A uniformly charged ball of radius a and charge ?Q is at the center of a hollow metal shell with inner radius b and outer radius c. The hollow sphere has net charge +2Q.

Part A

Determine the electric field strength in the region r?a. Give your answer as a multiple of Q/?0.

Express your answer in terms of some or all of the variables a, b, c, r, and the constant ?.

Park B

Determine the electric field strength in the region a
Express your answer in terms of some or all of the variables a, b, c, r, and the constant ?.

Part C

Determine the electric field strength in the region b
Part D

Determine the electric field strength in the region c
Express your answer in terms of some or all of the variables a, b, c, r, and the constant ?.

A)  E = 1/4π ε₀  r/3a³ ,  B)  E = 1/4πε₀  Q / r²

, C) E = 0

, D)   E = 1/4πε₀   3Q / r²

Explanation:

For this exercise let's use Gauss's law

           Ф = ∫ E. dA = / ε₀

Since the two objects are spherical we will use a sphere as our Gaussian surface

           E A = q_{int} /ε₀  

           E 4π r² = q_{int} /ε₀

Part A

The radius is r <a

In this case the chage inside of the Gaussian surface can be found using the concept of density

             ρ = q / V  

            q_{int} = ρ 4/3 π r³

We replace

            E 4π r² = rho 4/3 π r³  1/ε₀  

            E = rho r/3ε₀    

            E = 1/4π ε₀  r/3a³             r <a

Part B

In this case they ask for the field between the sphere and the shell

           a <r <b

In this case the charge inside is the charge of the sphere, we replace

             E (4π r²) = Q /ε₀  

            E = 1 / 4πε₀  Q / r²

Part C

The electric filed in the region inside the spherical shell

          b <r <c

An electric conductor in equilibrium all the charge is on its surface so the charge inside the conductor is zero

            q_{int}= 0

             E = 0

Part D

In the region outside ofthe hollow shell

             c <r

The charge inside is

         q_{int} = Q + 2Q = 3 Q

The electric field is

        E = 1 / 4πε₀   3Q / r²

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