From bd2a69bf40abb3ca01a7527cf56c223a128d9e98 Mon Sep 17 00:00:00 2001 From: fYmious Date: Mon, 15 Dec 2025 15:36:14 +0300 Subject: [PATCH] rdy w/ hw --- course2/sem3/homework/assets/12.png | Bin 0 -> 121739 bytes course2/sem3/homework/assets/7.png | Bin 31481 -> 0 bytes course2/sem3/homework/assets/7.svg | 2 + course2/sem3/homework/assets/9.png | Bin 47106 -> 0 bytes course2/sem3/homework/assets/9.svg | 2 + course2/sem3/homework/solutions.pdf | Bin 1259234 -> 1171885 bytes course2/sem3/homework/solutions.typ | 331 ++++++++++------------------ 7 files changed, 123 insertions(+), 212 deletions(-) create mode 100644 course2/sem3/homework/assets/12.png delete mode 100644 course2/sem3/homework/assets/7.png create mode 100644 course2/sem3/homework/assets/7.svg delete mode 100644 course2/sem3/homework/assets/9.png create mode 100644 course2/sem3/homework/assets/9.svg diff --git a/course2/sem3/homework/assets/12.png b/course2/sem3/homework/assets/12.png new file mode 100644 index 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Постоянный ток.] -#align(center)[===== №1] +#align(center)[===== №1] // ready Система состоит из полусферы несущей равномерно распределённый заряд с поверхностной плотностью $sigma eq 5 " нКл/м"^2$. Рассчитать модуль напряжённости электростатического поля, создаваемого полусферой в её центре. @@ -59,7 +59,7 @@ $ d E eq k frac(d q, R^2) eq k frac(sigma R^2 sin theta d theta d phi, R^2) eq k sigma sin theta d theta d phi. $ -Расписав составляющие: +Расписав составляющие (так как поле направлено к центру, значение с минусом): $ d E_x eq -k sigma sin^2 theta cos phi d theta d phi, \ @@ -87,9 +87,9 @@ $ *Ответ*: $E approx 0.14 "кВ/м"$. -#align(center)[===== №2] +#align(center)[===== №2] // ready -Система представляет собой область пространства заполненного зарядом с объёмной плотностью $rho = rho_0 exp(-alpha r^3)$, где $rho_0$ и $alpha$ -- положительные постоянные, а $r$ -- расстояние от центра системы. Найти модуль напрёжённости электростатического поля, как функцию $r$. +Система представляет собой область пространства заполненного зарядом с объёмной плотностью $rho = rho_0 exp(-alpha r^3)$, где $rho_0$ и $alpha$ -- положительные постоянные, а $r$ -- расстояние от центра системы. Найти модуль напряжённости электростатического поля, как функцию $r$. *Решение*: @@ -104,10 +104,16 @@ $ По закону Гаусса: $ -integral.cont_S bold(E) dot d bold(S) eq frac(Q_"вн", epsilon_0) +integral.cont_S arrow(E) dot d arrow(S) eq frac(Q_"вн", epsilon_0) $ -Система обладает сферической симметрией. + + +Система обладает сферической симметрией. Модуль $E(r)$ одинаков по всей сфере радиуса $r$, тогда (площадь поверхности сферы $4 pi r^2$): + +$ +integral.cont_S arrow(E) dot d arrow(S) eq E(r) integral.cont_S d S eq E(r) dot S eq E(r) dot 4 pi r^2 +$ $ E(r) 4 pi r^2 eq frac(Q_"вн" (r), epsilon_0) arrow.double E(r) eq frac(Q_"вн" (r), 4 pi epsilon_0 r^2) @@ -132,12 +138,12 @@ $ Подставим в закон Гаусса: $ -E(r) eq frac(rho_0, 3 epsilon_0 alpha r^2) (1 - e^(-alpha r^3)) +E(r) eq frac(Q_"вн" (r), 4 pi epsilon_0 r^2) eq frac(frac(4 pi rho_0, 3 alpha)(1 minus e(minus alpha r^3)), 4 pi epsilon_0 r^2) eq frac(rho_0, 3 epsilon_0 alpha r^2) (1 - e^(-alpha r^3)) $ *Ответ*: $E(r) eq frac(rho_0, 3 epsilon_0 alpha r^2) (1 - exp(-alpha r^3))$. -#align(center)[===== №3] +#align(center)[===== №3] // ready Система состоит из равномерно заряженного шара радиуса $R eq 20 "см"$. Рассчитать разность потенциалов между точками, лежащими на расстоянии $r_1 eq 1 "см"$ и $r_2 eq 15 "см"$ от центра шара. Объёмная плотность заряда $rho eq 10 " нКл/м"^3$. Диэлектрическая проницаемость вещества из которого состоит шар $epsilon = 1$. @@ -161,7 +167,7 @@ $ Поток через сферу радиуса $r$ равен: $ -integral.cont bold(E) dot d bold(S) eq E(r) dot 4 pi r^2 eq frac(Q_"вн", epsilon_0). +integral.cont arrow(E) dot d arrow(S) eq E(r) dot 4 pi r^2 eq frac(Q_"вн", epsilon_0). $ Отсюда можно выразить $E(r)$: @@ -190,7 +196,7 @@ $ *Ответ*: $Delta phi approx 4.2 "B"$. -#align(center)[===== №4] +#align(center)[===== №4] // ready Зазор между пластинами плоского конденсатора полностью плоская слюдяная пластинка ($epsilon_1 eq 7$) толщиной $d_1 eq 2 "мм"$, и слой парафина ($epsilon_1 eq 2$) толщиной $d_2 eq 1 "мм"$. Рассчитать модули напряжённости электрического поля в обоих диэлектриках, если разность потенциалов между пластинами $U eq 200 В$. @@ -204,7 +210,7 @@ $ ) ] -При статическом поле в плоском конденсаторе нормальная компонента вектора электрической индукции $bold(D)$ одинакова во всех слоях: +При статическом поле в плоском конденсаторе нормальная компонента вектора электрической индукции $arrow(D)$ одинакова во всех слоях: $ D eq epsilon_0 epsilon_(r 1) E_1 eq epsilon_0 epsilon_(r 2) E_2. @@ -242,7 +248,7 @@ $ *Ответ*: $E_1 approx 36 "кВ/м", E_1 approx 0.13 "МВ/м"$. -#align(center)[===== №5] +#align(center)[===== №5] // ready На расстоянии $l eq 1.5 "см"$ от проводящей плоскости расположен точечный заряд $q eq 100 "мкКл"$. Рассчитайте работу, которую необходимо совершить против электрических сил, чтобы медленно удалить этот заряд от плоскости на бесконечность. @@ -258,29 +264,40 @@ $ В системе СИ: $l eq 1.5 "см" eq 0.015 "м", space q eq 100 "мкКл" eq 1.0 dot 10^(-4) "Кл"$. +По закону сохранения энергии: + +$ +Delta E_K plus Delta E_P eq A_"тр" plus A_"вн" +$ + +Так как мы удаляем заряд медленно, то $Delta E_K eq 0$. Про трение ничего не сказано, поэтому $A_"тр" eq 1$. Тогда: + +$ +A eq Delta E_P eq E_(P 2) minus E_(P 1) +$ + +На бесконечности ($E_(P 2)$) равна нулю так как $r eq infinity$, и в формуле $E_P eq k frac(q, r)$ стоит в знаменателе. + Реальный заряд $q$ находится на расстоянии $l$ от плоскости, а мнимый заряд $q' eq -q$ находится на расстоянии $l$ по другую сторону плоскости. Тогда обозначим за $r eq 2 l eq 0.03 "м"$. -Потенциальная энергия взаимодействия двух зарядов $U$ равна: +Потенциальная энергия взаимодействия двух зарядов $E_P$ равна: $ -U eq k frac(q q', r) eq -k frac(q^2, 2 l) +E_P eq k frac(q q', r) eq -k frac(q^2, 2 l) $ -Чтобы удалить заряд в бесконечность, нужно сделать работу $A$: +Нужно учесть, что получившееся значение -- работа по удалению не одного, а двух зарядов. Тогда поделим значение на 2. -$ -A eq -U eq k frac(q^2, 2 l) -$ Подставив числа, получим: $ -A approx 0.15 dot 10^3 "Дж" +A eq frac(q^2, 16 pi epsilon_0 l) eq frac(10^4 dot 10^(-12), 16 pi dot 8.85 dot 10^(-12) dot 1.5 dot 10^(-2)) approx 0.15 dot 10^3 "Дж" $ *Ответ*: $A approx 0.15 dot 10^3 "Дж"$. -#align(center)[===== №6] +#align(center)[===== №6] // ready По прямому проводнику длина которого $l eq 400 "м"$ течёт постоянный ток, сила которого $I eq 10 "А"$. Рассчитать суммарный импульс электронов в проводнике. @@ -294,72 +311,34 @@ $ ) ] -Пусть $v_d$ - дрейфовая скорость электронов. Тогда импульс всех электронов $p$ равен: +По формуле плотности тока: $ -p eq M v_d +j eq rho U ", где" rho eq frac(q, l dot S) arrow.double j eq frac(q U, l S) $ -где $M$ - суммарная масса всех электронов в проводнике. - -За время $Delta t$ электроны сдвигаются вдоль провода на расстояние: +Так как $j eq I/S$ по определению, то можно выразить заряд: $ -Delta x eq v_d Delta t. +j eq I/S eq frac(q U, l S) arrow.double q eq frac(I l, U) $ -Тогда объем, прошедший через сечение: +Масса всех электронов равна произведению их количества на массу одного электрона: $ -V eq S Delta x eq S v_d Delta t. +m eq n_e dot m_e eq q/e m_e eq frac(I l, U e) m_e $ -Если $n$ - концентрация свободных электронов на метр кубический, то: +По формуле импульса: $ -N eq n V eq n S v_d Delta t +p eq m U eq frac(I l m_e, e) $ -это число электронов, прошедших через сечение. Каждый электрон имеет заряд $e$ по модулю, поэтому полный заряд $Delta Q$: +Подставим числа из условия: $ -Delta Q eq N e eq n e S v_d Delta t. -$ - -По определению силы тока: - -$ -I eq frac(Delta Q, Delta t) eq frac(n e S v_d Delta t, Delta t) eq n e S v_d. -$ - -Пусть площадь сечения - $S$. Тогда объем $V$ равен: - -$ -V eq S l -$ - -и число электронов $N$: - -$ -N eq n V eq n S l. -$ - -Выразив дрейфовую скорость из $I eq n e S v_d$, получим: - -$ -v_d eq frac(I, n e S) -$ - -Тогда, подставив это в $p eq N m_e v_d$: - -$ -p eq N m_e v_d eq (n S l) m_e dot frac(I, n e S) eq frac(m_e I l, e) -$ - -Подставим числа и получим: - -$ -p eq frac(3.644 dot 10^(-27), 1.6 dot 10^(-19)) approx 2.28 dot 10^(-8) "Н/c" +p eq frac(10 dot 400 dot 9.1 dot 10^(-31), 1.6 dot 10^(-19)) approx 2.3 dot 10^(-8). $ *Ответ*: $p eq 2.3 dot 10^(-8) "Н/с"$. @@ -367,7 +346,7 @@ $ #align(center)[=== Магнитостатика. Закон электромагнитной индукции Фарадея.] -#align(center)[===== №1] +#align(center)[===== №1] // ready Замкнутый контур с током имеет вид прямоугольника с диагональю $d eq 16 "см"$, угол между диагоналями $alpha eq 30 degree$. Сила тока, протекающего по контуру $I eq 5 "A"$. Рассчитать модуль индукции магнитного поля в центре контура. @@ -376,154 +355,73 @@ $ #align(center)[ #figure( - image("assets/7.png"), + image("assets/7.svg"), supplement: [Рис.], caption: [Прямоугольный контур с током и его центр.] ) ] -В системе СИ: $d eq 16 "см" eq 0.16 "м"$. Обозначим стороны прямоугольника $a$ и $b$, тогда длина диагонали $d$ равна: +В системе СИ: $d eq 16 "см" eq 0.16 "м"$. + +По принципу суперпозиции для магнитного поля: $ -d eq sqrt(a^2 + b^2). +arrow(B) eq arrow(B)_1 plus arrow(B)_2 plus arrow(B)_3 plus arrow(B)_4. $ -Угол между диагоналями $alpha$ выражается через $a$ и $b$. Для векторов диагоналей $bold(d)_1 eq (a, b), bold(d)_2 eq (a, -b)$ получим: +Так как все $arrow(B)_i$ сонаправлены, то $B eq 2(B_1 plus B_2)$. По закону Био-Савара-Лапласа: $ -cos alpha eq frac(bold(d)_1 dot bold(d)_2, |bold(d)_1||bold(d)_2|) eq frac(a^2 - b^2, a^2 + b^2) +B eq 2(frac(mu_0 I, 2 pi) (frac(cos frac(pi minus phi, 2), d/2 cos phi/2) + frac(cos phi/2, d/2 sin phi/2))) eq \ +eq frac(mu_0 I, pi) (frac(1, d/2 sin phi/2 cos phi/2)) eq frac(4 mu_0 I, pi d sin phi) $ -или +Подставив числа из условия, получим: $ -b/a eq tan alpha/2 -$ - -Отсюда: - -$ -a eq d cos alpha/2, space.quad b eq d sin alpha/2 -$ - -$ -a eq d cos 15 degree, space.quad b eq d sin 15 degree -$ - -По закону Био-Савара: - -$ -d B eq frac(mu_0 I, 4 pi) frac(d l sin phi, r^2), -$ - -где $d l$ - элемент проводника, $r$ - расстояние до точки наблюдения, $phi$ - угол между направлением тока и направлением на точку наблюдения. Проводник лежит вдоль оси $x$, точка наблюдения на оси $y$. Тогда расстояние $r eq sqrt(x^2 + y^2)$. - -Угол $phi$ между током и направлением на точку: - -$ -sin phi eq frac(y, sqrt(x^2 + y^2)) eq y/r -$ - -После подстановки в закон Био-Савара: - -$ -d B eq frac(mu_0 I, 4 pi) frac(d x, (x^2 + y^2)) dot frac(y, sqrt(x^2 + y^2)) eq frac(mu_0 I, 4 pi) frac(y d x, (x^2 + y^2)^(3/2)). -$ - -проинтегрировав по длине проводника: - -$ -B eq frac(mu_0 I, 4 pi) y integral_(x_A)^(x_B) frac(d x, (x^2 + y^2)^(3/2)) eq frac(mu_0 I, 4 pi y) (frac(x_B, sqrt(x^2_B + y^2)) - frac(x_A, sqrt(x^2_A + y^2))) -$ - -Обозначим углы до концов проводника: - -$ -cos theta_1 eq frac(x_A, sqrt(x_A^2 + y^2)), space.quad cos theta_2 eq frac(x_B, sqrt(x_B^2 + y^2)) -$ - -тогда - -$ -B eq frac(mu_0 I, 4 pi y) (cos theta_2 - cos theta_1). -$ - -По формуле разности косинусов: - -$ -cos theta_2 - cos theta_1 eq 2 sin frac(theta_2 + theta_1, 2) sin frac(theta_2 - theta_1, 2) -$ - -Но в точке на перпендикуляре можно возпользоваться более простым выражением: - -$ -cos theta_2 - cos theta_1 eq sin theta_1 + sin theta_2 -$ - -Так как точка находится напротив середины проводника, то $theta_1 eq theta_2 eq theta$. Тогда: - -$ -B eq frac(mu_0 I, 2 pi y) sin theta -$ - -Для симметричного случая можно расписать $sin theta$ как: - -$ -sin theta eq frac(L/2, sqrt((L/2)^2 + y^2)). -$ - -Для отрезка длины $L$ на расстоянии $y$ от его середины: - -$ -B_"отр" eq frac(mu_0 I L, 4 pi y sqrt((L/2)^2 + y^2)). -$ - -Сумма вкладов двух противоположных сторон длины $a$: - -$ -B_a^"sum" eq 2 dot frac(mu_0 I a, 4 pi (b/2) sqrt((a/2)^2 + (b/2)^2)) eq frac(mu_0 I a, pi b sqrt((a/2)^2 + (b/2)^2)) -$ - -Аналогично для сторон длины $b$: - -$ -B_b^"sum" eq frac(mu_0 I b, pi a sqrt((a/2)^2 + (b/2)^2)) -$ - -Сложив, получим: - -$ -B eq B_a^"sum" + B_b^"sum" eq frac(mu_0 I, pi sqrt((a/2)^2 + (b/2)^2)) (a/b + b/a) eq frac(mu_0 I, pi d/2) (a/b + b/a) eq \ -eq frac(2 mu_0 I, pi d) (a/b + b/a) eq frac(2 mu_0 I, pi d) frac(2, sin alpha) eq frac(4 mu_0 I, pi d sin alpha) -$ - -Подставив числа, получим: - -$ -B eq frac(4(4 pi dot 10^(-7)) dot 5, pi dot 0.16 dot 0.5) approx 1.0 dot 10^(-4) "Т" eq 0.1 "мТ". +B eq frac(4 dot 4 pi dot 10^(-7) dot 5, pi dot 0.16 dot sin 30 degree) approx 0.1 $ *Ответ*: $B approx 0.1 "мТл"$. -#align(center)[===== №2] +#align(center)[===== №2] // ready Два бесконечных прямых параллельных проводника разделены расстоянием $d eq 20 "см"$. По проводникам в противоположных направлениях текут токи $I_1 eq I_2 eq 10 "А"$. Рассчитать модуль напряжённости магнитного поля в точке, равноудалённой от обоих проводников на расстояние $a eq 20 "см"$. -*Решение*: хз +*Решение*: #align(center)[ #figure( - image("assets/9.png"), + image("assets/9.svg"), supplement: [Рис.], caption: [Два параллельных проводника с противоположными токами.] ) ] + +По формуле напряженности магнитного поля для прямого тока: + +$ +H eq 2 pi r I +$ + +Результирующее поле направлено вниз: + +$ +H eq (2I)/(2 pi r) cos pi/3 eq I/(2 pi r) +$ + +Подставив числа из условия, получим: + +$ +frac(10, 2 pi dot 0.2) approx 8 "А/м". +$ + *Ответ*: $H approx 8 "А/м"$. -#align(center)[===== №3] +#align(center)[===== №3] // ready -По проводу бесконечной длины, имеющего форму цилиндра радиуса $R$ течёт постоянный ток, плотность которого зависит от расстояния до центра провода как $j eq alpha r bold(e)_z$. Рассчитать вектор магнитной индукции создаваемый током внутри и вне провода, как функцию $r$ (магнитная проницаемость всюду равна 1). +По проводу бесконечной длины, имеющего форму цилиндра радиуса $R$ течёт постоянный ток, плотность которого зависит от расстояния до центра провода как $j eq alpha r arrow(e)_z$. Рассчитать вектор магнитной индукции создаваемый током внутри и вне провода, как функцию $r$ (магнитная проницаемость всюду равна 1). *Решение*: @@ -538,13 +436,13 @@ $ Для осесимметричного распределения удобно взять круговой контур радиуса $r$, с центром на оси цилиндра. Интеграл по контуру: $ -integral.cont bold(B) dot d bold(l) eq B_phi (r) (2 pi r) +integral.cont arrow(B) dot d arrow(l) eq B_phi (r) (2 pi r) $ Закон Ампера: $ -integral.cont bold(B) dot d bold(l) eq mu_0 I_"вн" (r) +integral.cont arrow(B) dot d arrow(l) eq mu_0 I_"вн" (r) $ где $I_"вн" (r)$ - суммарный ток, пронизывающий поверхность, ограниченную контуром. @@ -583,28 +481,28 @@ $ B_phi (r) eq frac(mu_0 I, 2 pi r) eq frac(mu_0, 2 pi r) dot frac(2 pi alpha R^3, 3) eq frac(mu_0 alpha R^3, 3 r) $ -*Ответ*: $bold(B) (r lt R) eq frac(mu_0 alpha r^2, 3) bold(e)_phi, space bold(B) (r gt R) eq frac(mu_0 alpha R^3, 3 r) bold(e)_phi$. +*Ответ*: $arrow(B) (r lt R) eq frac(mu_0 alpha r^2, 3) arrow(e)_phi, space arrow(B) (r gt R) eq frac(mu_0 alpha R^3, 3 r) arrow(e)_phi$. -#align(center)[===== №4] +#align(center)[===== №4] // ready В однородное магнитное поле с магнитной индукцией $B eq 0.4 "Тл"$ перпендикулярно полю с постоянной скоростью влетает заряженная частица. В течении $6 "мкс"$ включается постоянное электрическое поле напряжённостью $E eq 300 "В/м"$ сонаправленно магнитному полю. Рассчитать шаг винтовой траектории частицы после выключения электрического поля. *Решение*: По формуле силы Лоренца: $ -bold(F) eq q(bold(E) + bold(v) times bold(B)) +arrow(F) eq q(arrow(E) + arrow(v) times arrow(B)) $ До включения электрического поля: $ -bold(E) eq 0, space.quad bold(F) eq q(bold(v) times bold(B)) +arrow(E) eq 0, space.quad arrow(F) eq q(arrow(v) times arrow(B)) $ -Поскольку $bold(b) perp bold(B)$, частица движется по окружности +Частица движется по окружности $ -F_"маг" = q v b. +F_"маг" = q v B. $ Сила Лоренца равна центростремительной силе: @@ -651,7 +549,7 @@ $ *Ответ*: $h eq 0.28 "м"$. -#align(center)[===== №5] +#align(center)[===== №5] // ready Квадратная рамка со стороной $a eq 70 "см"$ помещена в магнитное поле так, что нормаль к рамке составляет угол $alpha eq 45 degree$ с направлением магнитного поля. Индукция магнитного поля меняется по закону $B eq B_0 cos omega t$, где $B_0 eq 0.2 "Тл", omega eq 6 " с"^(-1)$. Рассчитать ЭДС индукции, возникающей в рамке в момент времени $t eq 3 "с"$. @@ -665,52 +563,61 @@ $ ) ] -ЭДС индукции определяется законом Фарадея: +По формуле магнитного потока через плоскость: $ -cal(E) eq -frac(d Phi, d t) +Phi eq B S cos alpha $ -где $Phi$ - магнитный поток через рамку: +Площадь рамки: $ -Phi eq bold(B) dot bold(S) eq B S cos alpha +S eq a^2 $ -Площадь рамки $S eq a^2$, $alpha$ - угол между вектором магнитной индукции и нормалью к рамке. - -Магнитный поток через рамку: +Так как $B eq B_0 cos(omega t)$: $ -Phi(t) eq B(t) S cos alpha eq B_0 cos (omega t) dot a^2 cos alpha +Phi eq B_0 a^2 cos beta cos (omega t) $ -$ -cal(E) eq -frac(d Phi, d t) eq -frac(d, d t) [B_0 a^2 cos alpha cos(omega t)] eq -B_0 a^2 cos alpha frac(d, d t) cos(omega t) \ -eq -B_0 a^2 cos alpha dot (-omega sin (omega t)) eq B_0 a^2 omega cos alpha sin(omega t) -$ - -Подставляем числа: +По закону Фарадея: $ -cal(E) eq 0.2 dot (0.7)^2 dot 6 dot cos 45 degree dot sin(6 dot 3) approx -0.31 "В" +cal(E) eq -Phi'(t) eq B_0 a^2 omega cos beta sin omega t +$ + +Подставив числа из условия, получим: + +$ +cal(E) eq 0.2 dot 0.7^2 dot 6 dot cos(45 degree) dot sin (6 dot 3) approx -0.31 "B". $ *Ответ*: $epsilon eq -0.31 "В"$. -#align(center)[===== №6] +#align(center)[===== №6] // ready Плотность витков в катушке $n eq 25 " см"^(-1)$. Рассчитать объёмную плотность энергии магнитного поля в катушке при токе $I eq 2 "А"$. -*Решение*: В системе СИ: $n eq 25 " см"^(-1) eq 2500 " м"^(-1)$. +*Решение*: + +#align(center)[ + #figure( + image("assets/12.png"), + supplement: [Рис.], + caption: [Катушка с током и однородным магнитным полем внутри.] + ) +] + +В системе СИ: $n eq 25 " см"^(-1) eq 2500 " м"^(-1)$. По закону Ампера: $ -integral.cont bold(B) dot d bold(l) eq mu_0 I_"внутри" +integral.cont arrow(B) dot d arrow(l) eq mu_0 I_"внутри" $ -Возьмем прямоугольный контур. Одна сторона внутри катушки длиной $l_"внутри"$, другая снаружи. Магнитное поле внутри $bold(B) dot d bold(l) eq B l_"внутри"$. Ток, охваченный контуром: $I_"внутри" eq I dot N_"охваченных витков" eq I n l_"внутри"$. Подставив в закон Ампера, получим: +Возьмем прямоугольный контур. Одна сторона внутри катушки длиной $l_"внутри"$, другая снаружи. Магнитное поле внутри $arrow(B) dot d arrow(l) eq B l_"внутри"$. Ток, охваченный контуром: $I_"внутри" eq I dot N_"охваченных витков" eq I n l_"внутри"$. Подставив в закон Ампера, получим: $ B l_"внутри" eq mu_0 (n I l_"внутри") arrow.double B eq mu_0 n I.