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 p 1. When the sphere is shifted to the point (2R, 0, 0), the
corresponding dipole moment with respect to the origin is p 2. The value of
|p 1|
|p 2|
(in integer) is _____
Official Answer Key:
1 to 1
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE PH 2025 Q.31)**
**1. Electrodynamics of Polarized Media**:
- A dielectric sphere of radius $R$ possesses uniform frozen-in polarization $\vec{P} = P_0 \hat{z}$.
**2. Step-by-Step Derivation**:
- Step 1: Compute total electric dipole moment $\vec{p}$:
$\vec{p} = \int_V \vec{P} dV = \vec{P} \int_V dV = (P_0 \hat{z}) \left(\frac{4}{3}\pi R^3\right) = \frac{4}{3}\pi R^3 P_0 \hat{z}$
- Step 2: Invariance under translation for neutral systems:
- Bound surface charge: $\sigma_b = \vec{P} \cdot \hat{r} = P_0 \cos\theta$.
- Bound volume charge: $\rho_b = -\nabla \cdot \vec{P} = 0$.
- Total bound charge:
$Q_{\text{total}} = \int_0^\pi (P_0 \cos\theta)(2\pi R^2 \sin\theta) d\theta = 0$
- Under a spatial translation of origin by $\vec{d}$, the transformed dipole moment is:
$\vec{p}' = \vec{p} - Q_{\text{total}} \vec{d} = \vec{p} - 0 = \vec{p}$
- Step 3: Compute the ratio of dipole moments:
$\text{Ratio} = \frac{|\vec{p}'|}{|\vec{p}|} = 1$
**3. Official Verification**:
The verified official IIT answer is **1** (Numerical range: `1 to 1`).
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