Question 23.81: A spherical ball of charged particles has a uniform charge d......
A spherical ball of charged particles has a uniform charge density. In terms of the ball’s radius R, at what radial distances (a) inside and (b) outside the ball is the magnitude of the ball’s electric field equal to \frac{1}{4} of the maximum magnitude of that field?
(a) The field maximum occurs at the outer surface:
E_{\max }=\left(\frac{|q|}{4 \pi \varepsilon_0 r^2}\right)_{\text {at } r=R}=\frac{|q|}{4 \pi \varepsilon_0 R^2}
Applying Eq. 23-20, we have
E=\left(\frac{q}{4 \pi \varepsilon_0 R^3}\right) r (uniform charge, field at r ≤ R). (23-20)
E_{\text {internal }}=\frac{|q|}{4 \pi \varepsilon_0 R^3} r=\frac{1}{4} E_{\max } \Rightarrow r=\frac{R}{4}=0.25 \,R .
(b) Outside sphere 2 we have
E_{\text {external }}=\frac{|q|}{4 \pi \varepsilon_0 r^2}=\frac{1}{4} E_{\max } \Rightarrow r=2.0 \,R \text {. }