GATE 2024 PH – Question 40
A parallel capacitor has potential zero at x = 0 and $V_0$ at x = d, filled by $\epsilon=\epsilon_0(1+x/d)$. Neglect edges. The field is
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Correct answer: (A) $-V_0\hat x/[(d+x)\ln2]$
Explanation
The capacitor is filled with a dielectric whose permittivity varies with position, $\epsilon=\epsilon_0(1+x/d)$, and there is no free charge between the plates.
**Gauss's law:** with no free charge, $\nabla\cdot\mathbf D=0$. For a field depending only on $x$, $D_x$ is **constant**: $D_x=D$.
**Electric field:**
$$E_x=\frac{D}{\epsilon}=\frac{D}{\epsilon_0(1+x/d)}=\frac{C}{d+x},\qquad C=\frac{Dd}{\epsilon_0}.$$
**Fix the constant with the potentials.** $V(0)=0$ and $V(d)=V_0$, so
$$V_0=-\int_0^dE_x\,dx=-C\ln\frac{2d}{d}=-C\ln2\;\Rightarrow\;C=-\frac{V_0}{\ln2}.$$
$$\mathbf E=-\frac{V_0}{(d+x)\ln2}\,\hat x\quad(\text{option A}).$$