Question

A spherical shell in the region a ≤ r ≤ b carries a charge per unit volume of rho = A / r , where A is constant. At the center (r = 0) of the enclosed cavity is a point charge Q. What should the value of A be so that the electric field in the region a ≤ r ≤ b has constant magnitude?

Answer 1

`E=(Q+2A\pi(r^2-a^2))/(4\pi r^2\epsilon_0)=k, A=((k4\pir^2\epsilon _0-Q))/(2\pi(r^2-a^2))`

Step-by-step explanation:

I will use the Gauss's Law to find the field:

`\int\vec{E}.d\vec{S}=(Q_(in))/(\epsilon_0)`}

the surface S is a sphere of radio r, the normal vector only has radial coordinates.

E(r)=E(r)r /*The field, based on spherical symmetry, only depends of the radius, and only has radial coordinate*/

if r<a

`\int\vec{E}.d\vec{S}=\int EdS=E\int dS=4\pi r^2E=(Q_(in))/(\epsilon_0)`

`\vec{E}=(Q)/(4\pi r^2\epsilon_0). \^r`

if a=<r<b

`\int\vec{E}.d\vec{S}=\int EdS=E\int dS=4\pi r^2E=(Q_(in))/(\epsilon_0)`

`Q_(in)=Q+\int dq', \rho=(dq')/(dVol), Q_(in)=Q+\int\rho dVol=Q+\int\limits^(2\pi)_0\int\limits^(\pi)_0\int\limits^r_a {(A)/(r)} \, r^2sin(\phi)dr d \phi d \theta`

`E=(Q+2A\pi(r^2-a^2))/(4\pi r^2\epsilon_0)=k, A=((k4\pir^2\epsilon _0-Q))/(2\pi(r^2-a^2))`

with K=constant