JEE PYQ: Capacitance Question 20
Question 20 - 2020 (07 Jan Shift 1)
A parallel plate capacitor has plates of area A separated by distance ’d’ between them. It is filled with a dielectric which has a dielectric constant that varies as $k(x) = K(1 + \alpha x)$ where ‘x’ is the distance measured from one of the plates. If $(\alpha d) « 1$, the total capacitance of the system is best given by the expression:
(1) $\frac{AK\epsilon_0}{d}\left(1 + \frac{\alpha d}{2}\right)$
(2) $\frac{A\epsilon_0 K}{d}\left[1 + \left(\frac{\alpha d}{2}\right)^2\right]$
(3) $\frac{A\epsilon_0 K}{d}\left(1 + \frac{\alpha^2 d^2}{2}\right)$
(4) $\frac{AK\epsilon_0}{d}(1 + \alpha d)$
Show Answer
Answer: (1)
Solution
$C_{el} = \frac{\epsilon_0 K(1+\alpha x)A}{dx}$. Integrating: $\frac{1}{C} = \frac{1}{\epsilon_0 KA}\int_0^d \frac{dx}{(1+\alpha x)}$. For $\alpha d « 1$: $C = \frac{\epsilon_0 KA}{d}\left(1 + \frac{\alpha d}{2}\right)$.
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