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701ddb7..048d3e6 100644 --- a/course2/sem3/labs/lab3.01/report.typ +++ b/course2/sem3/labs/lab3.01/report.typ @@ -1,3 +1,4 @@ +#import "@preview/tablex:0.0.9": tablex, colspanx, rowspanx #set text(size: 1.3em) #set page(footer: context { if counter(page).get().first() > 1 [ @@ -34,45 +35,177 @@ === 1. Цель работы. +Построение сечений эквипотенциальных поверхностей и силовых линий электростатического поля на основе экспериментального моделирования распределения потенциала в слабопроводящей среде. + === 2. Задачи, решаемые при выполнении работы. +- Экспериментально построить сечения эквипотенциальных поверхностей и силовых линий электростатического поля для плоского конденсатора и поскости с дополнительным проводящим кольцом. + +- Измерить распределение потенциала в слабопроводящей среде и по данным построить эквипотенциальные линии. + +- По свойству ортогональности эквипотенциалей и линий напряжённости построить картину силовых линий и указать их направление. + +- Рассчитать величины напряжённости поля в центре ванны и вблизи электрода. + +- По данным измерений оценить поверхностную плотность зарядов на электродах. + +- Для эксперимента с кольцом определить области минимальной и максимальной напряжённости и оценить $E_min$ и $E_max$. + +- Построить и сравнить графики $phi(X)$ для горизонтали $Y = 10 "см"$ для двух экспериментов. + === 3. Объект исследования. +Электростатическое поле между двумя плоскими электродами в однородной слабопроводящей среде и изменение распределения потенциала при установке в ванну проводящего кольца. + === 4. Метод экспериментального исследования. +Моделирование электростатического поля в слабопроводящей среде с использованием двух плоских электродов, подключённых к генератору переменного напряжения. Потенциал внутри ванны измеряют зондом, подключённым к вольтметру. По набору точечных измерений потенциала строят эквипотенциальные линии, затем по ортогональности строят силовые линии. + === 5. Рабочие формулы и исходные данные. +#align(center)[ + #table(columns: 2, inset: 15pt)[*Формула*][*Пояснения*][$arrow(E)(arrow(r)) = frac(arrow(F)(arrow(r)), q)$][Вектор напряженности электрического поля. $arrow(F)$ - сила, действующая на неподвижный заряд $q$, помещенный в данную точку. Заряд $q$ - пробный. $arrow(r)$ - радиус-вектор точки.][$phi(arrow(r)) = frac(W_"П" (arrow(r)), q)$][Потенициал в данном точке поля. $W_"П"$ - потенциальная энергия заряда $q$, помещенного в данную точку.][$A_(12) = q(phi_1 - phi_2)$.][Работа сил электростатического поля над зарядом $q$ при его перемещении из точки с потенциалом $phi_1$ в точку с потенциалом $phi_2$.][$arrow(E) = -"grad" phi eq.triple -arrow(gradient) phi \ phi_2 - phi_1 = -integral_1^2 arrow(E) space d arrow(l)$][Связь напряженности и потенциала электростатического поля.][$arrow(gradient) phi = hat(e)_x frac(diff phi, diff x) + hat(e)_y frac(diff phi, diff y) + hat(e)_z frac(diff phi, diff z)$][Вектор градиента потенциала. $x, y, z$ - декартовы координаты. $hat(e)_x, hat(e)_y, hat(e)_z$ - единичные вектора положительных направлений (орты) координатных осей $O x, O y, O z$][$angle.l E_(12) angle.r approx.eq frac(phi_1 - phi_2, l_(12))$][Средняя напряженность между точками на одной силовой линии с потенциалами $phi_1$ и $phi_2$, где $l_(12)$ - длина участка силовой линии между точками.][$arrow(j) = sigma arrow(E)$][Закон Ома в дифференциальной форме, где $arrow(j)$ - вектор плотности тока в проводящей среде, $sigma$ - удельная электропроводность среды.][$arrow(gradient) dot arrow(j) eq.triple "div" arrow(j) = frac(diff j_x, diff x) + frac(diff j_y, diff y) + frac(diff j_z, diff z) = -frac(diff rho, diff t)$][Плотность тока в любой проводащей среде удовлетворяет уравнению неразрывности. $rho$ - объемная плотность заряда. Для стационарного тока $rho = "const", space frac(diff rho, diff t) = 0$ и в этом случае $arrow(gradient) dot arrow(j) = 0$.][$sigma(arrow(gradient) dot arrow(E)) = 0 arrow.double arrow(gradient) dot arrow(E) = 0$][Следует из однородности $sigma$.][$"rot" arrow(j) eq.triple arrow(gradient) times arrow(j) = 0$][Получено путем применения к $arrow(j) = sigma arrow(E)$ операцию нахождения ротора и учитывая безвихревой характер постоянного тока.][$arrow(gradient) times arrow(E) = 0$][Подставили $arrow(j) = sigma arrow(E)$ в $"rot" arrow(j) eq.triple arrow(gradient) times arrow(j) = 0$] +] + +Исходные данные: + +- Межэлектродная установленная амплитуда напряжения $U = 14 "В"$. + +- Частота переменного напряжения генератора $f = 400 plus.minus 50 "Гц"$ + +- Диапазон вольтметра $0 div 20 "В"$. + +- Координатная сетка на миллиметровой бумаге шаги по $Y$ используются: $2, 6, 10, 14, 18 "см"$; при конфигурации с кольцом рекомендуется уменьшить шаг потенциала и шаг $Y$ рядом с кольцом до $1–2 "см"$. + +- Шаг изменения потенциала для первого эксперимента $delta phi = 2 "В"$ +- Для эксперимента с кольцом $Delta phi = 1 "В"$ + +- Погрешности измерения координат $Delta X = plus.minus 1 "мм", Delta Y = plus.minus 0.5 "мм"$. + === 6. Измерительные приборы -#table(columns: 5)[№ п/п][Наименование][Тип прибора][Используемый диапазон][Погрешность прибора][1][][][][][2][][][][][3][][][][] +#table(columns: 5)[№ п/п][Наименование][Тип прибора][Используемый диапазон][Погрешность прибора][1][Вольтметр][AB1][0-20 В][$plus.minus 0.5 %$][2][Амперметр][AB1][0-5 А][$plus.minus 1.0% $][3][Резистор][ГН1][0-10 к$Omega$][$plus.minus 5%$] === 7. Схема установки (перечень схем, которые составляют Приложение 1). -=== 8. Результаты прямых измерений и их обработки (таблицы, примеры расчетов). +=== 8. Результаты прямых измерений. -=== 9. Расчет результатов косвенных измерений (таблицы, примеры расчетов). - -=== 10. Расчет погрешностей измерений (для прямых и косвенных измерений). - -=== 11. Графики (перечень графиков, которые составляют Приложение 2). - -=== 12. Окончательные результаты. - -=== 13. Выводы и анализ результатов работы. - -=== 14. Дополнительные задания. - -=== 15. Выполнение дополнительных заданий. - -=== 16. Замечания преподавателя (исправления, вызванные замечаниями преподавалтеля, также помещают в этот пункт). +Без диска. #align(center)[ - #table(columns: 2, stroke: none, fill: none)[ - *Примечание:* - ][ - 1. Пункты 1-6, 8-13 Протокола-отчета *обязательны* для заполнения. - 2. Необходимые исправления выполняют непосредственно в протоколе-отчете. - 3. При ручном построении графиков рекомендуется использовать миллиметровую бумагу. - 4. Приложения 1 и 2 вкладывают в бланк протокола-отчета. - ] + #tablex( + columns: 9, + [*$phi_1$*], [*$X_1$*], [*$Y_1$*], [*$phi_2$*], [*$X_2$*], [*$Y_2$*], [*$phi_3$*], [*$X_3$*], [*$Y_3$*], + + rowspanx(5)[*$1.89$*], $2.0$, $2$, rowspanx(5)[*$3.89$*], [$6.8$], [$2$], rowspanx(5)[*$5.89$*], [$11.8$], [$2$], + + (), [$2.5$], [$6$], (), [$6.8$], [$6$], (), [$12.2$], [$6$], + + (), [$2.8$], [$10$], (), [$7.0$], [$10$], (), [$12.5$], [$10$], + + (), [$2.7$], [$14$], (), [$6.9$], [$14$], (), [$12.8$], [$14$], + + (), [$2.0$], [$18$], (), [$7.2$], [$18$], (), [$12.6$], [$18$] + + ) ] + +#align(center)[ + #tablex( + columns: 9, + [*$phi_4$*], [*$X_4$*], [*$Y_4$*], [*$phi_5$*], [*$X_5$*], [*$Y_5$*], [*$phi_6$*], [*$X_6$*], [*$Y_6$*], + + rowspanx(5)[*$7.89$*], [$16.7$], [$2$], rowspanx(5)[*$9.89$*], [$21.3$], [$2$], rowspanx(5)[*$11.89$*], [$26.1$], [$2$], + + (), [$16.8$], [$6$], (), [$21.3$], [$6$], (), [$25.7$], [$6$], + + (), [$16.5$], [$10$], (), [$21.3$], [$10$], (), [$25.6$], [$10$], + + (), [$16.3$], [$14$], (), [$21.1$], [$14$], (), [$25.7$], [$14$], + + (), [$16.3$], [$18$], (), [$21.0$], [$18$], (), [$26.0$], [$18$] + ) +] + +С диском. + +#align(center)[ + #tablex( + columns: 12, + [*$phi_1$*], [*$X_1$*], [*$Y_1$*], [*$phi_2$*], [*$X_2$*], [*$Y_2$*], [*$phi_3$*], [*$X_3$*], [*$Y_3$*], [*$phi_4$*], [*$X_4$*], [*$Y_4$*], + + rowspanx(9)[*$2.4$*], [$2.0$], [$2$], rowspanx(9)[*$3.4$*], [$4.1$], [$2$], rowspanx(9)[*$4.4$*], [$6.1$], [$2$], rowspanx(9)[*$5.4$*], [$8.4$], [$2$], + + (), [$2.1$], [$4$], (), [$4.0$], [$4$], (), [$5.9$], [$4$], (), [$7.8$], [$4$], + + (), [$2.6$], [$6$], (), [$4.1$], [$6$], (), [$5.9$], [$6$], (), [$7.4$], [$6$], + + (), [$2.8$], [$8$], (), [$4.0$], [$8$], (), [$5.5$], [$8$], (), [$7.2$], [$8$], + + (), [$3.0$], [$10$], (), [$4.2$], [$10$], (), [$5.8$], [$10$], (), [$7.2$], [$10$], + + (), [$2.8$], [$12$], (), [$4.2$], [$12$], (), [$5.7$], [$12$], (), [$7.7$], [$12$], + + (), [$3.1$], [$14$], (), [$4.6$], [$14$], (), [$6.3$], [$14$], (), [$8.2$], [$14$], + + (), [$2.7$], [$16$], (), [$4.5$], [$16$], (), [$7.0$], [$16$], (), [$-$], [$16$], + + (), [$2.8$], [$18$], (), [$4.8$], [$18$], (), [$7.5$], [$18$], (), [$9.8$], [$18$] + ) +] + +#pagebreak() + +#align(center)[ + #tablex( + columns: 12, + [*$phi_5$*], [*$X_5$*], [*$Y_5$*], [*$phi_6$*], [*$X_6$*], [*$Y_6$*], [*$phi_7$*], [*$X_7$*], [*$Y_7$*], [*$phi_8$*], [*$X_8$*], [*$Y_8$*], + + rowspanx(9)[*$6.4$*], [$11.2$], [$2$], rowspanx(9)[*$7.4$*], [$16.0$], [$2$], rowspanx(9)[*$8.4$*], [$19.8$], [$2$], rowspanx(9)[*$9.4$*], [$22.2$], [$2$], + + (), [$-$], [$4$], (), [$-$], [$4$], (), [$20.3$], [$4$], (), [$-$], [$4$], + + (), [$9.0$], [$6$], (), [$-$], [$6$], (), [$21.3$], [$6$], (), [$22.8$], [$6$], + + (), [$-$], [$8$], (), [$-$], [$8$], (), [$21.7$], [$8$], (), [$-$], [$8$], + + (), [$8.8$], [$10$], (), [$-$], [$10$], (), [$21.8$], [$10$], (), [$22.9$], [$10$], + + (), [$-$], [$12$], (), [$-$], [$12$], (), [$21.3$], [$12$], (), [$-$], [$12$], + + (), [$11.0$], [$14$], (), [$-$], [$14$], (), [$20.7$], [$14$], (), [$22.5$], [$14$], + + (), [$-$], [$16$], (), [$-$], [$16$], (), [$19.7$], [$16$], (), [$-$], [$16$], + + (), [$12.3$], [$18$], (), [$15.5$], [$18$], (), [$18.0$], [$18$], (), [$21.7$], [$18$] + ) +] + +#align(center)[ + #tablex( + columns: 6, + [*$phi_9$*], [*$X_9$*], [*$Y_9$*], [*$phi_10$*], [*$X_10$*], [*$Y_10$*], + + rowspanx(9)[*$10.4$*], [$24.5$], [$2$], rowspanx(9)[*$11.4$*], [$26.7$], [$2$], + + (), [$-$], [$4$], (), [$-$], [$4$], + + (), [$24.4$], [$6$], (), [$26.2$], [$6$], + + (), [$-$], [$8$], (), [$-$], [$8$], + + (), [$24.6$], [$10$], (), [$26.2$], [$10$], + + (), [$-$], [$12$], (), [$-$], [$12$], + + (), [$24.2$], [$14$], (), [$26.0$], [$14$], + + (), [$-$], [$16$], (), [$-$], [$16$], + + (), [$23.9$], [$18$], (), [$26.1$], [$18$], + ) +] + +=== 9. Построение эквипотенциальных линий. + +Сначала точки с миллиметровой бумаги были перенесены в компьютер при помощи программы в Приложении. diff --git a/course2/sem3/labs/lab3.01/scripts/1.py b/course2/sem3/labs/lab3.01/scripts/1.py new file mode 100644 index 0000000..383323a --- /dev/null +++ b/course2/sem3/labs/lab3.01/scripts/1.py @@ -0,0 +1,59 @@ +import matplotlib.pyplot as plt + + +def main() -> None: + + plt.figure(figsize=(8.27, 11.69)) + plt.xlim(0, 30) + plt.ylim(0, 20) + plt.gca().set_aspect("equal", adjustable="box") + plt.grid(True, linewidth=0.3) + plt.xticks(range(0, 31, 2)) + plt.yticks(range(0, 21, 2)) + + points = [ + (2, 2, "1.89"), + (2.5, 6, "1.89B"), + (2.8, 10, "1.89"), + (2.7, 14, "1.89"), + (2.0, 18, "1.89"), + (6.8, 2, "3.89"), + (6.8, 6, "3.89"), + (7.0, 10, "3.89"), + (6.9, 14, "3.89"), + (7.2, 18, "3.89"), + (11.8, 2, "5.89"), + (12.2, 6, "5.89"), + (12.5, 10, "5.89"), + (12.8, 14, "5.89"), + (12.6, 18, "5.89"), + (16.7, 2, "7.89"), + (16.8, 6, "7.89"), + (16.5, 10, "7.89"), + (16.3, 14, "7.89"), + (16.3, 18, "7.89"), + (21.3, 2, "9.89"), + (21.3, 6, "9.89"), + (21.3, 10, "9.89"), + (21.1, 14, "9.89"), + (21.0, 18, "9.89"), + (26.1, 2, "11.89"), + (25.7, 6, "11.89"), + (25.6, 10, "11.89"), + (25.7, 14, "11.89"), + (26.0, 18, "11.89"), + ] + + for x, y, label in points: + plt.scatter(x, y, color="red", s=15) + plt.text(x + 0.3, y + 0.3, label, fontsize=9) + + plt.xlabel("X (см)") + plt.ylabel("Y (см)") + plt.title("Эквипотенциальные точки") + plt.tight_layout() + plt.savefig("points.png", dpi=300) + + +if __name__ == "__main__": + main() diff --git a/course2/sem3/labs/lab3.01/scripts/2.py b/course2/sem3/labs/lab3.01/scripts/2.py new file mode 100644 index 0000000..841720f --- /dev/null +++ b/course2/sem3/labs/lab3.01/scripts/2.py @@ -0,0 +1,113 @@ +import matplotlib.pyplot as plt +import numpy as np + + +def main() -> None: + plt.figure(figsize=(8.27, 11.69)) + plt.xlim(0, 30) + plt.ylim(0, 20) + plt.gca().set_aspect("equal", adjustable="box") + plt.grid(True, linewidth=0.3) + plt.xticks(range(0, 31, 2)) + plt.yticks(range(0, 21, 2)) + + points = [ + # phi = 2.4 + (2.0, 2, "2.4"), + (2.1, 4, "2.4"), + (2.6, 6, "2.4"), + (2.8, 8, "2.4"), + (3.0, 10, "2.4"), + (2.8, 12, "2.4"), + (3.1, 14, "2.4"), + (2.7, 16, "2.4"), + (2.8, 18, "2.4"), + # phi = 3.4 + (4.1, 2, "3.4"), + (4.0, 4, "3.4"), + (4.1, 6, "3.4"), + (4.0, 8, "3.4"), + (4.2, 10, "3.4"), + (4.2, 12, "3.4"), + (4.6, 14, "3.4"), + (4.5, 16, "3.4"), + (4.8, 18, "3.4"), + # phi = 4.4 + (6.1, 2, "4.4"), + (5.9, 4, "4.4"), + (5.9, 6, "4.4"), + (5.5, 8, "4.4"), + (5.8, 10, "4.4"), + (5.7, 12, "4.4"), + (6.3, 14, "4.4"), + (7.0, 16, "4.4"), + (7.5, 18, "4.4"), + # phi = 5.4 + (8.4, 2, "5.4"), + (7.8, 4, "5.4"), + (7.4, 6, "5.4"), + (7.2, 8, "5.4"), + (7.2, 10, "5.4"), + (7.7, 12, "5.4"), + (8.2, 14, "5.4"), + (9.8, 18, "5.4"), + # phi = 6.4 + (11.2, 2, "6.4"), + (9.0, 6, "6.4"), + (8.8, 10, "6.4"), + (11.0, 14, "6.4"), + (12.3, 18, "6.4"), + # phi = 7.4 + (16.0, 2, "7.4"), + (15.5, 18, "7.4"), + # phi = 8.4 + (19.8, 2, "8.4"), + (20.3, 4, "8.4"), + (21.3, 6, "8.4"), + (21.7, 8, "8.4"), + (21.8, 10, "8.4"), + (21.3, 12, "8.4"), + (20.7, 14, "8.4"), + (19.7, 16, "8.4"), + (18.0, 18, "8.4"), + # phi = 9.4 + (22.2, 2, "9.4"), + (22.8, 6, "9.4"), + (22.9, 10, "9.4"), + (22.5, 14, "9.4"), + (21.7, 18, "9.4"), + # phi = 10.4 + (24.5, 2, "10.4"), + (24.4, 6, "10.4"), + (24.6, 10, "10.4"), + (24.2, 14, "10.4"), + (23.9, 18, "10.4"), + # phi = 11.4 + (26.7, 2, "11.4"), + (26.2, 6, "11.4"), + (26.2, 10, "11.4"), + (26.0, 14, "11.4"), + (26.1, 18, "11.4"), + ] + + for x, y, label in points: + plt.scatter(x, y, color="red", s=10) + plt.text(x + 0.3, y + 0.3, label, fontsize=9) + + theta = np.linspace(0, 2 * np.pi, 200) + cx, cy = 15, 9 + + for r in [5, 6]: + x = cx + r * np.cos(theta) + y = cy + r * np.sin(theta) + plt.plot(x, y, color="blue", linewidth=1) + + plt.xlabel("X (см)") + plt.ylabel("Y (см)") + plt.title("Эквипотенциальные точки с кольцом") + plt.tight_layout() + plt.savefig("points.png", dpi=300) + + +if __name__ == "__main__": + main() diff --git a/course2/sem3/labs/lab3.01/scripts/points.png b/course2/sem3/labs/lab3.01/scripts/points.png new file mode 100644 index 0000000000000000000000000000000000000000..bd48edf7a34d6f4cbc230bd0f2d2abaab793eb60 GIT binary patch literal 230632 zcmeFZcTkmC*FA`-Z4{-=VgLh4Nh*S5R8+VkK@gFwph(U+8c__O7X+0gl5@^c0TB_9 zjATVba*&`PFzaAEXKfM3@paLVhk+Cz+wz6#=v3>EXKfM3@paLVhk+C zz+wz6#=v3>EXKfM3@paLVhk+Cz+wz6#=v3>EXKfM3@paLVhk+Cz+w#i|0o8&Cfjy= zI=g9bzEpho?&Zg~93S4LboBZ7+EwgJYnFTOmpOK>_|%4p$1e+))f_Y0xLRr*Bx^0wjE;Uux}kfAOac7I6 zV)bUdkI7e>3tS71OGs>JW3wGBqhhT6LE_2$oio&BBq%t_=2DGTKU&H8Ig zmZj=d@5xA4i&N#R2ol#SxwpyEFo(SB!q>=--w=X1i$&Te(z?IFJpb$PaKdvw!)V;k ze$~{-h1rI}jrz6W_1~&P(sWAK)<>Q#>a_`U91=9{$k(`1 z?!#Ai=l1O=#Ryr4k+SE&2dv_agDoPkXto5FB~imSdZC zwTe-rz(wPTY5O@Tsk^xRC!&rvqB8?2`xV0<9J*R*Wb-uKp?PT8sFb|?qsgJRm=`a0 ztyr<5tffV5$&w{_M846=33GK!Td<%uVrPI=Ko=|!@(-5Ov 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