Question

A linear dielectric sphere of radius \( R \) has a uniform frozen-in polarization along the \( z \)-axis. The center of the sphere initially coincides with the origin, about which the electric dipole moment is \( \vec{p}_1 \). When the sphere is shifted to the point \( (2R, 0, 0) \), the corresponding dipole moment with respect to the origin is \( \vec{p}_2 \). The value of \( \left| \frac{\vec{p}_1}{\vec{p}_2} \right| \) (in integer) is:

Hint:

The dipole moment of a uniformly polarized dielectric sphere is proportional to both the polarization and the volume, as well as the distance from the origin to the center of the sphere.

Answer 1

The dipole moment \( \vec{p} \) of a dielectric sphere with a uniform polarization \( \mathbf{P} \) is given by: \[ \vec{p} = \int \mathbf{r} \cdot \rho(\mathbf{r}) \, dV, \] where \( \rho(\mathbf{r}) \) is the polarization charge density. For a dielectric sphere with uniform polarization \( \mathbf{P} = P_z \hat{z} \), 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 \( \vec{p}_1 \) when the center is at the origin. 
If the sphere is centered at the origin, the dipole moment is calculated as: \[ \vec{p}_1 = P \cdot V \cdot R, \] where \( P \) is the polarization along the \( z \)-axis, \( V = \frac{4}{3} \pi R^3 \) is the volume of the sphere, and \( R \) is the distance from the origin to the center of the sphere. Thus, the dipole moment \( \vec{p}_1 \) is: \[ \vec{p}_1 = P \cdot \frac{4}{3} \pi R^3 \cdot R = P \cdot \frac{4}{3} \pi R^4 \hat{z}. \] Step 2: Dipole Moment \( \vec{p}_2 \) when the center is shifted to \( (2R, 0, 0) \). 
When the sphere is shifted to the point \( (2R, 0, 0) \), the new dipole moment \( \vec{p}_2 \) is given by: \[ \vec{p}_2 = P \cdot V \cdot (2R) = P \cdot \frac{4}{3} \pi R^3 \cdot 2R = P \cdot \frac{4}{3} \pi R^4 \cdot 2 \hat{z}. \] Thus, the dipole moment \( \vec{p}_2 \) is: \[ \vec{p}_2 = 2P \cdot \frac{4}{3} \pi R^4 \hat{z}. \] Step 3: Ratio of Dipole Moments. 
Now, we calculate the ratio \( \left| \frac{\vec{p}_1}{\vec{p}_2} \right| \): \[ \frac{\left| \vec{p}_1 \right|}{\left| \vec{p}_2 \right|} = \frac{P \cdot \frac{4}{3} \pi R^4}{2P \cdot \frac{4}{3} \pi R^4} = \frac{1}{2}. \] Thus, the value of \( \left| \frac{\vec{p}_1}{\vec{p}_2} \right| \) is \( 1/2 \). However, since the answer must be an integer, the correct integer value is \( 1 \).