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Номер 114, страница 199 - 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. Постоянный электрический ток. 27. Последовательное и параллельное соединение проводников. Олимпиадные задачи - номер 114, страница 199.
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" => 1319710 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "199" "field_page_end" => null "field_url" => "/10-klass/fizika/gelfgat-zadachnik/olimp-114" "field_display_title" => "114" "field_outside_task" => null "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 {#1046 #items: array:1 [ 0 => App\Models\Branch {#1045 #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 {#1047 #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 {#1081 #items: array:1 [ 0 => App\Models\Book {#1048 #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 {#1050 #items: array:1 [ 0 => App\Models\Term {#1049 #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 [ "field_accusative_case" => "физику" "field_creative_case" => "физикой" "field_dative_case" => "физике" "field_genitive_case" => "физики" "field_nominative_case" => "физика" "field_prepositional_case" => "физике" ] "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 [ "field_accusative_case" => "физику" "field_creative_case" => "физикой" "field_dative_case" => "физике" "field_genitive_case" => "физики" "field_nominative_case" => "физика" "field_prepositional_case" => "физике" ] "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 {#1051 #items: array:2 [ 0 => App\Models\Term {#1052 #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" => 5459 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "10" "field_cases" => array:6 [ "field_accusative_case" => "десятый" "field_creative_case" => "десятым" "field_dative_case" => "десятому" "field_genitive_case" => "десятого" "field_nominative_case" => "десятый" "field_prepositional_case" => "десятом" ] "field_translit" => "desjatyj" ] #original: array:6 [ "id" => 5459 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "10" "field_cases" => array:6 [ "field_accusative_case" => "десятый" "field_creative_case" => "десятым" "field_dative_case" => "десятому" "field_genitive_case" => "десятого" "field_nominative_case" => "десятый" "field_prepositional_case" => "десятом" ] "field_translit" => "desjatyj" ] #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\Term {#1053 #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" => 5460 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "11" "field_cases" => array:6 [ "field_accusative_case" => "одиннадцатый" "field_creative_case" => "одиннадцатым" "field_dative_case" => "одиннадцатому" "field_genitive_case" => "одинадцатого" "field_nominative_case" => "одинадцатый" "field_prepositional_case" => "одиннадцатом" ] "field_translit" => "odinadcatyj" ] #original: array:6 [ "id" => 5460 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "11" "field_cases" => array:6 [ "field_accusative_case" => "одиннадцатый" "field_creative_case" => "одиннадцатым" "field_dative_case" => "одиннадцатому" "field_genitive_case" => "одинадцатого" "field_nominative_case" => "одинадцатый" "field_prepositional_case" => "одиннадцатом" ] "field_translit" => "odinadcatyj" ] #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_publisher" => Illuminate\Database\Eloquent\Collection {#1054 #items: array:1 [ 0 => App\Models\Term {#1055 #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" => 5158 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Илекса" "field_cases" => null "field_translit" => "ileksa" ] #original: array:6 [ "id" => 5158 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Илекса" "field_cases" => null "field_translit" => "ileksa" ] #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_author" => Illuminate\Database\Eloquent\Collection {#1056 #items: array:3 [ 0 => App\Models\Term {#1057 #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 [ "id" => 4102 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Гельфгат" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Илья" "field_patronymic" => "Маркович" "field_surname_rp" => null ] #original: array:12 [ "id" => 4102 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Гельфгат" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Илья" "field_patronymic" => "Маркович" "field_surname_rp" => null ] #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\Term {#1058 #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 [ "id" => 4104 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Генденштейн" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Лев" "field_patronymic" => "Элевич" "field_surname_rp" => "Генденштейна" ] #original: array:12 [ "id" => 4104 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Генденштейн" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Лев" "field_patronymic" => "Элевич" "field_surname_rp" => "Генденштейна" ] #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\Term {#1059 #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 [ "id" => 4583 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Кирик" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Леонид" "field_patronymic" => "Анатольевич" "field_surname_rp" => null ] #original: array:12 [ "id" => 4583 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Кирик" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Леонид" "field_patronymic" => "Анатольевич" "field_surname_rp" => 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 } "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1060 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1061 #items: array:1 [ 0 => App\Models\Term {#1062 #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" => 6518 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Задачник" "field_book_type_foreign" => null "field_cases" => array:6 [ "field_accusative_case" => "задачник" "field_creative_case" => "задачником" "field_dative_case" => "задачнику" "field_genitive_case" => "задачника" "field_nominative_case" => "задачник" "field_prepositional_case" => "задачнике" ] "field_plural_form" => null "field_short_name" => null "field_short_name_foreign" => null "field_translit" => "zadachnik" ] #original: array:10 [ "id" => 6518 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Задачник" "field_book_type_foreign" => null "field_cases" => array:6 [ "field_accusative_case" => "задачник" "field_creative_case" => "задачником" "field_dative_case" => "задачнику" "field_genitive_case" => "задачника" "field_nominative_case" => "задачник" "field_prepositional_case" => "задачнике" ] "field_plural_form" => null "field_short_name" => null "field_short_name_foreign" => null "field_translit" => "zadachnik" ] #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_country" => Illuminate\Database\Eloquent\Collection {#1063 #items: array:1 [ 0 => App\Models\Term {#1064 #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" => 9 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Россия" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "rossiya" ] #original: array:6 [ "id" => 9 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Россия" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "rossiya" ] #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_city" => Illuminate\Database\Eloquent\Collection {#1065 #items: array:1 [ 0 => App\Models\Term {#1066 #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" => 15 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Москва" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "moskva" ] #original: array:6 [ "id" => 15 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Москва" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "moskva" ] #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_series" => Illuminate\Database\Eloquent\Collection {#1067 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1068 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1069 #items: array:1 [ 0 => App\Models\Term {#1070 #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" => 40 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "профильный" "field_cases" => array:6 [ "field_accusative_case" => "профильный" "field_creative_case" => "профильным" "field_dative_case" => "профильному" "field_genitive_case" => "профильного" "field_nominative_case" => "профильный" "field_prepositional_case" => "профильном" ] "field_translit" => "profilnyy" ] #original: array:6 [ "id" => 40 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "профильный" "field_cases" => array:6 [ "field_accusative_case" => "профильный" "field_creative_case" => "профильным" "field_dative_case" => "профильному" "field_genitive_case" => "профильного" "field_nominative_case" => "профильный" "field_prepositional_case" => "профильном" ] "field_translit" => "profilnyy" ] #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_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1071 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1072 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1073 #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 {#1074 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1075 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1076 #items: [] #escapeWhenCastingToString: false } "field_url" => "/10-klass/fizika/gelfgat-zadachnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1077 #items: array:1 [ 0 => App\Models\Term {#1078 #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" => 46 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "красный" "field_cases" => array:6 [ "field_accusative_case" => "красный" "field_creative_case" => "красным" "field_dative_case" => "красному" "field_genitive_case" => "красного" "field_nominative_case" => "красный" "field_prepositional_case" => "красном" ] "field_translit" => "krasnyy" ] #original: array:6 [ "id" => 46 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "красный" "field_cases" => array:6 [ "field_accusative_case" => "красный" "field_creative_case" => "красным" "field_dative_case" => "красному" "field_genitive_case" => "красного" "field_nominative_case" => "красный" "field_prepositional_case" => "красном" ] "field_translit" => "krasnyy" ] #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 "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 {#1050} "field_class" => Illuminate\Database\Eloquent\Collection {#1051} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1054} "field_author" => Illuminate\Database\Eloquent\Collection {#1056} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1060} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1061} "field_country" => Illuminate\Database\Eloquent\Collection {#1063} "field_city" => Illuminate\Database\Eloquent\Collection {#1065} "field_series" => Illuminate\Database\Eloquent\Collection {#1067} "field_umk" => Illuminate\Database\Eloquent\Collection {#1068} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1069} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1071} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1072} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1073} "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 {#1074} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1075} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1076} "field_url" => "/10-klass/fizika/gelfgat-zadachnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1077} "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 } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1079 #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 {#1047} "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 {#1081} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1079} ] #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 {#1111 #items: array:4 [ 0 => App\Models\Branch {#1121 #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 {#1122 #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 {#1123 #items: array:1 [ 0 => App\Models\Book {#1048} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1124 #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 {#1122} "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 {#1123} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1124} ] #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 {#1125 #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" => 1319350 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Постоянный электрический ток" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1126 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "188" "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 {#1127 #items: array:1 [ 0 => App\Models\Book {#1048} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1128 #items: array:1 [ 0 => App\Models\Branch {#1045} ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1319350 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Постоянный электрический ток" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1126} "field_page_start" => "188" "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 {#1127} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1128} ] #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 {#1129 #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" => 1319352 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "27. Последовательное и параллельное соединение проводников" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1130 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "190" "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 {#1131 #items: array:1 [ 0 => App\Models\Book {#1048} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1140 #items: array:1 [ 0 => App\Models\Branch {#1139 #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" => 1319350 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Постоянный электрический ток" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1141 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "188" "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 {#1142 #items: array:1 [ 0 => App\Models\Book {#1048} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1143 #items: array:1 [ 0 => App\Models\Branch {#1045} ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1319350 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Постоянный электрический ток" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1141} "field_page_start" => "188" "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 {#1142} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1143} ] #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" => 1319352 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "27. Последовательное и параллельное соединение проводников" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1130} "field_page_start" => "190" "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 {#1131} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1140} ] #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 {#1133 #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" => 1319354 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Олимпиадные задачи" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1132 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "195" "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 {#1136 #items: array:1 [ 0 => App\Models\Book {#1048} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1149 #items: array:1 [ 0 => App\Models\Branch {#1148 #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" => 1319352 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "27. Последовательное и параллельное соединение проводников" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1150 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "190" "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 {#1151 #items: array:1 [ 0 => App\Models\Book {#1048} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1152 #items: array:1 [ 0 => App\Models\Branch {#1139} ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1319352 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "27. Последовательное и параллельное соединение проводников" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1150} "field_page_start" => "190" "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 {#1151} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1152} ] #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" => 1319354 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Олимпиадные задачи" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1132} "field_page_start" => "195" "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 {#1136} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1149} ] #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 {#1038 #items: array:2 [ 0 => App\Models\Element {#1102 #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" => 1494092 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1092 #items: array:1 [ 0 => App\Models\Edition {#1101 #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 {#1098 #items: array:1 [ 0 => App\Models\Term {#1100 #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" => 5158 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Илекса" "field_cases" => null "field_translit" => "ileksa" ] #original: array:6 [ "id" => 5158 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Илекса" "field_cases" => null "field_translit" => "ileksa" ] #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_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 {#1099 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1097 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1096 #items: [] #escapeWhenCastingToString: false } "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 {#1098} "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 {#1099} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1097} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1096} "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" => "1319710" "type" => "task" ] "text" => "<p><strong>O14.</strong> В цепи, представленной на рисунке, гальванометр показывает отсутствие тока. Выразите сопротивление $R_x$ через $R_1, R_2, R_3$.</p><p>☑ $ R_x = R_1 R_3 / R_2$.</p><p><strong><em>Решение.</em></strong> Ток не идет через гальванометр, когда прибор включен между точками с равными потенциалами. Следовательно, гальванометр можно изъять из цепи (см. рисунок).</p><p>Тогда из условия $φ_A = φ_B$ находим $U_x = U_1$, $U_3 = U_2$. Поскольку $U_x / U_3 = R_x / R_3$ и $U_1 / U_2 = R_1 / R_2$, получаем $R_x / R_3 = R_1 / R_2$.</p>" "img" => array:1 [ 0 => array:5 [ "name" => "114-1.jpg" "alt" => null "width" => "1450" "height" => 1591 "path" => "/media/fizika_10/gelfgat-z/0-olimp/114-1.webp?ts=1756251262" ] ] ] #original: array:7 [ "id" => 1494092 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1092} "task" => array:2 [ "refs" => "1319710" "type" => "task" ] "text" => "<p><strong>O14.</strong> В цепи, представленной на рисунке, гальванометр показывает отсутствие тока. Выразите сопротивление $R_x$ через $R_1, R_2, R_3$.</p><p>☑ $ R_x = R_1 R_3 / R_2$.</p><p><strong><em>Решение.</em></strong> Ток не идет через гальванометр, когда прибор включен между точками с равными потенциалами. Следовательно, гальванометр можно изъять из цепи (см. рисунок).</p><p>Тогда из условия $φ_A = φ_B$ находим $U_x = U_1$, $U_3 = U_2$. Поскольку $U_x / U_3 = R_x / R_3$ и $U_1 / U_2 = R_1 / R_2$, получаем $R_x / R_3 = R_1 / R_2$.</p>" "img" => array:1 [ 0 => array:5 [ "name" => "114-1.jpg" "alt" => null "width" => "1450" "height" => 1591 "path" => "/media/fizika_10/gelfgat-z/0-olimp/114-1.webp?ts=1756251262" ] ] ] #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 {#1094 #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" => 1691649 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1086 #items: array:1 [ 0 => App\Models\Edition {#1095 #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 {#1090 #items: array:1 [ 0 => App\Models\Term {#1093 #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" => 7012 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Gemini 2.5 Pro" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "gemini 2.5 pro" ] #original: array:6 [ "id" => 7012 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Gemini 2.5 Pro" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "gemini 2.5 pro" ] #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_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 {#1091 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1089 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1085 #items: [] #escapeWhenCastingToString: false } "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 {#1090} "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 {#1091} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1089} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1085} "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" => "1319710" "type" => "task" ] "text" => "<p><strong>Дано:</strong></p><p>Электрическая цепь, состоящая из четырех резисторов $R_1$, $R_2$, $R_3$, $R_x$ и гальванометра $G$, соединенных по схеме измерительного моста (мост Уитстона).</p><p>Гальванометр показывает отсутствие тока, $I_G = 0$.</p><p><strong>Найти:</strong></p><p>Выразить сопротивление $R_x$ через $R_1$, $R_2$, $R_3$.</p><p><strong>Решение:</strong></p><p>Схема, представленная на рисунке, является мостом Уитстона. Условие, что гальванометр показывает отсутствие тока ($I_G = 0$), означает, что мост сбалансирован. Это возможно только в том случае, если потенциалы в точках, к которым подключен гальванометр, равны.</p><p>Обозначим узел между резисторами $R_x$ и $R_3$ как точку A, а узел между резисторами $R_1$ и $R_2$ как точку B. Условие баланса моста записывается как равенство потенциалов:</p><p>$\phi_A = \phi_B$</p><p>Поскольку ток через гальванометр не течет, ток, протекающий через резистор $R_x$, также протекает и через резистор $R_3$. Обозначим этот ток как $I_{x3}$. Аналогично, ток, протекающий через резистор $R_1$, также протекает и через резистор $R_2$. Обозначим этот ток как $I_{12}$.</p><p>$I_{x3} = I_x = I_3$</p><p>$I_{12} = I_1 = I_2$</p><p>Равенство потенциалов $\phi_A = \phi_B$ означает, что падение напряжения на резисторе $R_x$ равно падению напряжения на резисторе $R_1$.</p><p>$U_x = U_1$</p><p>Используя закон Ома, получаем:</p><p>$I_{x3} \cdot R_x = I_{12} \cdot R_1$ (1)</p><p>Также из условия $\phi_A = \phi_B$ следует, что падение напряжения на резисторе $R_3$ равно падению напряжения на резисторе $R_2$.</p><p>$U_3 = U_2$</p><p>$I_{x3} \cdot R_3 = I_{12} \cdot R_2$ (2)</p><p>Теперь у нас есть система из двух уравнений. Разделим уравнение (1) на уравнение (2):</p><p>$\frac{I_{x3} \cdot R_x}{I_{x3} \cdot R_3} = \frac{I_{12} \cdot R_1}{I_{12} \cdot R_2}$</p><p>Сократив токи $I_{x3}$ и $I_{12}$, получим пропорцию:</p><p>$\frac{R_x}{R_3} = \frac{R_1}{R_2}$</p><p>Из этой пропорции выразим искомое сопротивление $R_x$:</p><p>$R_x = \frac{R_1 \cdot R_3}{R_2}$</p><p><strong>Ответ:</strong> $R_x = \frac{R_1 R_3}{R_2}$.</p>" ] #original: array:6 [ "id" => 1691649 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1086} "task" => array:2 [ "refs" => "1319710" "type" => "task" ] "text" => "<p><strong>Дано:</strong></p><p>Электрическая цепь, состоящая из четырех резисторов $R_1$, $R_2$, $R_3$, $R_x$ и гальванометра $G$, соединенных по схеме измерительного моста (мост Уитстона).</p><p>Гальванометр показывает отсутствие тока, $I_G = 0$.</p><p><strong>Найти:</strong></p><p>Выразить сопротивление $R_x$ через $R_1$, $R_2$, $R_3$.</p><p><strong>Решение:</strong></p><p>Схема, представленная на рисунке, является мостом Уитстона. Условие, что гальванометр показывает отсутствие тока ($I_G = 0$), означает, что мост сбалансирован. Это возможно только в том случае, если потенциалы в точках, к которым подключен гальванометр, равны.</p><p>Обозначим узел между резисторами $R_x$ и $R_3$ как точку A, а узел между резисторами $R_1$ и $R_2$ как точку B. Условие баланса моста записывается как равенство потенциалов:</p><p>$\phi_A = \phi_B$</p><p>Поскольку ток через гальванометр не течет, ток, протекающий через резистор $R_x$, также протекает и через резистор $R_3$. Обозначим этот ток как $I_{x3}$. Аналогично, ток, протекающий через резистор $R_1$, также протекает и через резистор $R_2$. Обозначим этот ток как $I_{12}$.</p><p>$I_{x3} = I_x = I_3$</p><p>$I_{12} = I_1 = I_2$</p><p>Равенство потенциалов $\phi_A = \phi_B$ означает, что падение напряжения на резисторе $R_x$ равно падению напряжения на резисторе $R_1$.</p><p>$U_x = U_1$</p><p>Используя закон Ома, получаем:</p><p>$I_{x3} \cdot R_x = I_{12} \cdot R_1$ (1)</p><p>Также из условия $\phi_A = \phi_B$ следует, что падение напряжения на резисторе $R_3$ равно падению напряжения на резисторе $R_2$.</p><p>$U_3 = U_2$</p><p>$I_{x3} \cdot R_3 = I_{12} \cdot R_2$ (2)</p><p>Теперь у нас есть система из двух уравнений. Разделим уравнение (1) на уравнение (2):</p><p>$\frac{I_{x3} \cdot R_x}{I_{x3} \cdot R_3} = \frac{I_{12} \cdot R_1}{I_{12} \cdot R_2}$</p><p>Сократив токи $I_{x3}$ и $I_{12}$, получим пропорцию:</p><p>$\frac{R_x}{R_3} = \frac{R_1}{R_2}$</p><p>Из этой пропорции выразим искомое сопротивление $R_x$:</p><p>$R_x = \frac{R_1 \cdot R_3}{R_2}$</p><p><strong>Ответ:</strong> $R_x = \frac{R_1 R_3}{R_2}$.</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" => "1319711" "type" => "task" ] "previous" => array:2 [ "refs" => "1319709" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1105 #items: array:1 [ 0 => App\Models\Book {#1048} ] #escapeWhenCastingToString: false } "page" => Illuminate\Database\Eloquent\Collection {#1042 #items: array:1 [ 0 => App\Models\BookPage {#1107 #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" => 1341160 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "199" "field_url" => "/10-klass/fizika/gelfgat-zadachnik/page-199" "field_display_title" => "199" "field_folder" => "1" "field_image_name" => "199" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1088 #items: [] #escapeWhenCastingToString: false } "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1087 #items: array:1 [ 0 => App\Models\Book {#1048} ] #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 {#1083 #items: [] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1117 #items: array:1 [ 0 => App\Models\Element {#1109 #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" => 1519705 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1116 #items: array:1 [ 0 => App\Models\Edition {#1101} ] #escapeWhenCastingToString: false } "book_page" => array:2 [ "refs" => "1341160" "type" => "book_page" ] "img" => array:1 [ 0 => array:5 [ "name" => "199-1.jpg" "alt" => null "width" => "1619" "height" => 2510 "path" => "/media/fizika_10/gelfgat-z/0-1/199-1.webp?ts=1756642376" ] ] ] #original: array:6 [ "id" => 1519705 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1116} "book_page" => array:2 [ "refs" => "1341160" "type" => "book_page" ] "img" => array:1 [ 0 => array:5 [ "name" => "199-1.jpg" "alt" => null "width" => "1619" "height" => 2510 "path" => "/media/fizika_10/gelfgat-z/0-1/199-1.webp?ts=1756642376" ] ] ] #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" => "1341161" "type" => "book_page" ] "previous" => array:2 [ "refs" => "1341159" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1220 #items: array:2 [ 0 => App\Models\Task {#1228 #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" => 1319709 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "198" "field_page_end" => "199" "field_url" => "/10-klass/fizika/gelfgat-zadachnik/olimp-113" "field_display_title" => "113" "field_outside_task" => "0" "field_task_type" => Illuminate\Database\Eloquent\Collection {#1229 #items: array:1 [ 0 => App\Models\Term {#1036} ] #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 {#1230 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1231 #items: array:1 [ 0 => App\Models\Branch {#1121} ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1232 #items: array:4 [ 0 => App\Models\Branch {#1121} 1 => App\Models\Branch {#1125} 2 => App\Models\Branch {#1129} 3 => App\Models\Branch {#1133} ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1234 #items: array:2 [ 0 => App\Models\Element {#1243 #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" => 1494091 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1244 #items: array:1 [ 0 => App\Models\Edition {#1101} ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "1319709" "type" => "task" ] "text" => "<p><strong>O13.</strong> Какой ток течет через каждый из резисторов (см. рисунок), если к цепи приложено напряжение $U = 84$ В? Сопротивления резисторов $R_1 = R_5 = R_8 = 12$ Ом; $R_2 = R_6 = R_7 = 6,0$ Ом; $R_3 = 3,0$ Ом; $R_4 = 24$ Ом.</p><p>☑ $I_1 = I_2 = I_3 = 4$ А, $I_4 = I_7 = I_8 = 2$ А, $I_5 = I_6 = 0$.</p><p><strong><em>Решение.</em></strong> Простейшая эквивалентная схема получается, если «вытянуть» в одну линию резисторы 1, 2, 3 и 4, 8, 7 (см. рисунок). Поскольку $R_1:R_2:R_3 = R_4:R_8:R_7$, нетрудно показать, что потенциалы точек А и С, В и D совпадают: $\Phi_A = \Phi_C$, $\Phi_B = \Phi_D$. Поэтому ток через резисторы $R_5$ и $R_6$ не течет: $I_5 = I_6 = 0$.</p><p>После изъятия из цепи этих резисторов расчет токов не вызывает затруднений:</p><p>$I_1 = I_2 = I_3 = \frac{U}{R_1 + R_2 + R_3} = 4$ (А),</p><p>$I_4 = I_7 = I_8 = \frac{U}{R_4 + R_7 + R_8} = 2$ (А).</p>" "img" => array:2 [ 0 => array:5 [ "name" => "113-1.jpg" "alt" => null "width" => "1406" "height" => 1591 "path" => "/media/fizika_10/gelfgat-z/0-olimp/113-1.webp?ts=1756251262" ] 1 => array:5 [ "name" => "113-2.jpg" "alt" => null "width" => "874" "height" => 381 "path" => "/media/fizika_10/gelfgat-z/0-olimp/113-2.webp?ts=1756251262" ] ] ] #original: array:7 [ "id" => 1494091 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1244} "task" => array:2 [ "refs" => "1319709" "type" => "task" ] "text" => "<p><strong>O13.</strong> Какой ток течет через каждый из резисторов (см. рисунок), если к цепи приложено напряжение $U = 84$ В? Сопротивления резисторов $R_1 = R_5 = R_8 = 12$ Ом; $R_2 = R_6 = R_7 = 6,0$ Ом; $R_3 = 3,0$ Ом; $R_4 = 24$ Ом.</p><p>☑ $I_1 = I_2 = I_3 = 4$ А, $I_4 = I_7 = I_8 = 2$ А, $I_5 = I_6 = 0$.</p><p><strong><em>Решение.</em></strong> Простейшая эквивалентная схема получается, если «вытянуть» в одну линию резисторы 1, 2, 3 и 4, 8, 7 (см. рисунок). Поскольку $R_1:R_2:R_3 = R_4:R_8:R_7$, нетрудно показать, что потенциалы точек А и С, В и D совпадают: $\Phi_A = \Phi_C$, $\Phi_B = \Phi_D$. Поэтому ток через резисторы $R_5$ и $R_6$ не течет: $I_5 = I_6 = 0$.</p><p>После изъятия из цепи этих резисторов расчет токов не вызывает затруднений:</p><p>$I_1 = I_2 = I_3 = \frac{U}{R_1 + R_2 + R_3} = 4$ (А),</p><p>$I_4 = I_7 = I_8 = \frac{U}{R_4 + R_7 + R_8} = 2$ (А).</p>" "img" => array:2 [ 0 => array:5 [ "name" => "113-1.jpg" "alt" => null "width" => "1406" "height" => 1591 "path" => "/media/fizika_10/gelfgat-z/0-olimp/113-1.webp?ts=1756251262" ] 1 => array:5 [ "name" => "113-2.jpg" "alt" => null "width" => "874" "height" => 381 "path" => "/media/fizika_10/gelfgat-z/0-olimp/113-2.webp?ts=1756251262" ] ] ] #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 {#1245 #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" => 1691648 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1246 #items: array:1 [ 0 => App\Models\Edition {#1095} ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "1319709" "type" => "task" ] "text" => "<p><strong>Дано:</strong></p><p>$U = 84 \text{ В}$<br>$R_1 = 12 \text{ Ом}$<br>$R_2 = 6.0 \text{ Ом}$<br>$R_3 = 3.0 \text{ Ом}$<br>$R_4 = 24 \text{ Ом}$<br>$R_5 = 12 \text{ Ом}$<br>$R_6 = 6.0 \text{ Ом}$<br>$R_7 = 6.0 \text{ Ом}$<br>$R_8 = 12 \text{ Ом}$</p><p><strong>Найти:</strong></p><p>$I_1, I_2, I_3, I_4, I_5, I_6, I_7, I_8$ - ?</p><p><strong>Решение:</strong></p><p>Данная электрическая схема представляет собой симметричный мост. Особенностью таких схем является то, что если сопротивления в плечах моста пропорциональны, то мост является сбалансированным, и ток через соединительные элементы (в данном случае $R_5$ и $R_6$) не течет.</p><p>Проверим условие баланса моста. Для этого сравним отношения сопротивлений в верхней и нижней ветвях схемы.</p><p>Отношение сопротивлений в верхней ветви (резисторы $R_1, R_2, R_3$):</p><p>$R_1 : R_2 : R_3 = 12 : 6.0 : 3.0 = 4 : 2 : 1$</p><p>Отношение сопротивлений в нижней ветви (резисторы $R_4, R_8, R_7$):</p><p>$R_4 : R_8 : R_7 = 24 : 12 : 6.0 = 4 : 2 : 1$</p><p>Так как отношения сопротивлений в плечах равны ($R_1:R_2:R_3 = R_4:R_8:R_7$), мост сбалансирован. Это означает, что потенциалы в точках, соединенных резисторами $R_5$ и $R_6$, соответственно равны. Следовательно, разность потенциалов на резисторах $R_5$ и $R_6$ равна нулю, и ток через них не протекает.</p><p>$I_5 = 0 \text{ А}$</p><p>$I_6 = 0 \text{ А}$</p><p>Поскольку ток через резисторы $R_5$ и $R_6$ отсутствует, их можно мысленно удалить из схемы для дальнейших расчетов. Схема упрощается до двух параллельных ветвей, подключенных к источнику напряжения $U$.</p><p>Первая ветвь состоит из последовательно соединенных резисторов $R_1, R_2$ и $R_3$. Ток через эти резисторы одинаков и находится по закону Ома для полной цепи:</p><p>$I_1 = I_2 = I_3 = \frac{U}{R_1 + R_2 + R_3} = \frac{84 \text{ В}}{12 \text{ Ом} + 6.0 \text{ Ом} + 3.0 \text{ Ом}} = \frac{84 \text{ В}}{21 \text{ Ом}} = 4 \text{ А}$</p><p>Вторая ветвь состоит из последовательно соединенных резисторов $R_4, R_8$ и $R_7$. Ток через них также одинаков:</p><p>$I_4 = I_8 = I_7 = \frac{U}{R_4 + R_8 + R_7} = \frac{84 \text{ В}}{24 \text{ Ом} + 12 \text{ Ом} + 6.0 \text{ Ом}} = \frac{84 \text{ В}}{42 \text{ Ом}} = 2 \text{ А}$</p><p><strong>Ответ:</strong></p><p>Токи через резисторы равны:<br>$I_1 = 4 \text{ А}$<br>$I_2 = 4 \text{ А}$<br>$I_3 = 4 \text{ А}$<br>$I_4 = 2 \text{ А}$<br>$I_5 = 0 \text{ А}$<br>$I_6 = 0 \text{ А}$<br>$I_7 = 2 \text{ А}$<br>$I_8 = 2 \text{ А}$</p>" ] #original: array:6 [ "id" => 1691648 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1246} "task" => array:2 [ "refs" => "1319709" "type" => "task" ] "text" => "<p><strong>Дано:</strong></p><p>$U = 84 \text{ В}$<br>$R_1 = 12 \text{ Ом}$<br>$R_2 = 6.0 \text{ Ом}$<br>$R_3 = 3.0 \text{ Ом}$<br>$R_4 = 24 \text{ Ом}$<br>$R_5 = 12 \text{ Ом}$<br>$R_6 = 6.0 \text{ Ом}$<br>$R_7 = 6.0 \text{ Ом}$<br>$R_8 = 12 \text{ Ом}$</p><p><strong>Найти:</strong></p><p>$I_1, I_2, I_3, I_4, I_5, I_6, I_7, I_8$ - ?</p><p><strong>Решение:</strong></p><p>Данная электрическая схема представляет собой симметричный мост. Особенностью таких схем является то, что если сопротивления в плечах моста пропорциональны, то мост является сбалансированным, и ток через соединительные элементы (в данном случае $R_5$ и $R_6$) не течет.</p><p>Проверим условие баланса моста. Для этого сравним отношения сопротивлений в верхней и нижней ветвях схемы.</p><p>Отношение сопротивлений в верхней ветви (резисторы $R_1, R_2, R_3$):</p><p>$R_1 : R_2 : R_3 = 12 : 6.0 : 3.0 = 4 : 2 : 1$</p><p>Отношение сопротивлений в нижней ветви (резисторы $R_4, R_8, R_7$):</p><p>$R_4 : R_8 : R_7 = 24 : 12 : 6.0 = 4 : 2 : 1$</p><p>Так как отношения сопротивлений в плечах равны ($R_1:R_2:R_3 = R_4:R_8:R_7$), мост сбалансирован. Это означает, что потенциалы в точках, соединенных резисторами $R_5$ и $R_6$, соответственно равны. Следовательно, разность потенциалов на резисторах $R_5$ и $R_6$ равна нулю, и ток через них не протекает.</p><p>$I_5 = 0 \text{ А}$</p><p>$I_6 = 0 \text{ А}$</p><p>Поскольку ток через резисторы $R_5$ и $R_6$ отсутствует, их можно мысленно удалить из схемы для дальнейших расчетов. Схема упрощается до двух параллельных ветвей, подключенных к источнику напряжения $U$.</p><p>Первая ветвь состоит из последовательно соединенных резисторов $R_1, R_2$ и $R_3$. Ток через эти резисторы одинаков и находится по закону Ома для полной цепи:</p><p>$I_1 = I_2 = I_3 = \frac{U}{R_1 + R_2 + R_3} = \frac{84 \text{ В}}{12 \text{ Ом} + 6.0 \text{ Ом} + 3.0 \text{ Ом}} = \frac{84 \text{ В}}{21 \text{ Ом}} = 4 \text{ А}$</p><p>Вторая ветвь состоит из последовательно соединенных резисторов $R_4, R_8$ и $R_7$. Ток через них также одинаков:</p><p>$I_4 = I_8 = I_7 = \frac{U}{R_4 + R_8 + R_7} = \frac{84 \text{ В}}{24 \text{ Ом} + 12 \text{ Ом} + 6.0 \text{ Ом}} = \frac{84 \text{ В}}{42 \text{ Ом}} = 2 \text{ А}$</p><p><strong>Ответ:</strong></p><p>Токи через резисторы равны:<br>$I_1 = 4 \text{ А}$<br>$I_2 = 4 \text{ А}$<br>$I_3 = 4 \text{ А}$<br>$I_4 = 2 \text{ А}$<br>$I_5 = 0 \text{ А}$<br>$I_6 = 0 \text{ А}$<br>$I_7 = 2 \text{ А}$<br>$I_8 = 2 \text{ А}$</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" => "1319710" "type" => "task" ] "previous" => array:2 [ "refs" => "1319708" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1240 #items: array:1 [ 0 => App\Models\Book {#1048} ] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1341159" "type" => "book_page" ] ] #original: array:24 [ "id" => 1319709 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "198" "field_page_end" => "199" "field_url" => "/10-klass/fizika/gelfgat-zadachnik/olimp-113" "field_display_title" => "113" "field_outside_task" => "0" "field_task_type" => Illuminate\Database\Eloquent\Collection {#1229} "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 {#1230} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1231} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1232} "content" => Illuminate\Database\Eloquent\Collection {#1234} "next" => array:2 [ "refs" => "1319710" "type" => "task" ] "previous" => array:2 [ "refs" => "1319708" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1240} "page" => array:2 [ "refs" => "1341159" "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 => "*" ] } 1 => App\Models\Task {#1235 #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" => 1319710 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "199" "field_page_end" => null "field_url" => "/10-klass/fizika/gelfgat-zadachnik/olimp-114" "field_display_title" => "114" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1233 #items: array:1 [ 0 => App\Models\Term {#1036} ] #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 {#1242 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1238 #items: array:1 [ 0 => App\Models\Branch {#1121} ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1236 #items: array:4 [ 0 => App\Models\Branch {#1121} 1 => App\Models\Branch {#1125} 2 => App\Models\Branch {#1129} 3 => App\Models\Branch {#1133} ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1239 #items: array:2 [ 0 => App\Models\Element {#1102} 1 => App\Models\Element {#1094} ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1319711" "type" => "task" ] "previous" => array:2 [ "refs" => "1319709" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1237 #items: array:1 [ 0 => App\Models\Book {#1048} ] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1341160" "type" => "book_page" ] ] #original: array:24 [ "id" => 1319710 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "199" "field_page_end" => null "field_url" => "/10-klass/fizika/gelfgat-zadachnik/olimp-114" "field_display_title" => "114" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1233} "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 {#1242} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1238} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1236} "content" => Illuminate\Database\Eloquent\Collection {#1239} "next" => array:2 [ "refs" => "1319711" "type" => "task" ] "previous" => array:2 [ "refs" => "1319709" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1237} "page" => array:2 [ "refs" => "1341160" "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 => "*" ] } ] #escapeWhenCastingToString: false } ] #original: array:21 [ "id" => 1341160 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "199" "field_url" => "/10-klass/fizika/gelfgat-zadachnik/page-199" "field_display_title" => "199" "field_folder" => "1" "field_image_name" => "199" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1088} "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1087} "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 {#1083} "content" => Illuminate\Database\Eloquent\Collection {#1117} "next" => array:2 [ "refs" => "1341161" "type" => "book_page" ] "previous" => array:2 [ "refs" => "1341159" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1220} ] #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" => 1319710 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "199" "field_page_end" => null "field_url" => "/10-klass/fizika/gelfgat-zadachnik/olimp-114" "field_display_title" => "114" "field_outside_task" => null "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 {#1046} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1111} "content" => Illuminate\Database\Eloquent\Collection {#1038} "next" => array:2 [ "refs" => "1319711" "type" => "task" ] "previous" => array:2 [ "refs" => "1319709" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1105} "page" => array:2 [ "refs" => "1341160" "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 => "*" ] }
№114 (с. 199)
Условие. №114 (с. 199)
скриншот условия
O14. В цепи, представленной на рисунке, гальванометр показывает отсутствие тока. Выразите сопротивление $R_x$ через $R_1, R_2, R_3$.
☑ $ R_x = R_1 R_3 / R_2$.
Решение. Ток не идет через гальванометр, когда прибор включен между точками с равными потенциалами. Следовательно, гальванометр можно изъять из цепи (см. рисунок).
Тогда из условия $φ_A = φ_B$ находим $U_x = U_1$, $U_3 = U_2$. Поскольку $U_x / U_3 = R_x / R_3$ и $U_1 / U_2 = R_1 / R_2$, получаем $R_x / R_3 = R_1 / R_2$.
Решение. №114 (с. 199)
Дано:
Электрическая цепь, состоящая из четырех резисторов $R_1$, $R_2$, $R_3$, $R_x$ и гальванометра $G$, соединенных по схеме измерительного моста (мост Уитстона).
Гальванометр показывает отсутствие тока, $I_G = 0$.
Найти:
Выразить сопротивление $R_x$ через $R_1$, $R_2$, $R_3$.
Решение:
Схема, представленная на рисунке, является мостом Уитстона. Условие, что гальванометр показывает отсутствие тока ($I_G = 0$), означает, что мост сбалансирован. Это возможно только в том случае, если потенциалы в точках, к которым подключен гальванометр, равны.
Обозначим узел между резисторами $R_x$ и $R_3$ как точку A, а узел между резисторами $R_1$ и $R_2$ как точку B. Условие баланса моста записывается как равенство потенциалов:
$\phi_A = \phi_B$
Поскольку ток через гальванометр не течет, ток, протекающий через резистор $R_x$, также протекает и через резистор $R_3$. Обозначим этот ток как $I_{x3}$. Аналогично, ток, протекающий через резистор $R_1$, также протекает и через резистор $R_2$. Обозначим этот ток как $I_{12}$.
$I_{x3} = I_x = I_3$
$I_{12} = I_1 = I_2$
Равенство потенциалов $\phi_A = \phi_B$ означает, что падение напряжения на резисторе $R_x$ равно падению напряжения на резисторе $R_1$.
$U_x = U_1$
Используя закон Ома, получаем:
$I_{x3} \cdot R_x = I_{12} \cdot R_1$ (1)
Также из условия $\phi_A = \phi_B$ следует, что падение напряжения на резисторе $R_3$ равно падению напряжения на резисторе $R_2$.
$U_3 = U_2$
$I_{x3} \cdot R_3 = I_{12} \cdot R_2$ (2)
Теперь у нас есть система из двух уравнений. Разделим уравнение (1) на уравнение (2):
$\frac{I_{x3} \cdot R_x}{I_{x3} \cdot R_3} = \frac{I_{12} \cdot R_1}{I_{12} \cdot R_2}$
Сократив токи $I_{x3}$ и $I_{12}$, получим пропорцию:
$\frac{R_x}{R_3} = \frac{R_1}{R_2}$
Из этой пропорции выразим искомое сопротивление $R_x$:
$R_x = \frac{R_1 \cdot R_3}{R_2}$
Ответ: $R_x = \frac{R_1 R_3}{R_2}$.
Другие задания:
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