From f22dde9fc45520b718a83a0a56c3beda728929ca Mon Sep 17 00:00:00 2001 From: me Date: Sun, 4 Jan 2026 21:12:10 +0300 Subject: [PATCH] upd --- "course2/sem3/exam/\\" | 2479 ----------------------------------- course2/sem3/exam/res.pdf | Bin 0 -> 9838757 bytes course2/sem3/exam/tasks.pdf | 8 +- 3 files changed, 4 insertions(+), 2483 deletions(-) delete mode 100644 "course2/sem3/exam/\\" create mode 100644 course2/sem3/exam/res.pdf diff --git "a/course2/sem3/exam/\\" "b/course2/sem3/exam/\\" deleted file mode 100644 index 76afa1e..0000000 --- "a/course2/sem3/exam/\\" +++ /dev/null @@ -1,2479 +0,0 @@ -#set math.equation(numbering: "1.") - -#set text( - size: 14pt, - font: "New Computer Modern", - lang: "ru" -) - -#set par( - justify: true -) - -=== Потенциальная энергия электрического диполя с моментом $arrow(p)$ в поле с напряженностью $arrow(E)$. - -*1. $- arrow(p) arrow(E)$* -2. $|arrow(p)| |arrow(E)|$ -3. $- |arrow(p)| |arrow(E)|$ -4. $- frac(|arrow(p)|, |arrow(E)|)$ -5. $frac(|arrow(E)|, |arrow(p)|)$ - -*Ответ*: Потенциальная энергия диполя в электрическом поле: - -$ -W eq - arrow(p) dot arrow(E) eq - p E(r) cos alpha, -$ - -где $alpha$ -- угол между $arrow(E) (arrow(r))$ и $arrow(p)$. - -#line(length: 100%) - -=== Точечный заряд $q$ помещен в центр пирамиды. Поток вектора напряженности через грань пирамиды равен - -1. $q/4$ -*2. $q/(4 epsilon_0)$* -3. $q/(6 epsilon epsilon_0)$ -4. $q/(epsilon_0)$ -5. $epsilon epsilon_0 q$ - -*Ответ*: Из-за симметрии задачи, потоки вектора напряженности электрического поля через каждую из 4 граней пирамиды одинаковы. По теореме Остроградского-Гаусса полный поток $Phi$ - -$ -Phi eq q/(epsilon_0) -$ - -Поток через одну грань - -$ -Phi_1 eq q/(4 dot epsilon_0) -$ - -#line(length: 100%) - -=== Элемент проводника с током $I$, длиной $d l$ создает в точке $A$, положение которой задано вектором $arrow(r)$, магнитное поле с индукцией - -#align(center)[ - #figure( - image("assets/1.png"), - caption: [Поясняющий рисунок.], - supplement: [Рис.] - ) -] - -*1. $frac(mu_0 I, 4 pi) frac([d arrow(l), arrow(r)], r^3)$* -2. $frac(mu_0 I, 4 pi) frac([d arrow(l), arrow(r)], l^3)$ -3. $frac(mu_0 I, 4 pi) frac([d arrow(l), arrow(r)], r^2)$ -4. $frac(mu_0 I, pi) frac([d arrow(l), arrow(r)], r^2)$ -5. $-frac(mu_0 I, pi) frac([d arrow(l), arrow(r)], l^2)$ - -*Ответ*: По закону Био-Савара-Лапласа для тонкого проводника: - -$ -d arrow(B) eq frac(mu_0 I, 4 pi) frac([d arrow(l), arrow(r)], r^3) -$ - -#line(length: 100%) - -=== Диполь с моментом $arrow(p)$ помещен в электрическое поле напряженностью $arrow(E)$. На диполь действует механическй момент $arrow(M)$. Укажите верное выражение. - -1. $arrow(M) eq |arrow(p)| arrow(E)$ -2. $arrow(M) eq |arrow(E)| arrow(p)$ -3. $arrow(M) eq [arrow(E), arrow(p)]$ -4. $M eq 0$ -*5. $arrow(M) eq [arrow(p), arrow(E)]$* - -*Ответ*: В однородном электрическом поле энергия $W$ изменяется за счет изменения угла $alpha$, при этом элементарная работа сил поля при повороте диполя равна: $d A eq M_alpha d alpha eq - d W$, где $arrow(M)_alpha eq [arrow(p) times arrow(E)]$ -- момент сил, действующий на диполь: - -$ -M_alpha eq -frac(partial W, partial alpha) eq -p E sin alpha -$ - -#line(length: 100%) - -=== По витку радиусом $R$ течет ток силой $I$. Индукция магнитного поля $B$ в центре витка равна - -1. $frac(mu_0 I , 2 pi R)$ -*2. $frac(mu_0 I , 2 R)$* -3. $frac(mu_0 I , pi R)$ -4. $frac(mu_0 I , 4 pi R)$ -5. $frac(mu_0 I , 8 pi R)$ - -*Ответ*: По теореме Био-Савара-Лапласа: - -$ -d arrow(B) eq frac(mu_0 I, 4 pi) frac([d arrow(l), arrow(r)], r^3) eq frac(mu_0 I, 4 pi) frac(d l, r^2) eq \ -eq frac(mu_0 I, 4 pi) frac(r d alpha, r^2) eq frac(mu_0 I, 4 pi) frac(d alpha, R). -$ - -$ -B eq frac(mu_0 I, 4 pi R) integral_0^(2 pi) d alpha eq frac(mu_0 I 2 pi, 4 pi R) eq bold(frac(mu_0 I, 2 R)) -$ - -#line(length: 100%) - -=== Поток вектора индукции электростатического поля через замкнутую поверхность - -1. Равен алгебраической сумме свободных зарядов, находящихся внутри поверхности. -2. Равен сумме абсолютных величин связанных зарядов, находящихся внутри поверхности. -3. Равен сумме абсолютных величин всех зарядов, находящихся внутри поверхности, деленной на электрическую постоянную. -*4. Равен алгебраической сумме всех зарядов, охваченных поверхностью, деленную на электрическую постоянную.* -5. Равен нулю. - -*Ответ*: По теореме Остроградского-Гаусса для вектора электрической индукции $arrow(D)$: - -$ -integral.surf_S arrow(D) d arrow(S) eq q_"внутр". -$ - -#line(length: 100%) - -=== Точечный заряд $q$ помещен в центр куба. Поток вектора напряженности через одну грань куба равен - -1. $q/6$ -*2. $q/(6 epsilon_0)$* -3. $q/(4 epsilon epsilon_0)$ -4. $q/(epsilon_0)$ -5. $epsilon epsilon_0 q$ - -*Ответ*: Из-за симметрии задачи, потоки вектора напряженности электрического поля через каждую из шести граней куба одинаковы. По теореме Остроградского-Гаусса полный поток $Phi$ - -$ -Phi eq q/(epsilon_0) -$ - -Поток через одну грань - -$ -Phi_1 eq q/(6 dot epsilon_0). -$ - -#line(length: 100%) - -=== Укажите все верные утверждения. В однородном изотропном диэлектрике, который помещен в однородное электрическое поле. - -1. $"div" arrow(E) eq rho_"своб"$ -2. $"div" arrow(P) eq -rho_"своб"$ -*3. $"div" arrow(D) eq rho_"своб"$* - -*4. $"div" arrow(P) eq -rho_"связ"$* -5. $"div" arrow(D) eq 0$ - -*Ответ*: Плотность связанных зарядов определяется формулой: - -$ -rho_"связ" eq - "div" arrow(P) -$ - -Вектор электрической индукции: - -$ -arrow(D) eq epsilon_0 arrow(E) + arrow(P) -$ - -Уравнения Гаусса для поля $E$ - -$ -"div" arrow(E) eq frac(rho_"полн", epsilon_0) -$ - -где - -$ -rho_"полн"eq rho_"своб" + rho_"связ" -$ - -Возьмем дивергенцию для - -$ -arrow(D) eq epsilon_0 arrow(E) + arrow(P) -$ - -Получим - -$ -"div" arrow(D) eq epsilon_0 "div" arrow(E) + "div" arrow(P) -$ - -Подставляем в уравнение Гаусса - -$ -eq epsilon_0 dot frac(rho_"своб" + rho_"связ", epsilon_0) + "div" arrow(P) eq \ -eq rho_"своб" + rho_"связ" + "div" arrow(P) -$ - -Но мы знаем, что - -$ -rho_"связ" eq -"div" arrow(P) -$ - -то есть - -$ -"div" arrow(D) eq rho_"своб" + rho_"связ" + "div" arrow(P) eq rho_"своб" -"div" arrow(P) + "div" arrow(P) eq rho_"своб" -$ - -В результате получим - -$ -"div" arrow(D) eq rho_"своб" -$ - -#line(length: 100%) - -=== Электрическое поле проходит через границу раздела двух незаряженных диэлектриков $epsilon_2 gt epsilon_1$. Укажите все верные утверждения. На границе раздела - -*1. $D_(1 n) eq D_(2 n)$* - -2. $D_(1 n) lt D_(2 n)$ -3. $D_(1 n) gt D_(2 n)$ -*4. $D_(1 tau) lt D_(2 tau)$* - -5. $D_(1 tau) gt D_(2 tau)$ - -*Ответ*: Так как - -$ -"div" arrow(D) eq rho_"своб" -$ - -Проинтегрировав, получим - -$ -D_(2 n) - D_(1 n) eq rho_"своб" -$ - -Так как диэлектрики незаряжены - -$ -rho_"своб" eq 0 -$ - -Тогда - -$ -D_(1 n) eq D_(2 n) -$ - -Векторы $arrow(D)$ и $arrow(E)$ связаны - -$ -arrow(D) eq epsilon arrow(E) -$ - -Из уравнения - -$ -"rot" arrow(E) eq 0 -$ - -Следует - -$ -E_(1 tau) eq E_(2 tau) -$ - -Теперь умножаем на $epsilon$ - -$ -D_(1 tau) eq epsilon_1 E_tau \ -D_(2 tau) eq epsilon_2 E_tau -$ - -Так как - -$ -epsilon_2 gt epsilon_1 -$ - -то - -$ -D_(2 tau) gt D_(1 tau) -$ - -#line(length: 100%) - -=== Источник внутренним спротивлением $r$ подключен к нагрузке, сопротивлением $R$. Какой из графиков правильно качественно отражает зависимость полезной мощности от $R$. - -*Ответ*: По закону Ома для замкнутой цепи: - -$ -U eq cal(E) - I r -$ - -Домножим на $I$. - -$ -I U eq cal(E) I - I^2 r -$ - -Переставим слагаемые и воспользуемся $U eq I R$ - -$ -I cal(E) eq I^2 R + I^2 r -$ - -где $I^2 R$ -- полезная мощность. - -Полное сопротивление - -$ -R_"полн" eq R + r -$ - -Ток - -$ -I eq frac(cal(E), R + r) -$ - -Тогда полезная мощность - -$ -P(R) eq I^2 dot R eq (frac(cal(E), R + r))^2 dot R eq frac(cal(E)^2 R, (R + r)^2) -$ - -#align(center)[ - #figure( - image("assets/2.png"), - supplement: [Рис.], - caption: [График $P(R)$.] - ) -] - -#line(length: 100%) - -=== Какая формула позволяет вычислить разность потенциалов между точками $A$ и $B$, расположенными на расстоянии $l$ друг от друга в однородном электрическом поле напряженностью $E$. - -1. $phi_A - phi_B eq - E dot l$ -2. $phi_A - phi_B eq E dot l dot tg alpha$ -*3. $phi_A - phi_B eq E dot l dot cos alpha$* - -4. $phi_A - phi_B eq -E dot l dot cos alpha$ -5. $phi_A - phi_B eq -E dot l dot tg alpha$ - -*Ответ*: По определению разности потенциалов между точками $A$ и $B$ - -$ -phi_A - phi_B eq integral_A^B arrow(E) dot d arrow(l) -$ - -Так как поле однородное, то $arrow(E) eq "const"$ и интеграл упрощается до - -$ -phi_A - phi_B eq arrow(E) dot d arrow(l) -$ - -И по определению скалярного произведения - -$ -phi_A - phi_B eq E l cos alpha -$ - -#line(length: 100%) - -=== Потенциальная энергия контура с магнитным моментом $arrow(P)_m$ в поле с индукцией $arrow(B)$ равна - -1. $bold(- arrow(P)_m arrow(B))$ -2. $- |arrow(P)_m| |arrow(B)|$ -3. $arrow(P)_m times arrow(B)$ -4. $arrow(P)_m arrow(B)$ -5. $|arrow(P)_m| |arrow(B)|$ - -*Ответ*: Для контура с током магнитный момент: - -$ -arrow(p)_m eq I arrow(S) -$ - -Для электрического диполя в электрическом поле - -$ -U eq -arrow(p) dot arrow(E) -$ - -Для контура с током в магнитном поле: - -$ -U eq -arrow(p)_m dot arrow(B) -$ - -#line(length: 100%) - -=== Магнитное поле проходит через границу раздела двух сред. Токи проводимости отсутствуют. $mu_2 gt mu_1$. Укажите все верные утверждения. На границе раздела - -*1. $B_(1 n) eq B_(2 n)$* -2. $B_(1 n) lt B_(2 n)$ -3. $B_(1 n) gt B_(2 n)$ -*4. $B_(1 tau) lt B_(2 tau)$* -5. $B_(1 tau) gt B_(2 tau)$ - -*Ответ*: Уравнение Максвелла для магнитного поля - -$ -"div" arrow(B) eq 0 -$ - -Проинтегрировав, получим - -$ -integral.cont arrow(B) dot d arrow(S) eq 0 -$ - -Переходя к пределу, получим граничное условие - -$ -B_(2 n) - B_(1 n) eq 0 arrow.double B_(2 n) eq B_(1 n) -$ - -Из уравнения Максвелла - -$ -"rot" arrow(H) eq arrow(j)_"пров" -$ - -По условию - -$ -arrow(j)_"пров" eq 0 -$ - -То есть - -$ -H_(1 tau) eq H_(2 tau) -$ - -Так как - -$ -arrow(B) eq mu arrow(H) -$ - -С учетом того, что $mu_2 gt mu_1$ - -$ -B_(2 tau) gt B_(1 tau) -$ - -#line(length: 100%) - -=== Укажите все выражения, которые входят в ток смещения - -*1. $frac(partial arrow(P), partial t)$* -2. $frac(partial arrow(J), partial t)$ -*3. $epsilon_0 frac(partial arrow(E), partial t)$* -4. $arrow(j)_"проводимости"$ -5. $mu_0 frac(partial arrow(H), partial t)$ - -*Ответ*: По определению Максвелла плотность тока смещения - -$ -arrow(j)_"см" eq frac(partial arrow(D), partial t) -$ - -По определению $arrow(D)$ - -$ -arrow(D) eq epsilon_0 arrow(E) + arrow(P) -$ - -Взяв производную по времени получим - -$ -frac(partial arrow(D), partial t) eq epsilon_0 frac(partial arrow(E), partial t) + frac(partial arrow(P), partial t) -$ - -#line(length: 100%) - -=== В реальном колебательном контуре резонанс по величине ЭДС индукции в катушке наступает при частоте внешней ЭДС - -1. намного меньше собственной частоты контура -2. намного больше собственной частоты контура -3. примерно равной собственной частоте контура -4. чуть меньше собственной частоты контура -*5. чуть больше собственной частоты контура* - -*Ответ*: хз. - -#line(length: 100%) - -=== Укажите все волновые уравнения - -*1. $Delta arrow(E) eq frac(1, c^2) frac(partial^2 arrow(E), partial t^2)$* -2. $integral.cont_L arrow(H) d arrow(l) eq integral_S (arrow(j) + frac(partial arrow(D), partial t)) d arrow(S)$ -3. $integral.cont_L arrow(E) d arrow(l) eq -integral_S frac(partial arrow(B), partial t) d arrow(S)$. -4. $c eq frac(1, sqrt(epsilon_0 epsilon mu_0 mu))$ -*5. $Delta arrow(H) eq frac(1, c^2) frac(partial^2 arrow(H), partial t^2)$* - -*Ответ*: Волновое уравнение -- это дифференциальное уравнение вида - -$ -Delta arrow(F) eq frac(1, v^2) frac(partial^2 arrow(F), partial t^2) -$ - -где $Delta$ -- оператор Лапласа. - -#line(length: 100%) - -=== Эквипотенциальные поверхности поля точечного положительного заряда имеют вид - -1. равноотстоящих друг от друга плоскостей -*2. концентрических сфер* -3. коаксиальных цилиндров -4. эллипсоидов вращения -5. пересекающихся плоскостей - -*Ответ*: Эквипотенциальная поверхность -- это поверхность, на которой - -$ -phi eq "const" -$ - -Для точечного положительного заряда $q$ - -$ -phi(r) eq frac(1, 4 pi epsilon_0) q/r -$ - -Если $phi eq "const"$, то из формулы следует - -$ -1/r eq "const" arrow.double r eq "const" -$ - -Множество точек, находящихся на одинаковом расстоянии от одной точки, это сфера. - -#line(length: 100%) - -=== Укажите все верные утверждения. Электрическое поле проходит через границу раздела двух незаряженных диэлектриков $epsilon_2 gt epsilon_1$. Укажите все верные утверждения. На границе раздела - -1. $E_(1 n) eq E_(2 n)$ -2. $E_(1 n) lt E_(2 n)$ -*3. $E_(1 n) gt E_(2 n)$* -4. $E_(1 tau) lt E_(2 tau)$ -*5. $E_(1 tau) eq E_(2 tau)$* - -*Ответ*: Закон Фарадея - -$ -"rot" arrow(E) eq 0 -$ - -Интегрируя по малому контуру, пересекающему границу, получаем - -$ -E_(1 tau) eq E_(2 tau) -$ - -Из уравнения Гаусса - -$ -"div" arrow(D) eq rho_"своб" -$ - -Интегрирование дает - -$ -D_(2 n) - D_(1 n) eq rho_"своб" -$ - -Так как диэлектрики незаряжены - -$ -rho_"своб" eq 0 arrow.double D_(1 n) eq D_(2 n) -$ - -Так как - -$ -arrow(D) eq epsilon epsilon_0 arrow(E) -$ - -Получим - -$ -epsilon_1 E_(1 n) eq epsilon_2 E_(2 n) -$ - -Тогда если - -$ -epsilon_1 lt epsilon_2 -$ - -То - -$ -E_(1 n) gt E_(2 n) -$ - -#line(length: 100%) - -=== Проводящий шар заряжен положительным зарядом. Внутри шара - -1. линии напряженности замкнуты -2. линии напряженности идут вдоль радиусов к поверхности -3. линии напряженности идут вдоль радиусов к центру -*4. напряженность поля равна нулю* -5. линии напряженности перпендикулярны радиусам шара - -*Ответ*: В электростатическом равновесии внутри проводника - -$ -arrow(E) eq 0 -$ - -#line(length: 100%) - -=== Укажите все верные утверждения - -1. Первый закон Кирхгофа является следствием закона Кулона -*2. Первый закон Кирхгофа является следствием закона сохранения заряда * -*3. Второй закон Кирхгофа является следствием закона Ома для неоднородного участка цепи.* -4. Второй закон Кирхгофа является следствием закона Джоуля-Ленца. -5. Второй закон Кирхгофа является следствием закона Ома для однородного участка цепи. - -*Ответ*: По первому закону Кирхгофа алгебраическая сумма токов в узле равна нулю - -$ -sum I eq 0 -$ - -то есть - -$ -sum I_"вход" eq sum I_"выход" -$ - -то есть заряд не накапливается в узле. - -По второму закону Кирхгофа в любом замкнутом контуре алгебраическая сумма ЭДС равна алгебраической сумме падений напряжений. - -$ -sum E eq sum I R -$ - -или эквивалентно: - -$ -sum U eq 0 -$ - -Закон сохранения заряда - -$ -frac(d q, d t) eq 0 -$ - -Закон Ома для неоднородного участка цепи - -$ -U eq I R minus cal(E) -$ - -или - -$ -I R eq U + cal(E) -$ - -#line(length: 100%) - -=== Укажите формулу, которая всегда окажется верной при вычислении объемной плотности энергии электричского поля - -1. $frac(arrow(E) arrow(D), 2)$ -2. $frac(|arrow(E)||arrow(D)|, 2)$ -3. $frac(epsilon_0 epsilon |arrow(E)|^2, 2)$ -4. $arrow(D) arrow(E)$ -5. $frac(|arrow(D)|^2, 2 epsilon_0 epsilon)$ - -*Ответ*: Объемная плотность энергии $w eq frac(arrow(E) arrow(D), 2)$ содержит в себе как собственную энергию электрического поля $frac(epsilon_0 E^2, 2)$, так и энергию поляризации диэлектрика $frac(arrow(E) arrow(P), 2)$. - -=== Укажите все верные утверждения. Магнитное поле создают - -*1. Электрический ток* -*2. Движущаяся заряженная частица* -3. Потенциальное электрическое поле -4. Вихревое электрическое поле -*5. Ток смещения* - -*Ответ*: По закону Био-Савара и Ампера - -$ -arrow(B) tilde arrow(j) -$ - -Движущийся заряд -- это микроскопический ток. Если заряд $q$ движется со скоростью $arrow(v)$, он создает магнитное поле: - -Если заряд $q$ движется со скоростью $arrow(v)$, он создает магнитное поле - -$ -arrow(B) prop q arrow(v) -$ - -Ток смещения - -$ -arrow(j)_"см" eq frac(partial arrow(D), partial t) -$ - -создает магнитное поле наравне с током проводимости. - -#line(length: 100%) - -=== Магнитное поле проходит через границу раздела двух однородных изотропных магнетиков $mu_2 gt mu_1$. Токи проводимости отсутствуют. Укажите все верные утверждения. На границе раздела - -1. $H_(1 n) eq H_(2 n)$ -2. $H_(1 n) lt H_(2 n)$ -*3. $H_(1 n) gt H_(2 n)$* -4. $H_(1 tau) lt H_(2 tau)$ -*5. $H_(1 tau) eq H_(2 tau)$* - -*Ответ*: Связь между $arrow(B)$ и $arrow(H)$ - -$ -arrow(B) eq mu arrow(H) -$ - -Из уравнения Максвелла - -$ -"div" arrow(B) eq 0 -$ - -Следует - -$ -B_(1 n) eq B_(2 n) -$ - -Подставим и получим - -$ -mu_1 H_(1 n) eq mu_2 H_(2 n) -$ - -Так как - -$ -mu_2 gt mu_1 -$ - -Тогда - -$ -H_(1 n) gt H_(2 n) -$ - -Из уравнения Максвелла - -$ -"rot" arrow(H) eq arrow(j)_"пров" -$ - -По условию - -$ -arrow(j)_"пров" eq 0 -$ - -Тогда - -$ -H_(1 tau) eq H_(2 tau) -$ - -#line(length: 100%) - -=== Данная система уравнений Максвелла соответствует случаю, когда - -$ -cases( - integral.cont_L arrow(E) d arrow(l) eq -integral_S frac(partial arrow(B), partial t) d arrow(S), - integral.cont_L arrow(H) d arrow(l) eq integral_S (arrow(j) + frac(partial arrow(D), partial t) d arrow(S)), - integral.cont_S arrow(D) d arrow(S) eq 0, - integral.cont_S arrow(B) d arrow(S) eq 0 -) -$ - -1. электрическое и магнитное поля не изменяются во времени -2. отсутствуют токи смещения -3. отсутствуют токи проводимости -*4. отсутствуют свободные заряды* -5. отсутствуют связанные заряды - -*Ответ*: В общем случае третье уравнение Максвелла выглядит следующим образом - -$ -integral.cont_S arrow(D) dot d arrow(S) eq Q_"своб" -$ - -В задаче - -$ -integral.cont_S arrow(D) dot d arrow(S) eq 0 -$ - -Значит - -$ -Q_"своб" eq 0 -$ - -#line(length: 100%) - -=== В изображенной на рисунке точке $A$ магнитное поле направлено по стрелке - -#align(center)[ - #figure( - image("assets/3.png"), - caption: [Поясняющий рисунок.], - supplement: [Рис.] - ) -] - -1. $1$ -*2. $2$* -3. $3$ -4. $4$ -5. $5$ - -*Ответ*: хз. - -#line(length: 100%) - -=== Зависимость смещения материальной точки от времени определяется уравнением $x eq 0.12 cos(20 t + 0.2)$. Определите период колебаний. - -*Ответ*: Общий вид уравнения колебаний - -$ -x(t) eq A cos(omega t + phi_0) -$ - -где $A$ -- амплитуда, $omega$ -- циклическая (угловая) частота, $phi_0$ -- начальная фаза. - -Для данного уравнения - -$ -omega eq 20 "рад/с" -$ - -По формуле - -$ -T eq frac(2 pi , omega) -$ - -Подставив число, получим - -$ -T approx 0.314 "с" -$ - -#line(length: 100%) - -=== Напряженность поля диполя при удалении от него - -1. не изменяется -2. убывает пропорционально первой степени расстояния до центра диполя -3. убывает пропорционально квадрату расстояния до центра диполя -*4. убывает пропорционально кубу расстояния до центра диполя* -5. убывает пропорционально корню квадратному из расстояния до центра диполя - -*Ответ*: - -$ -E tilde frac(k q l, r^3) -$ - -где $q l eq p$ -- дипольный момент. - -$ -E tilde frac(p, r^3) -$ - -=== Пластина из однородного изотропного диэлектрика с проницаемостью $epsilon$ помещена параллельно пластинам в заряженный плоский конденсатор. Как связаны между собой векторы электрической индукции $D$ и поляризации диэлектрика $P$. - -1. $arrow(D) eq epsilon_0 epsilon arrow(P)$ -*2. $arrow(D) eq frac(epsilon arrow(P), epsilon - 1)$* -3. $arrow(D) eq -frac(epsilon arrow(P), epsilon - 1)$ -4. $arrow(D) eq -(epsilon - 1) arrow(P)$ -5. $arrow(D) eq (epsilon - 1) arrow(P)$ - -*Ответ*: По определению электрической индукции - -$ -arrow(D) eq epsilon_0 arrow(E) + arrow(P) -$ - -Связь поляризации (поляризованности) с полем - -$ -arrow(P) eq epsilon_0 (epsilon - 1) arrow(E) -$ - -Выразим $arrow(E)$ через $arrow(P)$ - -$ -arrow(E) eq frac(arrow(P), epsilon_0 (epsilon - 1)) -$ - -Подставляем в формулу для $arrow(D)$ - -$ -arrow(D) eq epsilon_0 arrow(E) + arrow(P) eq epsilon_0 dot frac(arrow(P), epsilon_0 (epsilon - 1)) + arrow(P) eq frac(arrow(P), epsilon - 1) + arrow(P) eq frac(epsilon, epsilon - 1) arrow(P) -$ - -#line(length: 100%) - -=== Плоский воздушный конденсатор заряжен и отключен от источника. Конденсатор заполняют диэлектриком. Выберите все верные утверждения. - -1. напряженность поля в конденсаторе увеличивается -*2. напряженность поля в конденсаторе уменьшается* -3. напряжение на конденсаторе увеличивается -4. заряд конденсатора увеличивается -*5. заряд конденсатора не изменится* - -*Ответ*: Если конденсатор отключен, то - -$ -Q eq "const" -$ - -При заполнении диэлектриком с $epsilon gt 1$: - -$ -C eq epsilon C_0 -$ - -емкость увеличивается - -Так как $Q eq "const"$, а $C$ увеличилось, то из формулы - -$ -Q eq C U -$ - -видно, что напряжение уменьшается - -Так как $U$ уменьшается, а $d$ не меняется, то из формулы - -$ -E eq U/d -$ - -$E$ уменьшается - -#line(length: 100%) - -=== Источник внутренним сопротивлением $r$ подключен к нагрузке сопротивлением $R$. Какой из графиков правильно качественно отражает зависимость КПД источника от $R$. - -*Ответ*: По закону Ома - -$ -U eq cal(E) - I r -$ - -Домножим на $I$ - -$ -cal(E) I eq I^2 R + I^2 r -$ - -$ -P_"общ" eq cal(E) I \ -P_"полезн" eq I^2 R -$ - -$ -eta eq frac(P_"полезн", P_"общ") eq frac(I^2 R, I cal(E)) eq frac(I R, cal(E)) -$ - -По закону Ома для полной цепи - -$ -I eq frac(cal(E), R + r) -$ - -Подставим и получим - -$ -eta(R) eq frac(E, R + r) dot R/E eq frac(R, R + r) -$ - -#align(center)[ - #figure( - image("assets/4.png"), - caption: [], - supplement: [Рис.] - ) -] - -#line(length: 100%) - -=== Силу, действующую на элемент проводника с током $I$ длиной $d l$ в магнитном поле с индукцией $B$, можно вычислить по формуле - -*1. $I [d arrow(l), arrow(B)]$* -2. $2 pi I (d arrow(l), arrow(B))$ -3. $1/pi I (d arrow(l), arrow(B))$ -4. $frac(mu_0 I B, d l)$ -5. $frac(mu_0 I B, 4 pi d l)$ - -*Ответ*: Силы, действующие на токи в магнитном поле, называют силами Ампера. Сила, действующая на элементарный объем $d V$ проводника с плотностью тока $arrow(j)$ равна - -$ -d arrow(F) eq [arrow(j), arrow(B)] d V. -$ - -Если проводник достаточно тонкий, то - -$ -d arrow(F) eq I [d arrow(l), arrow(B)]. -$ - -#line(length: 100%) - -=== Напряженность поля прямого проводника с током при удалении от него - -1. не изменяется -*2. убывает пропорционально первой степени расстояния до проводника* -3. убывает пропорционально квадрату расстояния до проводника -4. убывает пропорционально кубу расстояния до проводника -5. убывает пропорционально корню квадратному из расстояния до проводника - -*Ответ*: По теореме о циркуляции - -$ -integral.cont arrow(H) dot d arrow(l) eq I -$ - -Берем окружность радиуса $r$: - -$ -integral.cont arrow(H) dot d arrow(l) eq H dot 2 pi r eq I -$ - -Выразив напряженность получим - -$ -H eq frac(I, 2 pi r) -$ - -#line(length: 100%) - -=== Укажите все верные утверждения. В однородном, изотропном магнетике - -*1. $arrow(B) eq mu_0 (arrow(H) + arrow(J))$* - -*2. $arrow(B) eq mu_0 mu arrow(H)$* -3. $arrow(B) eq mu_0 arrow(H) + arrow(J)$ -*4. $mu eq 1 + chi$* \ -*5. $arrow(J) eq chi arrow(H)$* - -*Ответ*: это все стандартные формулы. - -#line(length: 100%) - -=== Данная система уравнений Максвелла соответствует случаю, когда - -$ -cases( - integral.cont_L arrow(E) d arrow(l) eq -integral_S frac(partial arrow(B), partial t) d arrow(S), - integral.cont_L arrow(H) d arrow(l) eq integral_S frac(partial arrow(D), partial t) d arrow(S), - integral.cont_S arrow(D) d arrow(S) eq integral_r rho_"стор" d V, - integral.cont_S arrow(B) d arrow(S) eq 0 -) -$ - -1. электрическое и магнитное поля не изменяются во времени -2. отсутствуют токи смещения -*3. отсутствуют токи проводимости* -4. отсутствуют свободные заряды -5. отсутствуют связанные заряды - -*Ответ*: Второе уравнение в общем виде - -$ -integral.cont_L arrow(H) dot d arrow(l) eq integral (arrow(J)_"пров" + frac(partial arrow(D), partial t)) d arrow(S) -$ - -Видно, что - -$ -arrow(j)_"пров" eq 0 -$ - -#line(length: 100%) - -=== Укажите все верные утверждения. Материальными уравнениями называются - -*1. $arrow(B) eq mu_0 mu arrow(H)$* \ -*2. $arrow(D) eq epsilon_0 epsilon arrow(E)$* \ -*3. $arrow(D) eq epsilon_0 arrow(E) + arrow(P)$* -4. $integral.cont_L B d l eq mu_0 I$ -5. $integral.cont_L arrow(E) d arrow(l) eq 0$ - -*Ответ*: Материальные уравнения -- это уравнения, которые связывают поля c откликом вещества. - -$ -arrow(B) eq mu_0 mu arrow(H) -$ - -связывает $arrow(B)$ и $arrow(H)$. - -$ -arrow(D) eq epsilon_0 epsilon arrow(E) -$ - -связывает $arrow(D)$ и $arrow(E)$. - -$ -arrow(D) eq epsilon_0 arrow(E) + arrow(P) -$ - -связывает $arrow(D)$ и $arrow(E)$, учитывая поляризацию $arrow(P)$. - -#line(length: 100%) - -=== Укажите все верные для световой волны утверждения - -*1. векторы $arrow(E)$ и $arrow(H)$ изменяются с одинаковой частотой* \ -*2. векторы $arrow(E)$ и $arrow(H)$ всегда перпендикулярны друг к другу* \ -*3. скорость распространения зависит от диэлектрической проницаемости среды* \ -*4. скорость распространения зависит от магнитной проницаемости среды* \ -*5. волна всегда переносит энергию в пространстве* - -*Ответ*: хз. - -#line(length: 100%) - -=== Укажите все верные утверждения - -*1. силовые линии электростатического поля не могут быть замкнуты* -2. силовые линии электростатического поля всегда замкнуты -*3. циркуляция напряженности электростатического поля по замкнутому контуру равна нулю* -4. циркуляция напряженности электростатического поля по любому замкнутому контуру отлична от нуля -5. циркуляция напряженности электростатического поля по замкнутому контуру зависит от формы контура - -*Ответ*: Для электростатического поля выполняется одно из уравнений максвелла - -$ -"rot" arrow(E) eq 0 -$ - -По определению циркуляции - -$ -integral.cont_L arrow(E) dot d arrow(l) -$ - -Из уравнения @eq1 - -$ -integral.cont_L arrow(E) dot d arrow(l) eq 0 -$ - -Из @eq2 следует то, что замкнутых силовых линий - быть не может. - - #line(length: 100%) - - === По длинному прямому проводнику течет электрический ток силой $I$. Индукция магнитного поля в вакууме, в точке $A$ на расстоянии $R$ от проводника равна - -1. $frac( - mu_0 I, - 4 pi R -)$ - -2. $frac( - I, - 2 pi R -)$ - -3. $frac( - mu_0 I, - 2 R -)$ - -*4. $frac( - mu_0 I, - 2 pi R -)$* - -5. $frac( - I pi, - 8 R -)$ - -*Ответ*: вывод был много раз. - -#line(length: 100%) - -=== Контур с током обладает магнитным моментом $P_m$. Механический момент, действующий на этот контур в поле с индукцией $B$, равен - -1. $bold([arrow(P)_m, arrow(B)])$ -2. $-[arrow(P)_m, arrow(B)]$ -3. $2 pi [arrow(P)_m, arrow(B)]$ -4. $frac( - [arrow(P)_m, arrow(B)], - 4 pi -)$ -5. $0$ - -*Ответ*: Магнитный момент контура - -$ -arrow(p)_m eq I arrow(S) -$ - -Формула механического момента - -$ -arrow(M) eq arrow(P)_m times arrow(B) -$ - -#line(length: 100%) - -=== Укажите, как изменяются потенциал $phi$ и напряженность $E$ внутри проводящей сферы, равномерно заряженной по поверхности - -1. $E eq "const", phi tilde 1/r$ -2. $E tilde frac(1, r^2), phi tilde 1/r$ -3. $E tilde 1/r, phi tilde frac(1, r^2)$ -*4. $E eq 0, phi eq "const"$* -5. $E tilde r, phi tilde r^2$ - -*Ответ*: В электростатическом равновесии - -$ -E_"внутри проводника" eq 0 -$ - -Связь потенциала и поля - -$ -arrow(E) eq - nabla phi -$ - -Если - -$ -arrow(E) eq 0 -$ - -то - -$ -nabla phi eq 0 -$ - -Потенциал не меняется в пространстве - -$ -phi eq "const" -$ - -#line(length: 100%) - -=== В некоторой точке однородного изотропного диэлектрика с проницаемостью $epsilon$ напряженность поля равна $arrow(E)$. Вектор поляризации $arrow(P)$ в этой точке определяется выражением - -1. $arrow(P) eq epsilon_0 (1 - epsilon) arrow(E)$ -*2. $arrow(P) eq epsilon_0 (epsilon - 1) arrow(E)$* -3. $arrow(P) eq epsilon_0 epsilon arrow(E)$ -4. $arrow(P) eq frac(epsilon, epsilon - 1) arrow(E)$ -5. $arrow(P) eq - frac(epsilon, epsilon - 1) arrow(E)$ - -*Ответ*: - -$ -arrow(P) eq epsilon_0 chi arrow(E) -$ - -$ -epsilon eq 1 + chi -$ - -$ -chi eq epsilon - 1 -$ - -$ -arrow(P) eq epsilon_0 (epsilon - 1) arrow(E) -$ - -#line(length: 100%) - -=== Поток вектора напряженности электростатического поля через замкнутую поверхность - -1. равен алгебраической сумме свободных зарядов, находящихся внутри поверхности -2. равен сумме абсолютных величин связанных зарядов, находящихся внутри поверхности -3. равен сумме абсолютных величин всех зарядов, находящихся внутри поверхности, деленной на электрическую постоянную -*4. равен алгебраической сумме всех зарядов, охваченных поверхностью, деленной на электрическую постоянную* -5. равен нулю - -*Ответ*: По закону Остроградского-Гаусса - -$ -integral.surf arrow(E) d arrow(S) eq frac( - sum q_"внутр", - epsilon_0 -) -$ - -#line(length: 100%) - -=== Укажите все верные утверждения. Магнитное поле создают - -1. неподвижные электрические заряды -*2. движущиеся электрические заряды* -3. потенциальное электрическое поле -4. вихревое электрическое поле -*5. изменяющееся во времени электрическое поле* - -*Ответ*: - -$ -"rot" arrow(H) eq arrow(j)_"проводимости" + frac(partial arrow(D), partial t) -$ - -#line(length: 100%) - -=== Данная система уравнений Максвелла соответствует случаю, когда - -$ -cases( - integral.cont_L arrow(E) d arrow(l) eq 0, - integral.cont_S arrow(D) d arrow(S) eq integral_V rho_"своб" d V, - integral.cont_L arrow(H) d arrow(l) eq integral_S arrow(j)_"пров" d arrow(S), - integral.cont_S arrow(B) d arrow(S) eq 0 -) -$ - -*1. электрическое и магнитное поля не изменяются во времени* -*2. отсутствуют токи смещения* -3. отсутствуют токи проводимости -4. отсутствуют свободные заряды -5. отсутствуют связанные заряды - -*Ответ*: Из первого уравнения системы: поля не изменяются во времени, из второго уравнения: ток смещения отсутствует. - -#line(length: 100%) - -=== Частица с зарядом $q$ движущаяся со скоростью $arrow(v)$ создает в точке $A$, положение которой задано вектором $arrow(r)$, магнитное поле с индукцией - -1. $frac( - mu_0 q, - 2 pi -) frac( - [arrow(v), arrow(r)], - r^2 -)$ - -*2. $frac( - mu_0 q, - 4 pi -) frac( - [arrow(v), arrow(r)], - r^3 -)$* - -3. $-frac( - mu_0 q, - 2 pi -) frac( - [arrow(v), arrow(r)], - r^2 -)$ - -4. $frac( - mu_0 q, - pi -) frac( - [arrow(v), arrow(r)], - r -)$ - -5. $-frac( - mu_0 q, - 4 pi -) frac( - [arrow(v), arrow(r)], - r^3 -)$ - -*Ответ*: По формуле магнитного поля движущегося заряда (Био-Савар) - -$ -arrow(B) (arrow(r)) eq frac(mu_0, 4 pi) q frac(arrow(v) times arrow(r), r^3) -$ - -#line(length: 100%) - -=== Укажите уравнения, справедливые для вихревого электрического поля - -*1. $integral.cont_L arrow(E) d arrow(l) eq -integral.cont_S frac(partial arrow(B), partial t) d S$* -*2. $E_(1 tau) eq E_(2 tau)$* -3. $"div" arrow(E) eq rho/epsilon_0$ -4. $integral.cont_L arrow(E) d arrow(l) eq 0$ -5. $arrow(E) eq -"grad" phi$ - -*Ответ*: Вихревое электрическое поле -- это поле, которое возникает при изменяющемся во времени магнитном поле. - -$ -"rot" arrow(E) eq.not 0 -$ - -Из первого выражения: если $frac(partial arrow(B), partial t) eq.not 0$, то циркуляция $arrow(E)$ не равна нулю. - -$ -E_(1 tau) eq E_(2 tau) -$ - -Это граничное условие для электрического поля. - -#line(length: 100%) - -=== Какое уравнение показывает, что не существует магнитных зарядов - -1. $B_(1 n) eq B_(2 n)$ -2. $integral.cont_L H_l d l eq integral_S j d S$ -3. $H_(1 tau) eq H_(2 tau)$ -*4. $"div" arrow(B) eq 0$* -5. $"div" arrow(j) eq -frac(partial rho, partial t)$ - -*Ответ*: $"div" arrow(B) eq 0$ это одно из уравнений Максвелла. И оно значит, что нет магнитных зарядов. - -#line(length: 100%) - -=== Укажите, как изменяются потенциал $phi$ и напряженность $E$ внутри шара, равномерно заряженного по объему - -1. $E eq "const", phi tilde 1/r$ -2. $E tilde frac(1, r^2), phi tilde frac(1, r^2)$ -3. $E tilde 1/r, phi tilde frac(1, r^2)$ -4. $E eq 0, phi eq "const"$ -*5. $E tilde r, phi tilde r^2$* - -*Ответ*: По закону Гаусса - -$ -integral.cont arrow(E) dot d arrow(S) eq frac(q_"внутр", epsilon_0) -$ - -$ -q_"внутр" eq rho dot 4/3 pi r^3 -$ - -$ -E dot 4 pi r^2 eq frac(rho 4/3 pi r^3, epsilon_0) -$ - -$ -E(r) eq frac(rho, 3 epsilon_0) r -$ - -$ -E tilde r -$ - -$ -E eq - frac(d phi, d r) -$ - -$ -phi(r) tilde - integral r d r tilde -r^2 -$ - -$ -phi tilde r^2 -$ - -#line(length: 100%) - -=== В некоторой точке однородного изотропного диэлектрика с проницаемостью $epsilon$ напряженность поля равна $E$, а вектор поляризации равен $P$. Индукция электрического поля в этой точке определяется выражением - -1. $arrow(P) + (1 - epsilon) arrow(E)$ -2. $arrow(P) + epsilon_0 epsilon arrow(E)$ -*3. $arrow(P) + epsilon_0 arrow(E)$* -4. $epsilon_0 arrow(E) + epsilon arrow(P)$ -5. $epsilon_0 epsilon arrow(E) - arrow(P)$ - -*Ответ*: По определению электрической индукции: - -$ -arrow(D) eq epsilon_0 arrow(E) + arrow(P) -$ - -#line(length: 100%) - -=== Поток вектора поляризации через замкнутую поверхность - -1. равен алгебраической сумме свободных зарядов, находящихся внутри поверхности -*2. равен алгебраической сумме связанных зарядов, находящихся внутри поверхности, взятой с обратным знаком* -3. равен сумме абсолютных величин всех зарядов, находящихся внутри поверхности, деленной на электрическую постоянную -4. равен алгебраической сумме всех зарядов, охваченных поверхностью, деленной на электрическую постоянную -5. равен нулю - -*Ответ*: - -$ -rho_"связ" eq -"div" arrow(P) -$ - -$ -integral.cont_S arrow(P) dot d arrow(S) -$ - -$ -integral.cont_S arrow(P) dot d arrow(S) eq integral_V "div" arrow(P) space d V -$ - -$ -"div" arrow(P) eq -rho_"связ" -$ - -$ -integral.cont_S arrow(P) dot d arrow(S) eq -integral_V rho_"связ" d V -$ - -$ -integral.cont_S arrow(P) dot d arrow(S) eq -Q_"связ" -$ - -#line(length: 100%) - -=== Слой однородного изотропного диэлектрика с проницаемостью $epsilon$ прижат к пластине, заряженной с поверхностной плоскостью $sigma$. Напряженность электрического поля в диэлектрике определяется выражением - -1. $E eq frac(sigma, epsilon_0 epsilon)$ -2. $E eq frac(sigma, epsilon_0)$ -3. $E eq epsilon_0 epsilon sigma$ -*4. $E eq frac(sigma, 2 epsilon_0 epsilon)$* -5. $E eq epsilon_0 sigma$ - -*Ответ*: хз. - -#line(length: 100%) - -=== Укажите все верные утверждения. Циркуляция вектора индукции магнитного поля вычисляется по формуле - -*1. $integral.cont_L B_tau d l$*\ -*2. $integral.cont_L arrow(B) d arrow(l)$* -3. $integral.cont_L arrow(B) times d arrow(l)$ -4. $integral.cont_L B 2 pi d r$ -5. $integral.cont_L arrow(B) times d arrow(r)$ - -*Ответ*: Второй вариант -- это точное определение циркуляции. Первый вариант -- это проекция на касательное направление. - -#line(length: 100%) - -=== По длинному прямому проводнику течет электрический ток силой $I$. Поток вектора магнитной индукции через поверхность сферы радиусом $R$, центр которой находится на расстоянии $a$ от проводника, равен - -1. $frac( - mu_0 I, - 4 pi R -)$ - -2. $frac( - I, - 2 pi R -)$ - -3. $frac( - mu_0 I a, - 4 pi R^2 -)$ - -4. $frac( - mu_0 I, - 2 pi a -)$ - -5. $0$ - -*Ответ*: для длинного прямого проводника магнитного поля: - -$ -B eq frac(mu_0 I, 2 pi r) -$ - -Поток магнитного поля - -$ -Phi_B eq integral.double_S arrow(B) dot d arrow(S) -$ - -$ -arrow(B) perp d arrow(S) arrow.double arrow(B) dot d arrow(S) eq 0 -$ - -#line(length: 100%) - -=== $I'$ -- алгебраическая сумма токов намагничивания, $I$ -- алгебраическая сумма токов проводимости. Циркуляцию вектора $J$ по замкнутому контуру $L$ можно определить по формуле - -1. $integral.cont_L arrow(J) d arrow(l) eq I' + I$ -*2. $integral.cont_L arrow(J) d arrow(l) eq I'$* -3. $integral.cont_L arrow(J) d arrow(l) eq I$ -4. $integral.cont_L arrow(J) d arrow(l) eq I' - I$ -5. $integral.cont_L arrow(J) d arrow(l) eq 0$ - -*Ответ*: Определение циркуляции тока намагничивания - -$ -integral.cont_L arrow(J) dot d arrow(l) eq sum "токов намагничивания внутри контура" -$ - -#line(length: 100%) - -=== Плотность тока смещения равна - -1. $frac(partial arrow(B), partial t)$ -2. $arrow(j)_"проводимости"$ -3. $epsilon_0 frac(partial arrow(E), partial t) + arrow(j)_"проводимости"$ -*4. $frac(partial arrow(D), partial t)$* -5. $mu_0 frac(partial arrow(H), partial t)$ - -*Ответ*: определение. - -#line(length: 100%) - -=== Укажите все верные утверждения. Циркуляция вектора напряженности магнитного поля вычисляется по формуле - -*1. $integral.cont_L H_l d l$* \ -*2. $integral.cont_L arrow(H) d arrow(l)$* -3. $integral.cont_L arrow(H) times d arrow(l)$ -4. $integral.cont_L H 2 pi d r$ -5. $integral.cont_L arrow(H) times d arrow(r)$ - -*Ответ*: второй вариант -- определение циркуляции. первый вариант -- проекция на касательную. - -#line(length: 100%) - -=== Посередине между двумя точечными зарядами $q_1 eq 6 "нКл"$ и $q_2 eq -2 "нКл"$ помещен заряд $q$. На этот заряд со стороны заряда $q_2$ действует сила $4$ мкН. Определить силу, действующую на заряд $q$ со стороны обоих зарядов $q_1$ и $q_2$. - -1. $36$ мкН -2. $24$ мкН -3. $18$ мкН -*4. $16$ мкН* -5. $12$ мкН - -*Ответ*: На заряд $q$ действует сила $F_(q_1)$ и $F_(q_2)$. - -$ -frac(F_(q_1), F_(q_2)) eq frac(k frac(q q_1, x^2), k frac(q q_2, x^2)) eq frac(q_1, q_2) eq |-3| eq 3 -$ - -$ -F_(q_1) eq 3 F_(q_2) eq 3 dot 4 "мкН" eq 12 "мкН" -$ - -Так как силы сонаправлены - -$ -F eq F_(q_1) + F_(q_2) eq 16 "мкН" -$ - -#line(length: 100%) - -=== Электростатическое поле создано двумя точечными зарядами $-q$ и $+4q$. Отношение потенциала поля, созданного вторым зарядом в точке $A$, к потенциалу результирующего поля в этой точке равно - -1. $2$ -2. $3$ -*3. $4$* -4. $-2$ -5. $-4$ - -*Ответ*: - -$ -phi_2 eq k frac(4 q, 3 a) -$ - -$ -phi_1 eq k frac(-q, a) -$ - -$ -phi_"рез" eq phi_1 + phi_2 eq k(frac(-q, a) + frac(4 q, 3 a)) eq k frac(q, 3 a) -$ - -$ -frac(phi_2, phi_"рез") eq frac(k frac(4 q, 3 a), k frac(q, 3 a)) eq 4 -$ - -#line(length: 100%) - -=== Укажите все верные утверждения. Вихревое электрическое поле создают - -*1. движущиеся с ускорением электрические заряды* -2. движущиеся равномерно точечные заряды -3. потенциальные, однородные электрические поля -*4. изменяющееся во времени магнитное поле* -5. стационарное, однородное магнитное поле - -*Ответ*: Ускоренный заряд создает переменное магнитное поле. - -$ -"ускорение заряда" arrow.double frac(partial arrow(B), partial t) eq.not 0 arrow.double "rot" arrow(E) eq.not 0 -$ - -$ -("rot" arrow(E) eq -frac(partial arrow(B), partial t)) -$ - -#line(length: 100%) - -=== Какой график представляет зависимость напряженности электрического поля $E(r)$ для равномерно заряженной сферы радиуса $R$ - -*Ответ*: По закону Гаусса: - -$ -integral.cont arrow(E) dot d arrow(S) eq frac(q_"внутр", epsilon_0) -$ - -$ -q(r) eq q frac(r^3, R^3) -$ - -$ -E(r) 4 pi r^2 eq frac(q r^3, epsilon_0 R^3) -$ - -$ -E(r) eq k frac(q, R^3) r -$ - -снаружи сферы - -$ -E(r) 4 pi r^2 eq q/(epsilon_0) -$ - -$ -E(r) eq k frac(q, r^2) -$ - -#align(center)[ - #figure( - image("assets/5.png"), - caption: [Напряженность электрического поля равномерно заряженной сферы], - supplement: [Рис.] - ) -] - -#line(length: 100%) - -=== Пластина из однородного изотропного диэлектрика с проницаемостью $epsilon$ вплотную прилегает к проводящей пластине, заряженной с поверхностной плотностью $sigma$. Поверхностная плотность связанных зарядов на поверхности диэлектрика $sigma'$ равна. - -1. $sigma' eq frac(epsilon - 1, epsilon) sigma$ -2. $sigma' eq -frac(epsilon, epsilon - 1) sigma$ -3. $sigma' eq frac(epsilon, epsilon - 1) sigma$ -4. $sigma' eq - sigma / epsilon$ -*5. $sigma' eq -frac(epsilon - 1, epsilon) sigma$* - -*Ответ*: - -$ -D eq sigma -$ - -$ -arrow(D) eq epsilon_0 epsilon arrow(E) -$ - -$ -E eq frac(sigma, epsilon_0 epsilon) -$ - -$ -arrow(P) eq epsilon_0 (epsilon - 1) arrow(E) -$ - -$ -arrow(P) eq epsilon_0 (epsilon - 1) frac(sigma, epsilon_0 epsilon) eq frac(epsilon - 1, epsilon) sigma -$ - -$ -sigma' eq -P eq -frac(epsilon - 1, epsilon) sigma -$ - -#line(length: 100%) - -=== В каком случае поток вектора напряженности однородного электрического поля через произвольную замкнутую поверхность равен нулю? - -1. только когда на поверхности находятся электрические заряды -*2. только если вектор напряженности перпендикулярен поверхности во всех точках* -3. всегда -4. никогда не равен нулю -5. только когда поверхность имеет сферическую форму - -*Ответ*: - -$ -Phi eq integral.cont_S arrow(E) dot d arrow(S) -$ - -если $arrow(E) perp d arrow(S)$, то $arrow(E) dot d arrow(S) eq 0$, соответственно $Phi eq 0$ - -#line(length: 100%) - -=== Электрическое поле создается заряженным равномерно по объему шаром из дэлектрика с проницаемостью $epsilon eq 3$. Как изменится напряженность электрического поля на некотором расстоянии от центра шара внутри него, при уменьшении диэлектрической проницаемости в 2 раза. - -1. увеличится в 2 раза -*2. увеличится в 1.33 раза* -3. не изменится -4. уменьшится в 4 раза -5. уменьшится в 1.33 раза - -*Ответ*: - -$ -epsilon eq 3, epsilon' eq 3/2 -$ - -$ -integral.cont arrow(D) dot d arrow(S) eq q_"своб, внутри" -$ - -внутри шара - -$ -D dot 4 pi r^2 eq rho dot 4/3 pi r^3 -$ - -$ -D eq frac(rho r, 3) -$ - -$ -arrow(D) eq epsilon_0 epsilon arrow(E) -$ - -$ -E eq frac(D, epsilon_0 epsilon) -$ - -$ -E_1 eq frac(D, epsilon_0 3) -$ - -$ -E_2 eq frac(D, epsilon_0 3/2) -$ - -$ -frac(E_2, E_1) eq 3/(3/2) eq 2 -$ - -#line(length: 100%) - -=== Электрический диполь помещен в электрическое поле так, что его дипольный момент перпендикулярен линиям напряженности поля. Что произойдет с диполем? - -1. останется неподвижным -2. развернется моментом по полю и будет выталкиваться в область слабого поля -*3. развернется моментом по полю и будет втягиваться в область сильного поля* -4. развернется моментом против поля и будет выталкиваться в область слабого поля -5. развернется моментом против поля и будет втягиваться в область сильного поля - -*Ответ*: Диполь всегда втягивается в область сильного поля. Поле всегда пытается расположить диполь так, чтобы плюс был по полю, минус против. - -#line(length: 100%) - -=== Сегнетоэлектрик, поляризованность которого равна нулю, помещен в незаряженный плоский конденсатор. Напряжение на конденсаторе начинают увеличивать от нулевого значения. Укажите все верные утверждения - -*1. диэлектрическая восприимчивость сегнетоэлектрика сначала растет, потом убывает* \ -*2. индукция поля в сегнетоэлектрике растет* -3. индукция поля в сегнетоэлектрике сначала растет, потом убывает -4. диэлектрическая восприимчивость сегнетоэлектрика растет -5. индукция поля в сегнетоэлектрике убывает - -*Ответ*: В сегнетоэлектрике поляризация нелинейна. При малых полях диполи легко поворачиваются. При больших полях наступает насыщение. - -Диэлектрическая восприимчивость - -$ -chi eq frac(P, epsilon_0 E) -$ - -Электрическая индукция - -$ -arrow(D) eq epsilon_0 arrow(E) + arrow(P) -$ - -поле $arrow(E)$ растет, поляризация $arrow(P)$ растет. - -#line(length: 100%) - -=== Плоский воздушный конденсатор подключен к источнику напряжения. Расстояния между обкладками конденсатор увеличивают. Выберите все верные утверждения - -*1. напряженность поля в конденсаторе не меняется* -2. заряд конденсатора не меняется -*3. напряжение на конденсаторе не меняется* -4. заряд конденсатора увеличивается -*5. заряд конденсатора уменьшается* - -*Ответ*: Так как конденсатор подключен к источнику, то источник поддерживает напряжение. - -Напряженность тоже не меняется. (хз) - -Емкость плоского конденсатора - -$ -C eq frac(epsilon_0 S, d) -$ - -Так как $d$ увеличивается, то $C$ уменьшается. - -$ -Q eq C U -$ - -Если $C$ уменьшается, а $U$ постоянно, тогда $Q$ уменьшается. - -#line(length: 100%) - -=== Вектор напряженности электростатического поля по отношению к эквипотенциальным поверхностям направлен - -*1. по нормали в сторону убывания потенциала* -2. по касательной в сторону убывания потенциала -3. по нормали в сторону возрастания потенциала -4. по касательной в сторону возрастания потенциала -5. по спирали охватывает силовые линии - -*Ответ*: По определению - -$ -arrow(E) eq - gradient phi -$ - -вектор напряженности направлен в сторону убывания потенциала. - -Эквипотенциальная поверхность -- это поверхность, где: - -$ -phi eq "const" -$ - -Тогда $arrow(E)$ направлен по нормали к эквипотенциальной поверхности. - -#line(length: 100%) - -=== При затухающих гармонических колебаниях частота колебаний - -1. намного меньше собственной частоты колебательной системы -2. намного больше собственной частоты колебательной системы -3. равной собственной частоте колебательной системы -*4. чуть меньше собственной частоты колебательной системы* -5. чуть больше собственной частоты колебательной системы - -*Ответ*: Для линейной колебательной системы - -$ -x'' + 2 beta x' + omega_0^2 x eq 0 -$ - -где $omega_0$ -- собственная частота, $beta$ -- коэффициент затухания. - -При слабом затухании ($beta lt omega_0$) - -$ -x(t) eq A_0 e^(-beta t) cos (omega t + phi) -$ - -где частота затухающих колебаний равна - -$ -omega eq sqrt(omega_0^2 - beta^2) -$ - -Так как - -$ -beta^2 gt 0 -$ - -то - -$ -omega_0^2 - beta^2 lt omega_0^2 -$ - -следовательно - -$ -omega lt omega_0 -$ - -Если колебания гармонические, затухание слабое - -$ -beta lt.double omega_0 -$ - -Тогда - -$ -omega eq omega_0 sqrt(1 - frac(beta^2, omega_0^2)) approx omega_0 (1 - frac(beta^2, 2 omega_0^2)) -$ - -$ -omega_0 - omega approx frac(beta^2, 2 omega_0) -$ - -#line(length: 100%) - -=== Плоский воздушный конденсатор подключен к источнику напряжения. Конденсатор заполняют диэлектриком. Выберите все верные утверждения - -1. напряженность поля в конденсаторе увеличивается -*2. напряженность поля в конденсаторе уменьшается* -*3. напряжение на конденсаторе не меняется* -*4. заряд конденсатора увеличивается* -5. заряд конденсатора уменьшается - - -*Ответ*: поскольку конденсатор подключен к источнику, $U$ остается постоянным. - -Напряженность поля в конденсаторе с диэлектриком - -$ -E eq U/d -$ - -Диэлектрик не изменяет внешнее напряжение $U$, но в нем создаются внутренние поляризационные заряды, которые частично компенсируют поле. Напряженность внутри диэлектрика меньше, чем в воздухе. - -Емкость конденсатора с диэлектриком увеличивается - -$ -C eq epsilon_r epsilon_0 S/d gt C_0 -$ - -А поскольку $U eq "const"$, заряд - -$ -Q eq C U -$ - -увеличивается - -#line(length: 100%) - -=== Плоский воздушный конденсатор заполнен диэлектриком с проницаемостью $epsilon$. Конденсатор подключен к источнику напряжения. Диэлектрика вынимают из конденсатора. Выберите верные утверждения - -*1. напряженность поля в конденсаторе увеличивается* -2. напряженность поля в конденсаторе уменьшается -*3. напряжение на конденсаторе не меняется* -4. заряд конденсатора увеличивается -*5. заряд конденсатора уменьшается* - -*Ответ*: Конденсатор подключен к источнику, поэтому $U$ остается постоянным. - -Напряженность внутри диэлектрика была меньше, чем в воздухе - -$ -E_"диэлектрик" eq frac(U, epsilon d) lt U/d eq E_"воздух" -$ - -После того как диэлектрик вынимают - -$ -E eq U/d gt E_"диэлектрик" -$ - -Емкость конденсатора с диэлектриком - -$ -C_"диэлектрик" eq epsilon C_0 -$ - -После вынимания диэлектрика - -$ -C_"воздух" eq C_0 lt C_"диэлектрик" -$ - -Поскольку $U eq "const"$ - -$ -Q eq C U -$ - -емкость уменьшилась, соответственно заряд уменьшился. - -#line(length: 100%) - -=== Укажите все верные утверждения. Для потенциального электрического поля - -*1. $integral_L arrow(E) d arrow(l) eq 0$* -2. на незаряженной границе диэлектриков $E_(1 n) eq E_(2 n)$ -*3. $"rot" arrow(E) eq 0$* -4. $integral_L arrow(E) d arrow(l) eq -integral_S frac(partial arrow(B), partial t) d S$ -*5. $arrow(E) eq -"grad" phi$* - -*Ответ*: Электрическое поле называется потенциальным, если существует скалярный потенциал $phi$, такой что: - -$ -arrow(E) eq - nabla phi -$ - -$ -arrow(E) eq -"grad" phi -$ - -это определение - -$ -"rot" arrow(E) eq 0 -$ - -Это свойство потенциального поля. Поле не завихрено, линии поля не образуют замкнутых петель. - -$ -integral.cont_L arrow(E) dot d arrow(l) eq 0 -$ - -Из теоремы Стокса - -$ -integral.cont_L arrow(E) dot d arrow(l) eq integral_S "rot" arrow(E) dot d arrow(S) -$ - -Для потенциального поля $"rot" arrow(E) eq 0$, значит интеграл по любому замкнутому контуру $eq 0$. - -#line(length: 100%) - -=== Собственная частота в колебательном контуре определяется выражением - -*1. $2 pi sqrt(L C)$* -2. $frac(2 pi, sqrt(L C))$ -3. $frac(R, 2 L)$ -4. $2 pi L C$ - -*Ответ*: хз. - -#line(length: 100%) - -=== Электрическое поле создается заряженным равномерно по объему шаром из диэлектрика с проницаемостью $epsilon eq 5$. Как изменится напряженность электрического поля на некотором расстоянии от центра шара внутри него при увеличении объемной плотности заряда внутри шара в 2 раза - -*1. увеличится в $2$ раза* -2. увеличится в $1.33$ раза -3. увеличится в $4$ раза -4. уменьшится в $4$ раза -5. уменьшится в $5$ раз - -*Ответ*: Для шара с объемной плотностью заряда $rho$ в диэлектрике с $epsilon$ напряженность внутри шара задается формулой - -$ -E eq frac(rho r, 3 epsilon_0 epsilon) -$ - -Если увеличиваем $rho$ в два раза - -$ -E arrow frac((2 rho) r, 3 epsilon_0 epsilon) eq 2 dot frac(rho r, 3 epsilon_0 epsilon) eq 2 E -$ - -#line(length: 100%) - -=== Как изменится напряженность поля внутри заряженного и отключенного от источника воздушного конденсатора, если увеличить расстояние между пластинами в 4 раза? - -1. увеличится в 4 раза -2. уменьшится в 4 раза -3. уменьшится в 2 раза -4. увеличится в 2 раза -*5. не изменится* - -*Ответ*: Напряженность поля в плоском конденсаторе - -$ -E eq frac(sigma, epsilon_0) eq frac(q, epsilon_0 S) -$ - -Емкость конденсатора - -$ -C eq frac(epsilon_0 S, d) -$ - -Напряжение на пластинах - -$ -U eq Q/C eq frac(Q d, epsilon_0 S) -$ - -Если $d arrow 4 d$, емкость уменьшается в 4 раза, а напряжение - -$ -U arrow 4 U -$ - -Напряжение увеличилось, но $E$ внутри, как плотность поля между пластинами, остается - -$ -E eq U/d eq frac(4 U, 4 d) eq U/d eq E_"исходное" -$ - -#line(length: 100%) - -=== Пластину из однородного изотропного диэлектрика внесли в заряженный конденсатор, параллельно его пластинам, но не касаясь их. Если пренебречь утечкой заряда с конденсатор, то - -*1. на поверхности диэлектрика появятся связанные заряды* -2. на поверхности и в объеме диэлектрика появятся свободные заряды -3. в объеме диэлектрика появятся связанные заряды -4. на поверхности диэлектрика появятся связанные заряды, а в объеме -- свободные -5. на поверхности диэлектрика появятся свободные заряды, а внутри -- связанные - -*Ответ*: хз. - -#line(length: 100%) - -=== При преломлении линий индукции электрического поля на границе двух однородных изотропных диэлектриков - -1. $frac(sin alpha_2, sin alpha_1) eq frac(epsilon_1, epsilon_2)$ -2. $frac(sin alpha_2, sin alpha_1) eq frac(epsilon_2, epsilon_1)$ -*3. $frac(tg alpha_2, tg alpha_1) eq frac(epsilon_2, epsilon_1)$* -4. $frac(tg alpha_2, tg alpha_1) eq frac(epsilon_1, epsilon_2)$ -5. $tg alpha_1 eq tg alpha_2$ - -*Ответ*: Тангенциальная компонента напряженности непрерывна - -$ -E_(1 tau) eq E_(2 tau) -$ - -Нормальная компонента электрической индукции непрерывна - -$ -D_(1 n) eq D_(2 n) -$ - -Так как - -$ -arrow(D) eq epsilon epsilon_0 arrow(E) -$ - -то - -$ -epsilon_1 E_(1 n) eq epsilon_2 E_(2 n) -$ - -$ -E_n eq E cos alpha \ -E_tau eq E sin alpha -$ - -Тангенциальная компонента - -$ -E_1 sin alpha_1 eq E_2 sin alpha_2 -$ - -Нормальная компонента - -$ -epsilon_1 E_1 cos alpha_1 eq epsilon_2 E_2 cos alpha_2 -$ - -Поделив друг на друга - -$ -frac(E_1 sin alpha_1, epsilon_1 E_1 cos alpha_1) eq frac(E_2 sin alpha_2, epsilon_2 E_2 cos alpha_2) -$ - -Сократив $E_1, E_2$ - -$ -frac(tg alpha_1, epsilon_1) eq frac(tg alpha_2, epsilon_2) -$ - -$ -frac(tg alpha_2, tg alpha_1) eq frac(epsilon_2, epsilon_1) -$ - -#line(length: 100%) - -=== При преломлении линий индукции магнитного поля на границе двух однородных изотропных диэлектриков - -1. $frac(sin alpha_2, sin alpha_1) eq frac(mu_1, mu_2)$ -2. $frac(sin alpha_2, sin alpha_1) eq frac(mu_2, mu_1)$ -*3. $frac(tg alpha_2, tg alpha_1) eq frac(mu_2, mu_1)$* -4. $frac(tg alpha_2, tg alpha_1) eq frac(mu_1, mu_2)$ -5. $tg alpha_1 eq tg alpha_2$ - -*Ответ*: Нормальная компонента непрерывна - -$ -B_(1 n) eq B_(2 n) -$ - -Тангенциальная компонента непрерывна - -$ -H_(1 tau) eq H_(2 tau) -$ - -Но - -$ -arrow(B) eq mu mu_0 arrow(H) -$ - -Следовательно - -$ -frac(B_(1 tau), mu_1) eq frac(B_(2 tau), mu_2) -$ - -$ -B_n eq B cos alpha \ -B_tau eq B sin alpha -$ - -$ -B_1 cos alpha_1 eq B_2 cos alpha_2 -$ - -$ -frac(B_1 sin alpha_1, mu_1) eq frac(B_2 sin alpha_2, mu_2) -$ - -$ -frac(B_1 sin alpha_1, mu_1 B_1 cos alpha_1) eq frac(B_2 sin alpha_2, mu_2 B_2 cos alpha_2) -$ - -$ -frac(tg alpha_1, mu_1) eq frac(tg alpha_2, mu_2) -$ - -$ -frac(tg alpha_2, tg alpha_1) eq frac(mu_2, mu_1) -$ - -#line(length: 100%) - -=== При преломлении линий напряженности электрического поля на границе двух однородных изотропных диэлектриков - -1. $frac(sin alpha_2, sin alpha_1) eq frac(epsilon_1, epsilon_2)$ -2. $frac(sin alpha_2, sin alpha_1) eq frac(epsilon_2, epsilon_1)$ -*3. $frac(tg alpha_2, tg alpha_1) eq frac(epsilon_2, epsilon_1)$* -4. $frac(tg alpha_2, tg alpha_1) eq frac(epsilon_1, epsilon_2)$ -5. $tg alpha_1 eq tg alpha_2$ - -*Ответ*: Тангенциальная компонента $arrow(E)$ непрерывна - -$ -E_(1 tau) eq E_(2 tau) -$ - -Нормальная компонента $arrow(D)$ непрерывна - -$ -D_(1 n) eq D_(2 n) -$ - -Но - -$ -arrow(D) eq epsilon epsilon_0 arrow(E) -$ - -следовательно - -$ -epsilon_1 E_(1 n) eq epsilon_2 E_(2 n) -$ - -$ -E_n eq E cos alpha \ -E_tau eq E sin alpha -$ - -Тангенциальная компонента - -$ -E_1 sin alpha_1 eq E_2 sin alpha_2 -$ - -Нормальная компонента - -$ -epsilon_1 E_1 cos alpha_1 eq epsilon_2 E_2 cos alpha_2 -$ - -$ -frac(E_1 sin alpha_1, epsilon_1 E_1 cos alpha_1) eq frac(E_2 sin alpha_2, epsilon_2 E_2 cos alpha_2) -$ - -Сократив - -$ -frac(tg alpha_1, epsilon_1) eq frac(tg alpha_2, epsilon_2) -$ - -$ -frac(tg alpha_2, tg alpha_1) eq frac(epsilon_2, epsilon_1) -$ - -#line(length: 100%) - -=== При преломлении линий напряженности магнитного поля на границе двух однородных изотропных магнетиков - -1. $frac(sin alpha_2, sin alpha_1) eq frac(mu_1, mu_2)$ -2. $frac(sin alpha_2, sin alpha_1) eq frac(mu_2, mu_1)$ -*3. $frac(tg alpha_2, tg alpha_1) eq frac(mu_2, mu_1)$* -4. $frac(tg alpha_2, tg alpha_1) eq frac(mu_1, mu_2)$ -5. $tg alpha_1 eq tg alpha_2$ - -*Ответ*: Тангенциальная компонента непрерывна - -$ -H_(1 tau) eq H_(2 tau) -$ - -Нормальная компонента непрерывна - -$ -B_(1 n) eq B_(2 n) -$ - -Но - -$ -arrow(B) eq mu mu_0 arrow(H) -$ - -следовательно - -$ -mu_1 H_(1 n) eq mu_2 H_(2 n) -$ - -$ -H_n eq H cos alpha \ -H_tau eq H sin alpha -$ - -Тангенциальная компонента - -$ -H_1 sin alpha_1 eq H_2 sin alpha_2 -$ - -Нормальная компонента - -$ -mu_1 H_1 cos alpha_1 eq mu_2 H_2 cos alpha_2 -$ - -$ -frac(H_1 sin alpha_1, mu_1 H_1 cos alpha_1) eq frac(H_2 sin alpha_2, mu_2 H_2 cos alpha_2) -$ - -$ -frac(tg alpha_1, mu_1) eq frac(tg alpha_2, mu_2) -$ - -$ -frac(tg alpha_2, tg alpha_1) eq frac(mu_2, mu_1) -$ diff --git a/course2/sem3/exam/res.pdf b/course2/sem3/exam/res.pdf new file 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