# Conducting Spherical Shell Capacitor

• BeBattey
In summary, the problem is to find the appropriate coefficients for the potential function Φ in a conducting spherical shell with a narrow insulating ring dividing it into two halves, with the top half at 10V and the bottom half at -10V. Using symmetry and the expected behavior at the origin, it is determined that the B coefficients must all be zero. To satisfy the boundary conditions at r = a, the remaining coefficients must be solved for using integral form. However, it is yet to be determined how to handle the boundary condition for different values of theta. This problem is similar to those given in a graduate level EM course.
BeBattey

## Homework Statement

A conducting spherical shell is divided into upper and lower halves with a narrow insulating ring between them. The top half is at 10V and the bottom half is at -10V. Write down the appropriate expansion for Φ and use symmetry and the expected behavior at the origin to identify which coefficients are zero. Then solve for the nonzero coefficients which make Φ satisfy the values given at r = a. You will undoubtedly have to express the coefficients in integral form.

## Homework Equations

No charge inside, so Laplace's equation applies:
$\nabla^{2}\phi=0$
Given the general solution for solving Laplaces equation in spherical coordinates:
$\phi (r,\theta,\varphi)= \sum^{\infty}_{n=0}(A_{n}r^{n}+\frac{B^{n}}{r^{n+1}})P_{n}(cos\theta)$

## The Attempt at a Solution

I've only concluded so far that the B coefficients must all be 0 due to requiring finite potential at r=0. Past that I'm at a loss on how to tackle the function. I know:
$\phi (r,\theta,\varphi)= \sum^{\infty}_{n=0}A_{n}r^{n}P_{n}(cos\theta)$
But I don't know how I can tackle the boundary condition of plus and minus 10 at radius a, depending on angle theta.

Also I'm not quite sure which forum to put this in. It's a fourth year undergrad course, but all I've been told about the professor is that he gives us grad school type problems like this one, as previous graduates have come back and told us that their graduate EM course was actually easier.

## What is a Conducting Spherical Shell Capacitor?

A conducting spherical shell capacitor is a type of capacitor that consists of two concentric spherical conductors separated by a dielectric material, such as air or vacuum. It is used to store and release electrical energy, similar to other types of capacitors.

## How does a Conducting Spherical Shell Capacitor work?

A conducting spherical shell capacitor works by storing electrical energy in the electric field that is created between the two conductive shells. When a voltage is applied to the capacitor, one shell becomes positively charged and the other becomes negatively charged, creating an electric field between them. The dielectric material between the shells helps to maintain this electric field and store the energy.

## What are the applications of Conducting Spherical Shell Capacitors?

Conducting spherical shell capacitors have various applications in electronic circuits, including in power supplies, filters, and timing circuits. They are also used in radio frequency and microwave applications, as well as in medical equipment.

## What factors affect the capacitance of a Conducting Spherical Shell Capacitor?

The capacitance of a conducting spherical shell capacitor is affected by several factors, including the distance between the shells, the size of the shells, and the type of dielectric material used. The dielectric constant of the material also plays a significant role in determining the capacitance.

## How is the capacitance of a Conducting Spherical Shell Capacitor calculated?

The capacitance of a conducting spherical shell capacitor can be calculated using the formula C = 4πεrε0r, where C is the capacitance, εr is the relative permittivity of the dielectric material, and r is the distance between the shells. This formula assumes that the shells are perfect spheres and that the electric field is uniform between them.

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