# Help with electrostatics problem (spherical shell charge distribution)

• sroot
In summary: That should be ##\varphi (r,t)##.In summary, according to my professor, the solution in this book (pages 20-21) for item (ii) is wrong. The electric field is not the same in all regions, and the integral is off.
sroot
Homework Statement
A static electric charge is distributed in a spherical shell of inner radius R1 and outer radius R2. The electric charge density is given by ρ=a+br, where r is the distance from the centre, and zero everywhere else.
(i) Find an expression for the electric field everywhere in terms of r.
(ii) Find expressions for the electric potentials and energy density for r < R1. Take the potential to be zero at r→∞.
Relevant Equations
ρ=a+br
sroot said:
Homework Statement:: A static electric charge is distributed in a spherical shell of inner radius R1 and outer radius R2. The electric charge density is given by ρ=a+br, where r is the distance from the centre, and zero everywhere else.
(i) Find an expression for the electric field everywhere in terms of r.
(ii) Find expressions for the electric potentials and energy density for r < R1. Take the potential to be zero at r→∞.
Relevant Equations:: ρ=a+br

According to my professor, the solution in this book (pages 20-21) for item (ii) is wrong: https://www.u-cursos.cl/usuario/754...roblems_and_Solutions_on_Electromagnetism.pdf
Welcome to PF.

Can you please just take screenshots of the pages that you want to discuss? Asking us to download the whole book PDF file is a bit too much. And please explain what is confusing you about the solution. Thank you.

berkeman said:
Welcome to PF.

Can you please just take screenshots of the pages that you want to discuss? Asking us to download the whole book PDF file is a bit too much. And please explain what is confusing you about the solution. Thank you.

This is the solution for item (b). According to the professor, the integral is completely off. I can't understand why.

Did you carry out the integrations? What did you get for the results of the individual integrals?

It looks like the solution asserts that the electric field is the same in all regions by showing it factored out of the integrals. That is simply not true. I started doing the integrals but I quit before I got to the bottom line when it became obvious that my expression would not simplify to the answer shown above.

I interpreted "The electric charge density is given by ρ=a+br, where r is the distance from the centre, and zero everywhere else" to mean that there is charge only in the region ##R_1\leq r \leq R_2## "everywhere else" being outside this region. Otherwise why bother mentioning a shell?

Also, it is misleading (but not an error) to write the left hand side as ##\varphi (r)## when it is actually independent of ##r##.

Last edited:
berkeman

## 1. How do I calculate the electric field at a point outside a spherical shell with a charge distribution?

To calculate the electric field at a point outside a spherical shell with a charge distribution, you can use the formula E = kQ/r^2, where k is the Coulomb's constant, Q is the total charge of the shell, and r is the distance from the center of the shell to the point.

## 2. What is the difference between a conducting and non-conducting spherical shell in terms of electrostatics?

A conducting spherical shell has the property of charge distribution being on the surface only, while a non-conducting spherical shell can have charge distribution throughout its volume. This affects the calculation of electric field and potential at a point outside the shell.

## 3. How do I determine the potential at a point outside a spherical shell with a charge distribution?

To determine the potential at a point outside a spherical shell with a charge distribution, you can use the formula V = kQ/r, where k is the Coulomb's constant, Q is the total charge of the shell, and r is the distance from the center of the shell to the point.

## 4. Can the electric field inside a spherical shell with a charge distribution be non-zero?

No, the electric field inside a spherical shell with a charge distribution is always zero. This is because the electric field inside a conductor is zero and a conducting shell has charge distribution only on its surface.

## 5. How does the charge distribution on a spherical shell affect the electric potential at a point outside the shell?

The charge distribution on a spherical shell affects the electric potential at a point outside the shell by changing the total charge Q in the formula V = kQ/r. A non-uniform charge distribution will result in a non-uniform potential, while a uniform charge distribution will result in a constant potential at all points outside the shell.

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