Calculate the electric field inside and outside the sphere.

Question

A sphere of radius a carries a volume charge density rho = rho-sub-zero(r/a)**2 for r < a. Calculate the electric
field inside and outside the sphere. (N.B.: rho-subscript-zero is a constant volume charge density coefficient of (r/a)**2) 

Answer 1

Dalia,

 

For this problem you need to consider two scenarios, (1) inside the sphere and (2) outside the sphere as separate problems.

 

Start with the general expression for the electric field:

 

E = kq/r².

 

We can then write that dE/dq = k/r² or dE = kdq/r²

 

However, we're not given q, we're given the charge density ρ.  The amount of charge enclosed by a portion of the sphere is then:

 

q_enclosed = ρ·(4/3)πr³ , since the amount of charge enclosed depends only on r ⇒

 

dq = (4/3)πρr²dr

 

Substituting into the expression for dE:  dE = (k/r²)(4/3)(πρr²)dr

 

r² in the numerator cancels with r² in the denominator, so:

 

E = 4πkρ∫dr, evaluated from r = 0 to r, which leaves us with:

 

(a)  E_r = (4/3)πkρr , for r < a

 

for part (b) we start by noting that all of the charge Q resides inside the sphere when r > a.

 

Then Q = ρ·(4/3)πa³

 

Then for E_r when r > a

 

E = kQ/r² which is simply the field of a point charge. 

 

To put this back in terms of ρ and a just substitute for Q

 

(b) E_r = 4πkρa³/3r² ,  r > a