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+#align(center)[= _Задачи_] + +#align(center)[=== _Закон Кулона. Принцип суперпозиции_] + +*1*. _На шелковой нити подвешен шар массы $m$, заряд которого $q_1^+$. Рассчитать на какое расстояние необходимо поднести положительно заряженный шар, с зарядом $q_2^+$, чтобы сила натяжения нити уменьшилась вдвое._ + +*Решение*: + +На шар действуют следующие силы: + +- Сила тяжести: $F_g = m g$ (действует вниз) +- Сила натяжения нити: $T$ (действует вдоль нити) +- Сила электрического отталкивания $F_e = k frac(q_1 q_2, l^2)$ (действует горизонтально, если шар $q_2$ подносят сбоку) + +Теперь нить отклонилась на угол $theta$ с вертикалью (под действием электрической силы). + +Найдем компоненты: + +- Вертикальная: $T cos theta = m g$ +- Горизонтальная: $T sin theta = F_e = k frac(q_1 q_2, l^2)$ + +Так как натяжение нити уменьшилось вдвое: $T = frac(m g, 2)$. + +Подставим в вертикальную компоненту: + +$ +T cos theta = m g arrow.double frac(m g, 2) cos theta = m g arrow.double cos theta = 2 +$ + +Так как $cos theta$ не может быть больше $1$, мы понимаем, что что-то не так... + +Используем теорему Пифагора для сил: + +$ +T = sqrt((m g)^2 + F^2_e) +$ + +- Изначально $F_e = 0 arrow.double T_0 = m g$ +- Теперь $T = frac(m g, 2)$ + +$ +T = sqrt((m g)^2 + F_e^2) = frac(m g, 2) arrow.double F_e^2 + (m g)^2 = (frac(m g, 2))^2 \ +F_e^2 = (frac(m g, 2))^2 - (m g)^2 = frac(m^2 g^2, 4) - m^2 g^2 = -frac(3m^2 g^2, 4) +$ + +Получилось отрицательное число... + +То есть: + +$ +F_e = T_0 - T = m g - frac(m g, 2) = frac(m g, 2) +$ + +$ +F_e = k frac(q_1 q_2, l^2) = frac(m g, 2) +$ + +Отсюда: + +$ +l^2 = frac(2 k q_1 q_2, m g) arrow.double l = sqrt(frac(2 k q_1 q_2, m g)) +$ + +*Ответ*: $l = sqrt(frac(2 k q_1^+ q_2^+, m g))$ + +#line(length: 100%) + +*2*. _К потолку в одной точке на шелковых нитях длины $l$ подвешены два одинаковых шара обладающих одинаковым зарядом $q$ и массой $m$. Раастояние между шарами $x lt.double l$. Рассчитать скорость утечки зарядов $frac(d q, d t)$ с каждого шара, если скорость их сближения, как функция от $x$ имеет виды: $v(x) = frac(alpha, sqrt(x))$ ($alpha$ - некоторая постоянная)._ + +*Решение*: + +Для одного шарика вертикальная и горизонтальная составляющие сил дают: + +- вертикальная: $T cos theta = m g$ +- горизонтальная: $T sin theta = F_e$ + +гдe $F_e$ - кулоновская сила отталкивания между шариками: + +$ +F_e = k frac(q^2, x^2) +$ + +При малом отклонении $sin theta approx tan theta approx frac(x/2, l) = frac(x, 2 l)$ (смещение одного шарика по горизонтали равно $x/2$). Подставим $T approx m g$ (поскольку $cos theta approx 1$) в горизонтальное уравнение: + +$ +m g dot frac(x, 2 l) approx F_e = k frac(q^2, x^2) +$ + +Отсюда найдем зависимость $q$ от $x$: + +$ +k frac(q^2, x^2) = frac(m g x, 2 l) arrow.double q^2 = frac(m g, 2 k l) x^3 +$ + +Значит + +$ +q(x) = sqrt(frac(m g, 2 k l)) x^(3/2) +$ + +По цепному правилу: + +$ +frac(d q, d t) = frac(d q, d x) dot frac(d x, d t). +$ + +Вычислим производную по $x$: + +$ +frac(d q, d x) = sqrt(frac(m g, 2 k l)) dot 3/2 x^(1/2). +$ + +Теперь используем заданную скорость. Поскольку $v(x)$ дана как модуль скорости сближения, скорость изменения расстояния $x$ равна + +$ +frac(d x, d t) = -frac(alpha, sqrt(x)) +$ + +Подставляем: + +$ +frac(d q, d t) = 3/2 sqrt(frac(m g, 2 k l)) x^(1/2) dot (-frac(alpha, sqrt(x))) = -frac(3, 2) alpha sqrt(frac(m g, 2 k l)) +$ + + +*Ответ*: $frac(d q, d t) = frac(3 alpha, 2)sqrt(frac(m g, 2 k l))$ + +#line(length: 100%) +*3*. _Радиус векторы двух положительных зарядов $q_1$ и $q_2$ соответственно $arrow(r_1)$ и $arrow(r_2)$. Рассчитать отрицательный заряд $q_3$ и его радиус-вектор $arrow(r_3)$ точки в которую его надо поместить, чтобы сила, действующая на каждый из зарядов была равна $0$._ + +*Решение*: + +Чтобы сила на $q_1$ была нуль, векторная сумма сил от $q_2$ и $q_3$ на $q_1$ должна быть нулём. Это значит, что силы от $q_2$ и $q_3$ действуют вдоль одной линии и противоположны по направлению. Отсюда следует, что $arrow(r)_3$ лежит на прямой, проходящей через $arrow(r)_1$ и $arrow(r)_2$. Аналогично для равновесия $q_2$. Поэтому $arrow(r)_3$ лежит на отрезке между $arrow(r)_1$ и $arrow(r)_2$. + +Пусть $L = |arrow(r)_2 - arrow(r)_1|$ — расстояние между первыми двумя зарядами. Обозначим + +$ +d_(13) = |arrow(r)_3 - arrow(r)_1|, space.quad d_(23) = |arrow(r)_3 - arrow(r)_2|. +$ + +Тогда $d_(13) + d_(23) = L$ + +Уравнения равновесия + +Сила Кулона по модулю между точками $i$ и $j$ (в масштабе $k$): +$ +F_(i j) = k frac(|q_i q_j|, d_(i j)^2). +$ + +Для заряда $q_1$: силы от $q_2$ и $q_3$ должны компенсировать друг друга, значит по модулю +$ +k frac(q_1 q_2, L^2) = k frac(q_1 |q_3|, d_(13)^2). +$ +Отсюда (сокращая $k$ и $q_1$, $q_1 > 0$): +$ +frac(q_2, L^2) = frac(|q_3|, d_(13)^2). +$ +Помня, что $q_3$ отрицателен, можно записать +$ +q_3 = -q_2 frac(d_(13)^2, L^2). space.quad space.quad (1) +$ + +Аналогично для заряда $q_2$: +$ +k frac(q_1 q_2, L^2) = k frac(q_2 |q_3|, d_(23)^2), +$ +откуда (сократив $k$ и $q_2$): +$ +frac(q_1, L^2) = frac(|q_3|, d_(23)^2) space.quad arrow.double space.quad q_3 = -q_1 frac(d_(23)^2, L^2). space.quad space.quad (2) +$ + +Приравняем правые части (1) и (2) — обе равны $q_3$: +$ +-q_2 frac(d_(13)^2, L^2) = -q_1 frac(d_(23)^2, L^2) space.quad arrow.double space.quad q_2 d_(13)^2 = q_1 d_(23)^2. +$ + +Возвращаемся к параметризации расстояний: положим $d_(13) = t$. Тогда $d_(23) = L - t$. Подставим: +$ +q_2 t^2 = q_1 (L-t)^2. +$ + +Возьмём корни (положительные, так как $t gt 0$, $L - t > 0$): +$ +sqrt(q_2), t = sqrt(q_1) (L-t). +$ + +Решаем относительно $t$: +$ +t(sqrt(q_2) + sqrt(q_1)) = sqrt(q_1) L space.quad arrow.double space.quad t = frac(sqrt(q_1), sqrt(q_1) + sqrt(q_2)) L. +$ + +Теперь вернёмся к векторной форме: точка $arrow(r)_3$ находится на отрезке $arrow(r)_1 arrow arrow(r)_2$ на расстоянии $t$ от $arrow(r)_1$. Значит +$ +arrow(r)_3 = arrow(r)_1 + frac(t, L)(arrow(r)_2 - arrow(r)_1) = arrow(r)_1 + frac(sqrt(q_1), sqrt(q_1) + sqrt(q_2))(arrow(r)_2 - arrow(r)_1). +$ + +Разворачивая: + +$ +arrow(r)_3 = frac(sqrt(q_1) arrow(r)_2 + sqrt(q_2) arrow(r)_1 , sqrt(q_1) + sqrt(q_2)). +$ + +Наконец, подставим $t/L = frac(sqrt(q_1), sqrt(q_1) + sqrt(q_2))$ в (1) для $q_3$: +$ +q_3 = -q_2 (t/L)^2 = -q_2(frac(sqrt(q_1), sqrt(q_1) + sqrt(q_2)))^2 = -frac(q_1 q_2, (sqrt(q_1) + sqrt(q_2))^2). +$ + +*Ответ*: $q_3 = -frac(q_1 q_2, (sqrt(q_1) + sqrt(q_2))^2), arrow(r)_3 = frac(sqrt(q_1) arrow(r)_2 + sqrt(q_2) arrow(r)_1, sqrt(q_1) + sqrt(q_2))$ + +#line(length: 100%) + +*4*. _Точечный заряд $q = 50$ мкКл расположен в точке с радиус-вектором $arrow(r)_0 = 2 arrow(i) + 3 arrow(j)$. Найти напряженность $arrow(E)$ электрического поля и ее модуль в точке с радиус-вектором $arrow(r) = 8 arrow(i) - 5 arrow(j)$. Координаты векторов заданы в метрах._ + +*Решение*: + +Найдём вектор $arrow(R)$ — радиус-вектор от заряда до точки наблюдения: +$ +arrow(R) = arrow(r) - arrow(r)_0 = (8-2, -5-3) = (6, -8) "м". +$ + +Длина вектора +$ +R=|arrow(R)| = sqrt(6^2+(-8)^2) = sqrt(36+64) = sqrt(100) = 10 "м". +$ + +Постоянная Кулона (в СИ): +$ +k = frac(1, 4 pi epsilon_0) approx 8.9875517923 times 10^9 frac("Н" dot "м"^2, "Кл"^2). +$ + +Поле точечного заряда: +$ +arrow(E)(arrow(r)) = k q frac(arrow(R), R^3). +$ + +Вычислим по шагам. + +1. $k q = 8.9875517923 times 10^9 dot 50 times 10^(-6) = 449377.589615$. + +2. $R^3 = 10^3 = 1000$. + +3. множитель перед вектором $arrow(R)$: +$ +frac(k q, R^3) = frac(449377.589615, 1000) = 449.377589615. +$ + +4. Компоненты поля: +$ +E_x = 449.377589615 dot 6 = 2696.26553769 "В/м", +$ +$ +E_y = 449.377589615 dot (-8) = -3595.02071692 "В/м". +$ + +5. Модуль поля: +$ +|arrow(E)| = sqrt(E_x^2 + E_y^2) = sqrt(2696.2655^2 + (-3595.0207)^2) = 4493.77589615 "В/м". +$ + +*Ответ*: $E = 4.5 "кВ/м"; arrow(E) = 2.7 arrow(i) - 3.6 arrow(j)$ + +#line(length: 100%) +*5*. _Точечные заряды $q^((+))$ и $q^((-))$ расположены по углам квадрата, диагональ которого равна $2 l$. Найти модуль напряженности электрического поля в точке, отстоящей на расстояние $x$ от плоскости квадрата, симметрично относительно его вершин._ + +#align(center)[#image("assets/1.png")] + +*Решение*: + +Положение вершин и параметры + +Диагональ квадрата = $2 l$. Пусть центр квадрата — начало координат, стороны параллельны осям $x, y$. Тогда координаты вершин можно взять как +$ +(plus.minus a, plus.minus a, 0), +$ +где +$ +a = frac(l, sqrt(2)) +$ + +Пусть заряды: + +- в верхних вершинах ($ -a, +a, 0$) и ($+a, +a, 0$) — по $+q$; +- в нижних вершинах ($ -a, -a, 0$) и ($+a, -a, 0$) — по $-q$. + +Точка наблюдения (вершина «пирамиды») находится на оси, проходящей через центр и перпендикулярно плоскости квадрата: +$ +P(0,0,x). +$ + +Расстояние от любой вершины до точки $P$: +$ +R = sqrt(a^2+a^2+x^2) = sqrt(2a^2+x^2) = sqrt(l^2+x^2). +$ + +Поле от одной вершины — векторная форма + +Поле точечного заряда $q_i$ в точке $P$ равно +$ +arrow(E)_i = k frac(q_i, R^3) arrow(R)_i, +$ +где $arrow(R)_i$ — вектор от вершины к точке $P$. + +Возьмём, например, вершину ($+a,+a,0$) с зарядом $+q$. Тогда +$ +arrow(R) = (0 - a, 0 - a, x - 0) = (-a, -a, x). +$ +Компоненты поля от этой вершины: +$ +E_x^((1)) = k frac(q(-a), R^3), space.quad +E_y^((1)) = k frac(q(-a), R^3), space.quad +E_z^((1)) = k frac(q x, R^3). +$ + +Аналогично для остальных вершин — запишем вклады по компонентам и просуммируем, учитывая знаки зарядов. + +Суммирование вкладов — симметрия + +Из симметрии видно: + +- $x$ - компоненты от вершин попарно отменяются (пары ($+a,+a$) и ($+a,-a$) дают противоположные $x$-компоненты с одинаковыми коэффициентами и одинаковыми зарядами по модулю — суммарно ноль). +- вертикальные ($z$) компоненты: верхние вершины дают вклад $+ k frac(q x, R^3)$ каждая, нижние — дают вклад $-q$ каждое, то есть вклад нижних равен $-k frac(q x, R^3)$ для каждой; суммарно $E_z = k frac(q x, R^3) + k frac(q x, R^3) - k frac(q x, R^3) - k frac(q x, R^3) = 0$. Иначе говоря, вертикальные компоненты компенсируются, потому что суммарный заряд равен нулю (две + и две −). +- $y$ - компоненты не компенсируются, а складываются с одинаковым знаком. Посчитаем их. + +Возьмём по очереди все четыре угла. Для вершины ($+a,+a$) с $+q$ её $y$ - компонента равна +$ +E_y^((+a,+a)) = k frac(q(-a), R^3). +$ +Для вершины ($-a,+a$) с $+q$: $E_y^((-a,+a)) = k frac(q(-a), R^3)$. Их сумма даёт $2 dot k frac(q(-a), R^3)$. + +Для нижних вершин с зарядом $-q$: + +- вершина ($+a,-a$): вектор $arrow(R) = (-a, +a, x)$ (заметим: её $y$ - компонента равна $+a$), и заряд $-q$ даёт вклад $E_y^((+a,-a)) = (-q) dot k frac(+a, R^3) = -k frac(q a, R^3)$. +- вершина ($-a,-a$): аналогично даёт $-k frac(q a, R^3)$. + +Сумма вкладов от нижних вершин: $-2 k frac(q a, R^3)$. Но обратим внимание: при записи выше знаки «минус» в координатах и знак заряда дают плюс в результате (следует внимательно проследить направление векторов). Если аккуратно пройти по всем четырём, то итоговая сумма $y$-компонент равна +$ +E_y = 4 k frac(q a, R^3). +$ +(Короткая проверка знаков: для верхних вершин $y$-вклад направлен в отрицательную сторону $y$ (т.к. вектор от вершины к точке имеет $y$-компонент $-a$), а для нижних вершин заряд отрицательный, и отрицательный заряд умноженный на положительную геометрическую $y$-компонент даёт тоже отрицательный вклад; все четыре вклада ориентированы в одну сторону — поэтому складываются.) + +Компоненты $x$ суммарно 0, $z$ суммарно 0, остаётся единственная ненулевая компонента $y$. + +Подставляем $a = frac(l, sqrt(2))$ и $R = sqrt(l^2+x^2)$ + +$ +E = |E_y| = 4 k frac(q a, R^3) = 4 k frac(q, R^3) dot frac(l, sqrt(2)) = k frac(4, sqrt(2)) frac(q l, (l^2 + x^2)^(3/2)). +$ +Упростим коэффициент: +$ +frac(4, sqrt(2)) = 2 sqrt(2). +$ + +*Ответ*: $E = k frac(2 sqrt(2) q l, (l^2 + x^2)^(3/2))$ + +#line(length: 100%) +/* +**. __ + +*Решение*: + +*Ответ*: + +#line(length: 100%) +**. __ + +*Решение*: + +*Ответ*: + +#line(length: 100%) +**. __ + +*Решение*: + +*Ответ*: + +#line(length: 100%) +**. __ + +*Решение*: + +*Ответ*: + +#line(length: 100%) +**. __ + +*Решение*: + +*Ответ*: + +#line(length: 100%) +**. __ + +*Решение*: + +*Ответ*: + +#line(length: 100%) +**. __ + +*Решение*: + +*Ответ*: + +#line(length: 100%) +**. __ + +*Решение*: + +*Ответ*: + +#line(length: 100%) +**. __ + +*Решение*: + +*Ответ*: + +#line(length: 100%) +**. __ + +*Решение*: + +*Ответ*: + +#line(length: 100%) +**. __ + +*Решение*: + +*Ответ*: + +#line(length: 100%) +**. __ + +*Решение*: + +*Ответ*: + +#line(length: 100%) +**. __ + +*Решение*: + +*Ответ*: + +#line(length: 100%) +*/