GATE 2025 PH – Question 31
A linear dielectric sphere of radius $R$ has a uniform frozen-in polarization along the z-axis. Its center initially coincides with the origin, about which the electric dipole moment is $\mathbf p_1$. When shifted to $(2R,0,0)$, the corresponding dipole moment is $\mathbf p_2$. The value of $|\mathbf p_1|/|\mathbf p_2|$ (in integer) is _____
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Correct answer: 1
Explanation
The **dipole moment** of a charge distribution about an origin is $\mathbf p=\int\mathbf r\,\rho\,dV$. If the origin is shifted by $\mathbf a$, the dipole moment changes by
$$\mathbf p^\prime=\mathbf p-Q\,\mathbf a,$$
where $Q$ is the **total charge**.
For a uniformly polarised sphere the bound charges are a surface charge $\sigma_b=P\cos\theta$ and no volume charge. The total bound charge is zero:
$$Q=\oint P\cos\theta\,dA=0 .$$
So the dipole moment does **not depend on the choice of origin**: $\mathbf p_2=\mathbf p_1$ (its value is $\tfrac43\pi R^3\mathbf P$ in either case).
$$\frac{|\mathbf p_1|}{|\mathbf p_2|}=\mathbf{1}.$$