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zz^^P3XIo+OYXl0e{~pK>Fj&OJQHYiP%pHwIqt>oA z21UfJ9gIPdF>4nV9*X$pDj^s&k+_Z@28RT|Z@QIL!lKuE06gDyo`Qp?{Kvq*EQ7^h zkns4;aJhlSqL9*Xt8h5jQ~nXW+=;_e(MW)=A|ztSNQke(;qdFNfFr{%m)FqYF*rDI zSC7Tx;QDN9ad;e@31B*S{(sDmCy>{f8c)Iz*RDIBj3ci#D*;7Xw|8yZ{C@54ORvBrZxFHlFHi$KApM{F6eA@i8Swvg2`b4f|Ge<0*!fX{{G6PbiA30$m?b4O H4Yd9TNFh0t diff --git a/course2/sem3/theory/qa.typ b/course2/sem3/theory/qa.typ index 64b4af4..f6ffe40 100644 --- a/course2/sem3/theory/qa.typ +++ b/course2/sem3/theory/qa.typ @@ -65,52 +65,130 @@ $ #align(center)[#image("assets/1.svg")] -Если ввести вектор ж +Если ввести вектор элеметнарной площадки $d arrow(S) = hat(n) d S$, то поток можно представить в форме: $d Phi = arrow(E) d arrow(S) = E_n d S$, где $E_n$ - проекция вектора $arrow(E)$ на нормаль $arrow(n)$. Для отдельной площадки $arrow(n)$ определено неоднозначно (2 варианта), но если $d S$ принадлежит замкнутой поверхности, то, как правило, вектор нормали $arrow(n)$ направляют наружу объема, охватываемого поверхностью. Полный поток, по его смыслу, равен + +$ +Phi = integral_S arrow(E) d arrow(S). +$ #line(length: 100%) *Q*: _*7*. Сформулируйте теорему Гаусса в интегральной форме._ -*A*: +*A*: Теорема Гаусса: + +$ +integral.surf_S arrow(E) d arrow(S) = frac(q_"внутр", epsilon_0) +$ + +*Поток вектора $arrow(E)$ сквозь замкнутую поверхность равен, с точностью до множителя $frac(1, epsilon_0)$, алгебраической сумме зарядов $q_"внутр"$, находящихся внутри этой поверхности.* + +Если заряд распределен непрерывно, то при вычислении $q_"внутр"$ сумма заменяется интегралом по объему, поверхности или линии, которые попали внутрь поверхности, соответственно: $integral rho d V, space integral sigma d S, space integral lambda d l$. #line(length: 100%) *Q*: _*8*. Сформулируйте теорему Гаусса в дифференциальной форме._ -*A*: +*A*: Пренебрежем дискретностью заряда, считая его распределенным в пространстве с плотностью $rho = rho(arrow(r))$. В этом случае теорема Гаусса имеет следующий вид: + +$ +integral.surf_S arrow(E) d arrow(S) = 1/(epsilon_0) integral_V rho d V. +$ + +Интеграл по поверхности, можно с помощью математической теоремы Остроградского-Гаусса преобразовать к форме + +$ +integral.cont arrow(E) d arrow(S) = integral_V "div" arrow(E) d V. +$ + +Так как это справедливо для любых по форме и величине объемов, то из сравнения интегралов, представленных выше, следует + +$ +"div" arrow(E) = frac(rho, epsilon_0). +$ #line(length: 100%) *Q*: _*9*. В чем заключается физический смысл $d i v arrow(E)$?_ -*A*: +*A*: Дивергенция $"div" arrow(E)$ является скалярной величиной. Формула вычисления $"div" arrow(E)$ в разных системах координат выглядит по-разному. В произвольной системе координат $"div" arrow(E)$ (это справедливо для любого векторного поля) определяется как + +$ +"div" arrow(E) = lim_(V arrow 0) frac(1, V) integral.surf arrow(E) d arrow(S) +$ + +В декартовых координатах: + +$ +"div" arrow(E) = frac(diff E_x, diff x) + frac(diff E_y, diff y) + frac(diff E_z, diff z) +$ + +Если использовать векторный дифференциальный оператор $arrow(gradient)$ ("набла"), который имеет вид $gradient = hat(i) frac(diff, diff x) + hat(j) frac(diff, diff y) + hat(k) frac(diff, diff z)$, то $"div" arrow(E)$ можно представить в виде скалярного "произведения": $"div" arrow(E) = arrow(gradient) dot arrow(E)$. #line(length: 100%) *Q*: _*10*. Дайте определение циркуляции вектора $arrow(E)$._ -*A*: +*A*: Из механики известно, что любое стационарное поле центральных сил является консервативным, т.е. работа сил этого поля не зависит от формы пути, а зависит только от положения его начальной и конечной точки. Именно таким свойством обладает электростатическое поле - поле, образованное системой неподвижных зарядов. Если в качестве пробного заряда, переносимого из точки 1 заданного поля $arrow(E)$ в точку 2, взять единичный положительный заряд, то элементарная работа сил поля на перемещении $d arrow(l)$ равна $arrow(E) d arrow(l)$, а вся работа сил поля на этом пути: $integral_1^2 arrow(E) d arrow(l)$. + +Этот интеграл берется по некоторой линии (пути), поэтому его называют линейным. Интеграл по замкнутому пути называют циркуляцией вектора $arrow(E)$ и обозначают $integral.cont$. #line(length: 100%) *Q*: _*11*. Сформулируйте теорему о циркуляции вектора $arrow(E)$?_ -*A*: +*A*: Циркуляция вектора $arrow(E)$ в любом электростатическом поле равна нулю, т.е. + +$ +integral.cont arrow(E) d arrow(l) = 0 +$ + +Это утверждение и называют теоремой о циркуляции вектора $arrow(E)$. #line(length: 100%) *Q*: _*12*. Дайте определение потенциального поля._ -*A*: +*A*: Поле, обладающее этим свойством, называют потенциальным. Значит, любое электростатическо поле является потенциальным. #line(length: 100%) *Q*: _*13*. Докажите, что линии электростатического поля $arrow(E)$ не могут быть замкнутыми._ -*A*: +*A*: Теорема о циркуляции вектора $arrow(E)$ позволяет сделать ряд важных выводов, практически не прибегая к расчетам. + +*Пример 1.* Линии электростатического поля $arrow(E)$ не могут быть замкнутыми. + +Если это не так и какая-то линия вектора $arrow(E)$ замкнута, то, взяв циркуляцию вектора $arrow(E)$ вдоль этой линии, мы сразу же придем к противоречию с теоремой, т.к. вдоль силовой линии $arrow(E) d arrow(r) gt 0$. Значит, действительно, в электростатическом поле замкнутых линий вектора $arrow(E)$ не существует: линии начинаются на положительных зарядах и заканчиваются на отрицательных (или уходят в бесконечность). #line(length: 100%) *Q*: _*14*. По какой формуле можно определить потенциальную энергию системы точечных зарядов?_ -*A*: +*A*: Электростатическое поле является потенциальным, т.е. работа его сил по перемещению заряда не зависит от форму пути. Работа сил поля при перемещении точечного заряда $q$ из точки 1 в точку 2 равна убыли его потенциальной энергии: + +$ +A = W_1 - W_2. +$ + +Потенциальная энергия заряда $q$ в системе зарядов $q_i$: + +$ +W = k sum_i frac(q dot q_i, r_i) +$ + +где $r_i$ - расстояние между $q$ и $q_i$. + +Полная потенциальная энергия системы точечных зарядов: + +$ +W = frac(k, 2) sum_i sum_(j eq.not i) frac(q_i dot q_j, r_(i j)), +$ + +где $r_(i j)$ - расстояние между зарядами $q_i$ и $q_j$. + #line(length: 100%) *Q*: _*15*. Дайте определение потенциалов._ -*A*: +*A*: Энергетическая характеристика электростатического поля - потенциал: + +$ +phi(arrow(r)) = frac(W(arrow(r)), q). +$ + #line(length: 100%) *Q*: _*16*. Чему равен потенциал системы точечных зарядов?_