# Pressure on uniformly charge spherical shell

• timhunderwood
In summary, the conversation discusses finding the pressure exerted on a uniformly charged spherical shell with charge Q and radius R, using the principle of virtual work. The solution involves calculating the energy stored in the electric field and using the formula P=(1/2)eps*E^2 to find the pressure. There was a small error in one of the equations, but the overall approach is correct.
timhunderwood

## Homework Statement

First part of the problem was to work out E-field of uniformly charged spherical shell with charge Q and Radius R.

This was fine : E = 0 for r<R
and E = Q/(4*Pi*eps*r2) for r>R

QUESTION:
Find the pressure exerted on the shell due to the charges on its surface.

## The Attempt at a Solution

I tried to do it using the principle of virtual work, I get an answer and was hoping someone could confirm it is right?

I did this:
u_s, Energy stored in Electric field = $$\int$$eps*E2/2 (integrate over all space)

I did this in spherical polar coordinates and get U_s = Q2/(8*Pi*eps*R)

I then said (principle of virtual work:)

F, Force*dR = $$\partial$$U_s/$$\partial$$R *dR
and solved this to get:
F = -Q2/(8*Pi*eps*R2)

dividing by area of shell then gives

pressure = Q2/(32*Pi2*eps*R2)

is this right? If there's a mistake I think I may have not used the principle of virtual work correctly...

Thanks very much.

I do believe you are making this a little more complicated than necessary. The electrostatic pressure on the surface is given by the formula

$$P=\frac{\varepsilon_0}{2}E^2$$

(cf. Griffiths Equation 2.52)

timhunderwood said:
pressure = Q2/(32*Pi2*eps*R2)

is this right? If there's a mistake I think I may have not used the principle of virtual work correctly...
You're missing a factor of R^2 in the denominator, but otherwise, yes, that is fine.

Thats good,

I mistyped my last equation- it should have read R^4.

Thanks for the help

## 1. What is the formula for calculating the pressure on a uniformly charged spherical shell?

The formula for calculating the pressure on a uniformly charged spherical shell is P = Q/4πr², where P is the pressure, Q is the total charge on the shell, and r is the radius of the shell.

## 2. How does the pressure on a uniformly charged spherical shell change with distance from the center?

The pressure on a uniformly charged spherical shell decreases with increasing distance from the center. This is because the force of the electric field, which is responsible for the pressure, decreases with distance according to the inverse square law.

## 3. What is the relationship between pressure and charge on a uniformly charged spherical shell?

The pressure on a uniformly charged spherical shell is directly proportional to the total charge on the shell. This means that as the charge increases, the pressure also increases.

## 4. Can the pressure on a uniformly charged spherical shell be negative?

No, the pressure on a uniformly charged spherical shell cannot be negative. This is because the electric field always exerts a repulsive force, resulting in a positive pressure.

## 5. How does the pressure on a uniformly charged spherical shell compare to the pressure on a uniformly charged solid sphere?

The pressure on a uniformly charged spherical shell is always greater than the pressure on a uniformly charged solid sphere of the same radius and charge. This is because the electric field is more concentrated on the surface of a spherical shell compared to a solid sphere, resulting in a higher pressure.

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