Question

A sphere of radius \( R \) has a uniform charge density \( \rho \). A sphere of smaller radius \( \frac{R}{2} \) is cut out from the original sphere, as shown in the figure below. The center of the cut-out sphere lies at \( z = \frac{R}{2} \). After the smaller sphere has been cut out, the magnitude of the electric field at \( z = - \frac{R}{2} \) is \( \frac{\rho R}{2 \epsilon_0} \). The value of the integer \( n \) is: 

Hint:

For spheres with uniform charge densities, the electric field inside is directly proportional to the radial distance. Cutting out a spherical region affects the overall charge distribution.

Answer 1

Correct Answer: 8

Step 1: Understanding the problem.
The problem involves a sphere with uniform charge density \( \rho \), from which a smaller sphere is cut out. The electric field at a point inside the sphere, after the cut, needs to be calculated. Step 2: Electric field due to a uniformly charged sphere.
The electric field inside a uniformly charged sphere with charge density \( \rho \) is given by: \[ E = \frac{\rho r}{3 \epsilon_0} \] where \( r \) is the radial distance from the center of the sphere. Step 3: Applying the cut-out sphere concept.
Since a smaller sphere of radius \( \frac{R}{2} \) is cut out from the original sphere, the electric field at the point where the cut was made must account for the effect of this absence. This results in the electric field expression: \[ E = \frac{\rho R}{2 \epsilon_0} \] Therefore, the integer \( n \) in the equation is 8.