Question

23.6 applying gauss law: spherical symmetry
charge is uniformly distributed throughout a spherical insulating volume of radius ( r = 4.00 , \text{cm} ). the charge per unit volume is ( -8.94 , mu\text{c/m}^3 ). find the magnitude of the electric field at the center of the sphere. enter a positive number if the field points radially out, negative if the field points radially in, or zero if there is no field.

Explanation:

Step1: Recall Gauss's Law for electric field inside a uniformly charged sphere

For a uniformly charged insulating sphere, the electric field at a distance \( r \) from the center (where \( r < R \), \( R \) is the radius of the sphere) is given by \( E=\frac{
ho r}{3\epsilon_0} \), where \(
ho \) is the volume charge density, \( r \) is the distance from the center, and \( \epsilon_0 = 8.85\times 10^{-12}\ C^2/(N\cdot m^2) \) is the permittivity of free space.

At the center of the sphere, \( r = 0 \).

Step2: Substitute \( r = 0 \) into the formula

Substituting \( r = 0 \) into \( E=\frac{
ho r}{3\epsilon_0} \), we get \( E=\frac{
ho\times0}{3\epsilon_0}=0 \).

Answer:

\( 0 \)