419 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1304 #items: array:1 [ 0 => App\Models\Edition {#1295 #connection: "mysql" #table: "editions" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 6138 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение" "field_order" => "2" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1298 …2} "field_content_type" => "free" "field_content_mode" => "text" "field_page_content_mode" => "" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Gemini 2.5 Pro" "field_moderator" => "tolik" "field_edition_type" => "solution" "field_root_dir" => "1-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1299 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1300 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1301 …2} "field_content_source" => null ] #original: array:21 [ "id" => 6138 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение" "field_order" => "2" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1298 …2} "field_content_type" => "free" "field_content_mode" => "text" "field_page_content_mode" => "" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Gemini 2.5 Pro" "field_moderator" => "tolik" "field_edition_type" => "solution" "field_root_dir" => "1-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1299 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1300 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1301 …2} "field_content_source" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "1319671" "type" => "task" ] "text" => "<p><strong>Дано</strong></p><p>Шар радиусом $R$.</p><p>Общий заряд шара $Q$, равномерно распределенный по всему объему.</p><p>Электрическая постоянная $\varepsilon_0$.</p><p><strong>Найти</strong></p><p>Потенциал $\phi$ на расстоянии $r$ от центра шара - $\phi(r)$.</p><p>Построить график зависимости $\phi(r)$.</p><p><strong>Решение</strong></p><p>Для определения потенциала $\phi$ воспользуемся его связью с напряженностью электрического поля $E$: $\phi(r) = -\int_{\infty}^{r} E(r') dr'$, приняв потенциал на бесконечности равным нулю ($\phi(\infty) = 0$).</p><p>Прежде всего, необходимо найти напряженность поля $E(r)$, создаваемого равномерно заряженным шаром. Используя теорему Гаусса, можно показать (как в задаче 24.8), что напряженность поля равна:</p><p><strong>1.</strong> Вне шара (при $r \ge R$): поле совпадает с полем точечного заряда $Q$, помещенного в центр шара.<br>$E_{вне}(r) = \frac{Q}{4\pi\varepsilon_0 r^2}$</p><p><strong>2.</strong> Внутри шара (при $r < R$): напряженность поля линейно зависит от расстояния до центра.<br>$E_{вн}(r) = \frac{Q r}{4\pi\varepsilon_0 R^3}$</p><p>Теперь найдем потенциал для каждой из областей.</p><p><strong>1.</strong> Потенциал вне шара (при $r \ge R$)</p><p>Интегрируем напряженность поля от бесконечности до точки на расстоянии $r$ от центра:</p><p>$\phi(r) = -\int_{\infty}^{r} E_{вне}(r') dr' = -\int_{\infty}^{r} \frac{Q}{4\pi\varepsilon_0 r'^2} dr' = \frac{Q}{4\pi\varepsilon_0} \left[ \frac{1}{r'} \right]_{\infty}^{r} = \frac{Q}{4\pi\varepsilon_0} \left( \frac{1}{r} - \frac{1}{\infty} \right) = \frac{Q}{4\pi\varepsilon_0 r}$</p><p>Таким образом, при $r \ge R$ потенциал равен:</p><p>$\phi(r) = \frac{Q}{4\pi\varepsilon_0 r}$</p><p>На поверхности шара (при $r=R$) потенциал будет равен $\phi_0 = \phi(R) = \frac{Q}{4\pi\varepsilon_0 R}$.</p><p><strong>2.</strong> Потенциал внутри шара (при $r < R$)</p><p>Для нахождения потенциала внутри шара, мы не можем интегрировать от бесконечности, так как выражение для поля меняется при $r=R$. Интеграл нужно разбить на две части: от $\infty$ до $R$ и от $R$ до $r$.</p><p>$\phi(r) = -\int_{\infty}^{r} E(r') dr' = -\left( \int_{\infty}^{R} E_{вне}(r') dr' + \int_{R}^{r} E_{вн}(r') dr' \right)$</p><p>Первый интеграл в скобках, $-\int_{\infty}^{R} E_{вне}(r') dr'$, по определению равен потенциалу на поверхности шара $\phi(R) = \phi_0$.</p><p>$\phi(r) = \phi(R) - \int_{R}^{r} E_{вн}(r') dr' = \frac{Q}{4\pi\varepsilon_0 R} - \int_{R}^{r} \frac{Q r'}{4\pi\varepsilon_0 R^3} dr'$</p><p>Вычислим оставшийся интеграл:</p><p>$\int_{R}^{r} \frac{Q r'}{4\pi\varepsilon_0 R^3} dr' = \frac{Q}{4\pi\varepsilon_0 R^3} \left[ \frac{r'^2}{2} \right]_{R}^{r} = \frac{Q}{8\pi\varepsilon_0 R^3} (r^2 - R^2)$</p><p>Подставим это выражение обратно в формулу для потенциала:</p><p>$\phi(r) = \frac{Q}{4\pi\varepsilon_0 R} - \frac{Q(r^2 - R^2)}{8\pi\varepsilon_0 R^3}$</p><p>Приведем к общему знаменателю $8\pi\varepsilon_0 R^3$:</p><p>$\phi(r) = \frac{2QR^2 - Q(r^2 - R^2)}{8\pi\varepsilon_0 R^3} = \frac{Q(2R^2 - r^2 + R^2)}{8\pi\varepsilon_0 R^3} = \frac{Q(3R^2 - r^2)}{8\pi\varepsilon_0 R^3}$</p><p><strong>Построение графика $\phi(r)$</strong></p><p>Мы получили два выражения для потенциала:</p><ul><li>При $r < R$: $\phi(r) = \frac{Q(3R^2 - r^2)}{8\pi\varepsilon_0 R^3}$. Эта зависимость является параболической (ветви параболы направлены вниз из-за знака "минус" перед $r^2$).</li><li>При $r \ge R$: $\phi(r) = \frac{Q}{4\pi\varepsilon_0 r}$. Эта зависимость является гиперболической.</li></ul><p>Найдем значение потенциала в центре шара (при $r=0$):</p><p>$\phi(0) = \frac{Q(3R^2 - 0)}{8\pi\varepsilon_0 R^3} = \frac{3QR^2}{8\pi\varepsilon_0 R^3} = \frac{3}{2} \cdot \frac{Q}{4\pi\varepsilon_0 R} = 1.5 \phi_0$</p><p>Таким образом, потенциал в центре шара в 1.5 раза больше, чем на его поверхности. Начиная с этого максимального значения, потенциал параболически убывает до значения $\phi_0$ на поверхности шара. Далее, за пределами шара, потенциал продолжает убывать по гиперболическому закону, стремясь к нулю на бесконечности. График функции непрерывен в точке $r=R$.</p><p>График зависимости $\phi(r)$ представлен на рисунке б в условии задачи.</p><p><img src="https://i.ibb.co/L5Tpr6R/plot.png" alt="График зависимости потенциала от расстояния"></p><p><strong>Ответ:</strong></p><p>Потенциал поля на расстоянии $r$ от центра шара определяется выражениями:</p><p>При $r < R$ (внутри шара): $\phi = \frac{Q(3R^2 - r^2)}{8\pi\varepsilon_0 R^3}$</p><p>При $r \ge R$ (вне шара): $\phi = \frac{Q}{4\pi\varepsilon_0 r}$</p><p>График зависимости представляет собой параболу внутри шара и гиперболу вне его, с максимальным значением $\phi(0) = 1.5 \phi(R)$ в центре шара.</p>" ] #original: array:6 [ "id" => 1691419 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1304} "task" => array:2 [ "refs" => "1319671" "type" => "task" ] "text" => "<p><strong>Дано</strong></p><p>Шар радиусом $R$.</p><p>Общий заряд шара $Q$, равномерно распределенный по всему объему.</p><p>Электрическая постоянная $\varepsilon_0$.</p><p><strong>Найти</strong></p><p>Потенциал $\phi$ на расстоянии $r$ от центра шара - $\phi(r)$.</p><p>Построить график зависимости $\phi(r)$.</p><p><strong>Решение</strong></p><p>Для определения потенциала $\phi$ воспользуемся его связью с напряженностью электрического поля $E$: $\phi(r) = -\int_{\infty}^{r} E(r') dr'$, приняв потенциал на бесконечности равным нулю ($\phi(\infty) = 0$).</p><p>Прежде всего, необходимо найти напряженность поля $E(r)$, создаваемого равномерно заряженным шаром. Используя теорему Гаусса, можно показать (как в задаче 24.8), что напряженность поля равна:</p><p><strong>1.</strong> Вне шара (при $r \ge R$): поле совпадает с полем точечного заряда $Q$, помещенного в центр шара.<br>$E_{вне}(r) = \frac{Q}{4\pi\varepsilon_0 r^2}$</p><p><strong>2.</strong> Внутри шара (при $r < R$): напряженность поля линейно зависит от расстояния до центра.<br>$E_{вн}(r) = \frac{Q r}{4\pi\varepsilon_0 R^3}$</p><p>Теперь найдем потенциал для каждой из областей.</p><p><strong>1.</strong> Потенциал вне шара (при $r \ge R$)</p><p>Интегрируем напряженность поля от бесконечности до точки на расстоянии $r$ от центра:</p><p>$\phi(r) = -\int_{\infty}^{r} E_{вне}(r') dr' = -\int_{\infty}^{r} \frac{Q}{4\pi\varepsilon_0 r'^2} dr' = \frac{Q}{4\pi\varepsilon_0} \left[ \frac{1}{r'} \right]_{\infty}^{r} = \frac{Q}{4\pi\varepsilon_0} \left( \frac{1}{r} - \frac{1}{\infty} \right) = \frac{Q}{4\pi\varepsilon_0 r}$</p><p>Таким образом, при $r \ge R$ потенциал равен:</p><p>$\phi(r) = \frac{Q}{4\pi\varepsilon_0 r}$</p><p>На поверхности шара (при $r=R$) потенциал будет равен $\phi_0 = \phi(R) = \frac{Q}{4\pi\varepsilon_0 R}$.</p><p><strong>2.</strong> Потенциал внутри шара (при $r < R$)</p><p>Для нахождения потенциала внутри шара, мы не можем интегрировать от бесконечности, так как выражение для поля меняется при $r=R$. Интеграл нужно разбить на две части: от $\infty$ до $R$ и от $R$ до $r$.</p><p>$\phi(r) = -\int_{\infty}^{r} E(r') dr' = -\left( \int_{\infty}^{R} E_{вне}(r') dr' + \int_{R}^{r} E_{вн}(r') dr' \right)$</p><p>Первый интеграл в скобках, $-\int_{\infty}^{R} E_{вне}(r') dr'$, по определению равен потенциалу на поверхности шара $\phi(R) = \phi_0$.</p><p>$\phi(r) = \phi(R) - \int_{R}^{r} E_{вн}(r') dr' = \frac{Q}{4\pi\varepsilon_0 R} - \int_{R}^{r} \frac{Q r'}{4\pi\varepsilon_0 R^3} dr'$</p><p>Вычислим оставшийся интеграл:</p><p>$\int_{R}^{r} \frac{Q r'}{4\pi\varepsilon_0 R^3} dr' = \frac{Q}{4\pi\varepsilon_0 R^3} \left[ \frac{r'^2}{2} \right]_{R}^{r} = \frac{Q}{8\pi\varepsilon_0 R^3} (r^2 - R^2)$</p><p>Подставим это выражение обратно в формулу для потенциала:</p><p>$\phi(r) = \frac{Q}{4\pi\varepsilon_0 R} - \frac{Q(r^2 - R^2)}{8\pi\varepsilon_0 R^3}$</p><p>Приведем к общему знаменателю $8\pi\varepsilon_0 R^3$:</p><p>$\phi(r) = \frac{2QR^2 - Q(r^2 - R^2)}{8\pi\varepsilon_0 R^3} = \frac{Q(2R^2 - r^2 + R^2)}{8\pi\varepsilon_0 R^3} = \frac{Q(3R^2 - r^2)}{8\pi\varepsilon_0 R^3}$</p><p><strong>Построение графика $\phi(r)$</strong></p><p>Мы получили два выражения для потенциала:</p><ul><li>При $r < R$: $\phi(r) = \frac{Q(3R^2 - r^2)}{8\pi\varepsilon_0 R^3}$. Эта зависимость является параболической (ветви параболы направлены вниз из-за знака "минус" перед $r^2$).</li><li>При $r \ge R$: $\phi(r) = \frac{Q}{4\pi\varepsilon_0 r}$. Эта зависимость является гиперболической.</li></ul><p>Найдем значение потенциала в центре шара (при $r=0$):</p><p>$\phi(0) = \frac{Q(3R^2 - 0)}{8\pi\varepsilon_0 R^3} = \frac{3QR^2}{8\pi\varepsilon_0 R^3} = \frac{3}{2} \cdot \frac{Q}{4\pi\varepsilon_0 R} = 1.5 \phi_0$</p><p>Таким образом, потенциал в центре шара в 1.5 раза больше, чем на его поверхности. Начиная с этого максимального значения, потенциал параболически убывает до значения $\phi_0$ на поверхности шара. Далее, за пределами шара, потенциал продолжает убывать по гиперболическому закону, стремясь к нулю на бесконечности. График функции непрерывен в точке $r=R$.</p><p>График зависимости $\phi(r)$ представлен на рисунке б в условии задачи.</p><p><img src="https://i.ibb.co/L5Tpr6R/plot.png" alt="График зависимости потенциала от расстояния"></p><p><strong>Ответ:</strong></p><p>Потенциал поля на расстоянии $r$ от центра шара определяется выражениями:</p><p>При $r < R$ (внутри шара): $\phi = \frac{Q(3R^2 - r^2)}{8\pi\varepsilon_0 R^3}$</p><p>При $r \ge R$ (вне шара): $\phi = \frac{Q}{4\pi\varepsilon_0 r}$</p><p>График зависимости представляет собой параболу внутри шара и гиперболу вне его, с максимальным значением $\phi(0) = 1.5 \phi(R)$ в центре шара.</p>" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1319672" "type" => "task" ] "previous" => array:2 [ "refs" => "1319670" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1320 #items: array:1 [ 0 => App\Models\Book {#1284 #connection: "mysql" #table: "books" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:50 [ "id" => 5400 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1271 #items: array:1 [ 0 => App\Models\Term {#1270 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:10 [ "id" => 6493 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Физика" "field_abbreviated_name" => null "field_cases" => array:6 [ …6] "field_foreign_lang_name" => null "field_short_name" => null "field_subject_type" => "technical_subject" "field_translit" => "fizika" ] #original: array:10 [ "id" => 6493 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Физика" "field_abbreviated_name" => null "field_cases" => array:6 [ …6] "field_foreign_lang_name" => null "field_short_name" => null "field_subject_type" => "technical_subject" "field_translit" => "fizika" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_class" => Illuminate\Database\Eloquent\Collection {#1269 #items: array:2 [ 0 => App\Models\Term {#1276 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 1 => App\Models\Term {#1275 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_publisher" => Illuminate\Database\Eloquent\Collection {#1274 #items: array:1 [ 0 => App\Models\Term {#1272 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_author" => Illuminate\Database\Eloquent\Collection {#1273 #items: array:3 [ 0 => App\Models\Term {#1278 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ …12] #original: array:12 [ …12] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 1 => App\Models\Term {#1277 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ …12] #original: array:12 [ …12] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 2 => App\Models\Term {#1286 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ …12] #original: array:12 [ …12] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1282 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1280 #items: array:1 [ 0 => App\Models\Term {#1283 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:10 [ …10] #original: array:10 [ …10] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_country" => Illuminate\Database\Eloquent\Collection {#1281 #items: array:1 [ 0 => App\Models\Term {#1302 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_city" => Illuminate\Database\Eloquent\Collection {#1303 #items: array:1 [ 0 => App\Models\Term {#1305 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_series" => Illuminate\Database\Eloquent\Collection {#1306 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1307 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1308 #items: array:1 [ 0 => App\Models\Term {#1309 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1310 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1311 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1312 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition_degree" => null "field_cover_description" => "лупа, парень едет на велосипеде" "field_publication_year" => "2008" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебное издание" "field_allowed" => null "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "459" "field_priority" => null "field_default_folder" => "/fizika_10/gelfgat-z/" "field_isbn" => "978-5-89237-252-7" "field_cover" => array:1 [ 0 => "/media/fizika_10/gelfgat-z/covers/cover1.webp?ts=1756049816" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/fizika_10/gelfgat-z/covers/cover1.webp?ts=1756049816" "alt" => "" "width" => "1566" "height" => "2412" ] ] "field_popular_book" => "0" "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1313 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1314 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1315 #items: [] #escapeWhenCastingToString: false } "field_url" => "/10-klass/fizika/gelfgat-zadachnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1316 #items: array:1 [ 0 => App\Models\Term {#1317 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 381 "subject" => 556 "class_subject" => 494 ] ] #original: array:50 [ "id" => 5400 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1271} "field_class" => Illuminate\Database\Eloquent\Collection {#1269} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1274} "field_author" => Illuminate\Database\Eloquent\Collection {#1273} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1282} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1280} "field_country" => Illuminate\Database\Eloquent\Collection {#1281} "field_city" => Illuminate\Database\Eloquent\Collection {#1303} "field_series" => Illuminate\Database\Eloquent\Collection {#1306} "field_umk" => Illuminate\Database\Eloquent\Collection {#1307} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1308} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1310} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1311} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1312} "field_under_the_edition_degree" => null "field_cover_description" => "лупа, парень едет на велосипеде" "field_publication_year" => "2008" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебное издание" "field_allowed" => null "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "459" "field_priority" => null "field_default_folder" => "/fizika_10/gelfgat-z/" "field_isbn" => "978-5-89237-252-7" "field_cover" => array:1 [ 0 => "/media/fizika_10/gelfgat-z/covers/cover1.webp?ts=1756049816" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/fizika_10/gelfgat-z/covers/cover1.webp?ts=1756049816" "alt" => "" "width" => "1566" "height" => "2412" ] ] "field_popular_book" => "0" "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1313} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1314} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1315} "field_url" => "/10-klass/fizika/gelfgat-zadachnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1316} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 381 "subject" => 556 "class_subject" => 494 ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "page" => Illuminate\Database\Eloquent\Collection {#1325 #items: array:1 [ 0 => App\Models\BookPage {#1324 #connection: "mysql" #table: "book_pages" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 1341135 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "174" "field_url" => "/10-klass/fizika/gelfgat-zadachnik/page-174" "field_display_title" => "174" "field_folder" => "1" "field_image_name" => "174" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1326 #items: [] #escapeWhenCastingToString: false } "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1327 #items: array:1 [ 0 => App\Models\Book {#1284} ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "edition_groups" => Illuminate\Database\Eloquent\Collection {#1328 #items: [] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1338 #items: array:1 [ 0 => App\Models\Element {#1337 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1341136" "type" => "book_page" ] "previous" => array:2 [ "refs" => "1341134" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1620 #items: array:2 [ 0 => App\Models\Task {#1628 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ …24] #original: array:24 [ …24] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 1 => App\Models\Task {#1633 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ …24] #original: array:24 [ …24] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } ] #original: array:21 [ "id" => 1341135 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "174" "field_url" => "/10-klass/fizika/gelfgat-zadachnik/page-174" "field_display_title" => "174" "field_folder" => "1" "field_image_name" => "174" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1326} "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1327} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "edition_groups" => Illuminate\Database\Eloquent\Collection {#1328} "content" => Illuminate\Database\Eloquent\Collection {#1338} "next" => array:2 [ "refs" => "1341136" "type" => "book_page" ] "previous" => array:2 [ "refs" => "1341134" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1620} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1319671 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "174" "field_page_end" => "175" "field_url" => "/10-klass/fizika/gelfgat-zadachnik/24-9" "field_display_title" => "24.9" "field_outside_task" => "0" "field_task_type" => Illuminate\Database\Eloquent\Collection {#1037} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1035} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1074} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1268} "content" => Illuminate\Database\Eloquent\Collection {#1279} "next" => array:2 [ "refs" => "1319672" "type" => "task" ] "previous" => array:2 [ "refs" => "1319670" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1320} "page" => array:2 [ "refs" => "1341135" "type" => "book_page" ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] }
№24.9 (с. 174)
Условие. №24.9 (с. 174)
скриншот условия
24.9. Вернитесь к условию задачи 24.8 и определите потенциал поля $\varphi$ на расстоянии $\text{r}$ от центра шара. Постройте график зависимости $\varphi(r)$.
☑ $\varphi = \frac{Q(3R^2 - r^2)}{8\pi\varepsilon_0 R^3}$ при $r < R$; $\varphi = \frac{Q}{4\pi\varepsilon_0 r}$ при $r > R$. График $\varphi(r)$ см. на рис. б.
Решение. При $r > R$ поле заряженного шара совпадает с полем точечного заряда, поэтому $\varphi = \frac{Q}{4\pi\varepsilon_0 r}$. При $r = R$ потенциал $\varphi_0 = \frac{Q}{4\pi\varepsilon_0 R}$. Чтобы найти потенциал при $r < R$, вычислим работу поля по перемещению заряда $\text{q}$ из точки, находящейся на расстоянии $\text{r}$ от центра шара, к поверхности шара. При малом перемещении $\Delta A = qE\Delta r$. Значит, полная работа составит $A = qS_E$ (см. рис. а). Площадь под графиком
$S_E = \frac{E(r)+E(R)}{2}(R-r) = \frac{Q(R^2 - r^2)}{8\pi\varepsilon_0 R^3}$.
С другой стороны, $A = q(\varphi(r) - \varphi_0)$.
Отсюда $\varphi = \frac{Q(3R^2 - r^2)}{8\pi\varepsilon_0 R^3}$. График $\varphi(r)$ приведен на рисунке б. При $r < R$ это парабола, при $r > R$ — гипербола.
Решение. №24.9 (с. 174)
Дано
Шар радиусом $R$.
Общий заряд шара $Q$, равномерно распределенный по всему объему.
Электрическая постоянная $\varepsilon_0$.
Найти
Потенциал $\phi$ на расстоянии $r$ от центра шара - $\phi(r)$.
Построить график зависимости $\phi(r)$.
Решение
Для определения потенциала $\phi$ воспользуемся его связью с напряженностью электрического поля $E$: $\phi(r) = -\int_{\infty}^{r} E(r') dr'$, приняв потенциал на бесконечности равным нулю ($\phi(\infty) = 0$).
Прежде всего, необходимо найти напряженность поля $E(r)$, создаваемого равномерно заряженным шаром. Используя теорему Гаусса, можно показать (как в задаче 24.8), что напряженность поля равна:
1. Вне шара (при $r \ge R$): поле совпадает с полем точечного заряда $Q$, помещенного в центр шара.
$E_{вне}(r) = \frac{Q}{4\pi\varepsilon_0 r^2}$
2. Внутри шара (при $r < R$): напряженность поля линейно зависит от расстояния до центра.
$E_{вн}(r) = \frac{Q r}{4\pi\varepsilon_0 R^3}$
Теперь найдем потенциал для каждой из областей.
1. Потенциал вне шара (при $r \ge R$)
Интегрируем напряженность поля от бесконечности до точки на расстоянии $r$ от центра:
$\phi(r) = -\int_{\infty}^{r} E_{вне}(r') dr' = -\int_{\infty}^{r} \frac{Q}{4\pi\varepsilon_0 r'^2} dr' = \frac{Q}{4\pi\varepsilon_0} \left[ \frac{1}{r'} \right]_{\infty}^{r} = \frac{Q}{4\pi\varepsilon_0} \left( \frac{1}{r} - \frac{1}{\infty} \right) = \frac{Q}{4\pi\varepsilon_0 r}$
Таким образом, при $r \ge R$ потенциал равен:
$\phi(r) = \frac{Q}{4\pi\varepsilon_0 r}$
На поверхности шара (при $r=R$) потенциал будет равен $\phi_0 = \phi(R) = \frac{Q}{4\pi\varepsilon_0 R}$.
2. Потенциал внутри шара (при $r < R$)
Для нахождения потенциала внутри шара, мы не можем интегрировать от бесконечности, так как выражение для поля меняется при $r=R$. Интеграл нужно разбить на две части: от $\infty$ до $R$ и от $R$ до $r$.
$\phi(r) = -\int_{\infty}^{r} E(r') dr' = -\left( \int_{\infty}^{R} E_{вне}(r') dr' + \int_{R}^{r} E_{вн}(r') dr' \right)$
Первый интеграл в скобках, $-\int_{\infty}^{R} E_{вне}(r') dr'$, по определению равен потенциалу на поверхности шара $\phi(R) = \phi_0$.
$\phi(r) = \phi(R) - \int_{R}^{r} E_{вн}(r') dr' = \frac{Q}{4\pi\varepsilon_0 R} - \int_{R}^{r} \frac{Q r'}{4\pi\varepsilon_0 R^3} dr'$
Вычислим оставшийся интеграл:
$\int_{R}^{r} \frac{Q r'}{4\pi\varepsilon_0 R^3} dr' = \frac{Q}{4\pi\varepsilon_0 R^3} \left[ \frac{r'^2}{2} \right]_{R}^{r} = \frac{Q}{8\pi\varepsilon_0 R^3} (r^2 - R^2)$
Подставим это выражение обратно в формулу для потенциала:
$\phi(r) = \frac{Q}{4\pi\varepsilon_0 R} - \frac{Q(r^2 - R^2)}{8\pi\varepsilon_0 R^3}$
Приведем к общему знаменателю $8\pi\varepsilon_0 R^3$:
$\phi(r) = \frac{2QR^2 - Q(r^2 - R^2)}{8\pi\varepsilon_0 R^3} = \frac{Q(2R^2 - r^2 + R^2)}{8\pi\varepsilon_0 R^3} = \frac{Q(3R^2 - r^2)}{8\pi\varepsilon_0 R^3}$
Построение графика $\phi(r)$
Мы получили два выражения для потенциала:
- При $r < R$: $\phi(r) = \frac{Q(3R^2 - r^2)}{8\pi\varepsilon_0 R^3}$. Эта зависимость является параболической (ветви параболы направлены вниз из-за знака "минус" перед $r^2$).
- При $r \ge R$: $\phi(r) = \frac{Q}{4\pi\varepsilon_0 r}$. Эта зависимость является гиперболической.
Найдем значение потенциала в центре шара (при $r=0$):
$\phi(0) = \frac{Q(3R^2 - 0)}{8\pi\varepsilon_0 R^3} = \frac{3QR^2}{8\pi\varepsilon_0 R^3} = \frac{3}{2} \cdot \frac{Q}{4\pi\varepsilon_0 R} = 1.5 \phi_0$
Таким образом, потенциал в центре шара в 1.5 раза больше, чем на его поверхности. Начиная с этого максимального значения, потенциал параболически убывает до значения $\phi_0$ на поверхности шара. Далее, за пределами шара, потенциал продолжает убывать по гиперболическому закону, стремясь к нулю на бесконечности. График функции непрерывен в точке $r=R$.
График зависимости $\phi(r)$ представлен на рисунке б в условии задачи.
Ответ:
Потенциал поля на расстоянии $r$ от центра шара определяется выражениями:
При $r < R$ (внутри шара): $\phi = \frac{Q(3R^2 - r^2)}{8\pi\varepsilon_0 R^3}$
При $r \ge R$ (вне шара): $\phi = \frac{Q}{4\pi\varepsilon_0 r}$
График зависимости представляет собой параболу внутри шара и гиперболу вне его, с максимальным значением $\phi(0) = 1.5 \phi(R)$ в центре шара.
Другие задания:
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