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+116575 +%%EOF \ No newline at end of file diff --git a/course2/sem3/exam/tasks.typ b/course2/sem3/exam/tasks.typ new file mode 100644 index 0000000..333afcb --- /dev/null +++ b/course2/sem3/exam/tasks.typ @@ -0,0 +1,133 @@ +#set text( + size: 14pt, + font: "New Computer Modern", +) + +#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. $q/(epsilon_0)$ +