Electro Dynamics

The dipole moment of a dielectric sphere with a uniform polarization is given by: where is the polarization charge density. For a dielectric sphere with uniform polarization , the dipole moment depends on the volume of the sphere and the distance of the center of the sphere from the point where we are calculating the dipole moment.
Step 1: Dipole Moment when the center is at the origin.
If the sphere is centered at the origin, the dipole moment is calculated as: where is the polarization along the -axis, is the volume of the sphere, and is the distance from the origin to the center of the sphere. Thus, the dipole moment is: Step 2: Dipole Moment when the center is shifted to .
When the sphere is shifted to the point , the new dipole moment is given by: Thus, the dipole moment is: Step 3: Ratio of Dipole Moments.
Now, we calculate the ratio : Thus, the value of is . However, since the answer must be an integer, the correct integer value is .