A thin spherical shell of radius a has total charge q distributed uniformaly over its surface(see the next figure)Find the electric field at a point outside the

Question

Find the electric field at a point outside the shell

Answer 1

By Gauss' law,

 

Φ_E = q_T/ε₀

 

where q_T is the total charge enclosed by the surface.  Consider a sphere of radius r > a, concentric with the sphere of radius a.  The uniform distribution of q over the inner sphere of radius a ensures the electric field is purely radial, with no angular dependence.  This lets you use Gauss' law to find the electric filed at a point on the surface of the sphere of radius r > a.

 

Φ_E = ∫∫E·dA = EA = E(r) (4πr²)

 

Φ_E = q_T/ε₀ = q/ε₀

 

Then

 

E(r) (4πr²) = q/ε₀

 

E(r) = q / (4πε₀r²) for r > a

 

Mathematically, it is the same as if all the charge was concentrated at the center of the spheres.