First slide

Electric field

Question

A sphere of radius R carries charge such that its volume charge density is proportional to the square of the distance from the center. What is the ratio of the magnitude of the electric field at a distance 2R from the center to the magnitude of the electric field at a distance of R/2 from the center?

Moderate

A

1
B

2
C

4
D

8

Solution

$\begin{aligned} \begin{aligned} ρ = C r^{2} \\ \begin{aligned} q = & \int_{0}^{r} 4 π r^{2} d r ρ = \int_{0}^{r} 4 π C r^{4} d r = \frac{4}{5} π C r^{5} \\ E_{r = 2 R} & = \frac{k q_{(r = R)}}{(2 R)^{2}} = \frac{k (4 / 5) π C R^{5}}{4 R^{2}} \\ = \frac{k q_{(r = R / 2)}}{(R / 2)^{2}} = \frac{k (4 / 5) π C (R / 2)^{5}}{(R / 2)^{2}} \\ Now solve to get \frac{E_{r = 2 R}}{E_{r = R / 2}} = 2 \end{aligned} \end{aligned} \end{aligned}$

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