Find A Such that the Electric Field is Constant

[NiendorfPhysics]

Homework Statement

The Spherical region a<r<b carries a charge per unit volume of $$\frac{A}{r}$$, where A is constant. At the center there is a point charge q. Find A such that the Electric field in a<r<b is constant.

Homework Equations

Law of superposition and $$E=\frac{kq}{r^2}$$

The Attempt at a Solution

$$E=k(\frac{q}{r^2}+\frac{\frac{A}{r}*\frac{4\pi(r^3-a^3)}{3}}{r^2})$$
$$\frac{dE}{dr}=0=\frac{-2q}{r^3}+0+\frac{4A{\pi}a^3}{r^4}$$
$$A=\frac{qr}{2{\pi}a^3}$$

Which means that I got that no constant would satisfy the stated condition. Answer in the back is $$A=\frac{q}{2{\pi}a^2}$$. Any hints are appreciated.

 

[kuruman]

Please fix the LaTeX code so that we can read what you wrote. You need two dollar signs to bracket the code. Also, it would help if you clicked "Preview..." to fix any LaTeX code errors before posting.

 

[NiendorfPhysics]

Please fix the LaTeX code so that we can read what you wrote. You need two dollar signs to bracket the code. Also, it would help if you clicked "Preview..." to fix any LaTeX code errors before posting.

Fixed it.

 

[kuruman]

Thank you, that's much better. How did you get the second term in the expression for the electric field?

 

[NiendorfPhysics]

Thank you, that's much better. How did you get the second term in the expression for the electric field?

I multiplied the charge density by the volume of the sphere that you have if you are at some point r between a and b. So you have to subtract the empty space between the point charge and when the cloud of charge begins at radius a. I cleaned it up a bit so that it is a little more clear.

 

[kuruman]

I multiplied the charge density by the volume of the sphere that you have ...

You can do that only if the charge density is constant. Here it depends on r. You need to use Gauss's law and actually do an integral to find the charge in the spherical region enclosed by your Gaussian surface. Don't forget to add the charge at the center to the enclosed charge.

 

[NiendorfPhysics]

You can do that only if the charge density is constant. Here it depends on r. You need to use Gauss's law and actually do an integral to find the charge in the spherical region enclosed by your Gaussian surface. Don't forget to add the charge at the center to the enclosed charge.

Wow, I'm stupid. Thanks.