Question

A nonconducting solid sphere of radius 8.40 cm has a uniform volume charge density. The magnitude of the electric field at 16.8 cm from the sphere's center is 2.04 x 103 N/C. (a) What is the sphere's volume charge density?

Answer 1

Answer:

The sphere's volume charge density is 2.58 μC/m³.

Explanation:

Given that,

Radius of sphere R= 8.40 cm

Electric field $E= 2.04\times10^{3}\ N/C$

Distance r= 16.8 cm

We need to calculate the sphere's volume charge density

Using Gauss's law

$\int{\vec{E}\cdot\vec{da}}=\dfrac{Q_{enc}}{\epsilon_{0}}$

$E\times 4\pi r^2=\dfrac{1}{\epsilon_{0}}\times\dfrac{4}{3}\piR^3\rho$

$E=\dfrac{\rho R^3}{3\epsilon_{0}r^2}$

$\rho=\dfrac{3\times E\times\epsilon_{0}r^2}{R^3}$

Put the value into the formula

$\rho=\dfrac{3\times2.04\times10^{3}\times8.85\times10^{-12}\times(16.8\times10^{-2})^2}{(8.40\times10^{-2})^3}$

$\rho=2.58\times10^{-6}\ C/m^3$

$\rho=2.58\ \mu C/m^3$

Hence, The sphere's volume charge density is 2.58 μC/m³.