- #1

GL_Black_Hole

- 21

- 0

## Homework Statement

A spherical capacitor has internal radius ##a## and external radius ##b##. At time ##t = 0##, the charge of the

capacitor is ##Q_0## Then the two shells are connected by a resistor in the radial direction of resistance ##R##. Find the Poynting vector and the energy dissipation in a small volume around the resistor and compare it to the time variation of the electrostatic energy.

## Homework Equations

##\begin{aligned} \nabla \times \vec{\mathbf{B}} - \frac{\partial\vec{\mathbf{E}}}{\partial t} & = \mu_0 \vec{\mathbf{j}} \\ \nabla \cdot \vec{\mathbf{E}} & = \frac{\rho}{\epsilon_0} \\ \nabla \times \vec{\mathbf{E}}\, +\, \frac1c\, \frac{\partial\vec{\mathbf{B}}}{\partial t} & = \vec{\mathbf{0}} \\ \nabla \cdot \vec{\mathbf{B}} & = 0 \end{aligned}##

## The Attempt at a Solution

When the resistor is connected to the shells the system becomes an RC circuit, so the charge on the capacitor is now ##Q(t) = Q_0 \exp{(-t/\tau)}## with ##\tau =RC = \frac{4\pi R}{\frac{1}{a} -\frac{1}{b}}##. This results in a flow of current but is the first source of confusion for me. Does the current flow through the resistor connecting the shells? If so then we have a radial current ##\vec{\mathbf{j}} =-\frac{dQ}{dt} \vec{\hat{r}}##. The electric field between the shells is also changing as the charge decreases so there is a displacement current determined by ##\vec{\mathbf{E(t)}} = \frac{Q(t)}{r^2 \epsilon_0} \vec{\hat{r}}##. Using these I could find the magnetic field around the resistor from Maxwell's equations and then Poynting's vector by taking a cross product and then integrate for a surface surrounding the resistor to compare against the time derivative of the electrostatic energy ##U = \frac{Q(t)^2}{2C}##. But before proceeding further I want to make sure that I'm understanding the physics here properly.