APhO 2013, theory — Задача 1

Problem Conductors in Conducting Liquid

Условие

A system consisted of two conductor bodies is immersed in a uniform dielectric and weakly conducting liquid. When a constant voltage difference is applied between both conductors, the system has both electric and magnetic fields. In this problem we will investigate this system.

1. (0.4 pts) First consider an infinitely long line with charge per unit length $\lambda$ in vacuum. Calculate the electric field $\mathbf{E}(\mathbf{r})$ due to the line.

2. (0.4 pts) The potential due to the line charge could be written as

$V(r) = f(r) + K,$ where $K$ is a constant. Determine $f(r)$.

3. (0.7 pts) Calculate the potential in all space $V(x,y,z)$ due to an infinitely long line with charge per unit length $\lambda$ at $x = -b, y = 0$ and another infinitely long line with charge per unit length $-\lambda$ at $x = b, y = 0$. Both lines are parallel to the $z$-axis. Take $V = 0$ at the origin. Sketch the equipotential surfaces.

For the following questions, ignore any edge effects.

4. (2.0 pts) Now consider two identical conducting cylinders, both with radius $R = 3a$ in vacuum. The length of each cylinders are the same and much larger than its radius ($l \gg R$). The axis of both cylinders are on the $xz$-plane and parallel to the $z$-axis, one at $x = -5a$, $y = 0$ and the other at $x = 5a$, $y = 0$. An electrical potential difference of $V_0$ is applied between the two cylinders (the cylinder at $x = -5a$ has the higher potential) by connecting them to a battery. Calculate the potential in all regions. Take $V = 0$ at the origin.

5. (0.5 pts) Calculate the capacitance $C$ of the system.

6. (1.0 pts) Now both cylinders are totally immersed in a weakly conducting liquid with conductivity $\sigma$. Calculate the total current that flows between both cylinders. Assume the permittivity of the liquid is equal to that of vacuum, $\epsilon = \epsilon_0$.

7. (0.5 pts) Calculate the resistance $R$ of the system. Calculate $RC$ of the system.

8. (1.5 pts) Calculate the magnetic field due to the current in question 6. Assume that the permeability of the liquid is equal to that of vacuum $\mu = \mu_0$.

Решение

Оригинал в Архива: 1.pdf · официални решения: 1_sol.pdf