Spheres of charge: Electric charge is uniformly distributed inside a nonconducting sphere of radius 0.30 m. The electric field at a point P, which is 0.50 m from the center of the sphere, is 15,000 N/C and is directed radially outward. At what distance from the center of the sphere does the electric field have the same magnitude as it has at P?
A) 0.11 m
B) 0.13 m
C) 0.15 m
D) 0.17 m
E) at no other point

Question

Quizplus · Accepted Answer

The electric field inside a uniformly charged sphere is given by:$$E = \frac{1}{4\pi\epsilon_0} \frac{Q}{r^2}$$where Q is the total charge, r is the distance from the center of the sphere, and $\epsilon_0$ is the permittivity of free space.At a point P, which is 0.50 m from the center of the sphere, the electric field is 15,000 N/C. Therefore, we have:$$15,000 N/C = \frac{1}{4\pi\epsilon_0} \frac{Q}{0.50^2 m^2}$$Solving for Q, we get:$$Q = 4\pi\epsilon_0 (15,000 N/C) (0.50^2 m^2) = 1.41 \mu C$$Now, we want to find the distance from the center of the sphere where the electric field has the same magnitude as it has at P. Let's call this distance r. Then, we have:$$15,000 N/C = \frac{1}{4\pi\epsilon_0} \frac{Q}{r^2}$$Substituting Q = 1.41 $\mu C$, we get:$$15,000 N/C = \frac{1}{4\pi\epsilon_0} \frac{1.41 \mu C}{r^2}$$Solving for r, we get:$$r = \sqrt{\frac{1}{4\pi\epsilon_0} \frac{1.41 \mu C}{15,000 N/C}} = 0.11 m$$Therefore, the electric field has the same magnitude as it has at P at a distance of 0.11 m from the center of the sphere.

Spheres of Charge: Electric Charge Is Uniformly Distributed Inside a Nonconducting