GATE 2026 PH – Question 30
A dielectric sphere carries a uniform polarization $P=26\ \mathrm{\mu C\,cm^{-2}}$. The magnitude of the electric field at the center of the sphere is $E\times10^9\ \mathrm{N\,C^{-1}}$. The value of $E$ (rounded off to one decimal place) is _____ ($\epsilon_0=8.85\times10^{-12}\ \mathrm{C^2N^{-1}m^{-2}}$)
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Correct answer: 9.75 to 9.85
Explanation
For a uniformly polarised sphere, the electric field **inside** is uniform:
$$\mathbf E=-\frac{\mathbf P}{3\epsilon_0}.$$
The field is opposite to the polarisation (the depolarising field).
**Numbers.** $P=26\ \mu\text{C/cm}^2=26\times10^{-6}\text{ C}/10^{-4}\text{ m}^2=0.26\text{ C/m}^2$:
$$E=\frac{0.26}{3\times8.85\times10^{-12}}=9.79\times10^{9}\text{ N/C}.$$
So $E=\mathbf{9.8}$ (in units of $10^9$ N/C).