Question 352: Electrostats & Capacitance

Question

A parallel plate capacitor consisting of two circular plates of radius 10 cm is being charged by a constant current of 0.15 A . If the rate of change of potential difference between the plates is $7 	imes 10^8 \mathrm{~V} / \mathrm{s}$ then the integer value of the distance between the parallel plates is $\left(ight.$ Take, $\left.\epsilon_0=9 	imes 10^{-12} \frac{\mathrm{~F}}{\mathrm{~m}}, \pi=\frac{22}{7}ight)$ ____________ $\mu \mathrm{m}$.

Accepted Answer

Given, $$r = 10\,cm = {1 \over {10}}\,m$$
$$I = 0.15\,A$$, and $${{dv} \over {dt}} = 7 	imes {10^8}v/s$$
We know, for a parallel plate capacitor,
$$c = {{{\varepsilon _0}A} \over d} \Rightarrow c = {{{\varepsilon _0}\pi {r^2}} \over d}$$ .... (1)
In a capacitor,
$$Q = CV$$
$$ \Rightarrow V = {Q \over C}$$
by differentiating w.r.t. t
$$ \Rightarrow {{dv} \over {dt}} = {1 \over C}{{dQ} \over {dt}}$$
$$ \Rightarrow {{dv} \over {dt}} = {d \over {{\varepsilon _0}\pi {r^2}}}I$$ (As $$I = {{dQ} \over {dt}}$$)
$$ \Rightarrow d = {{{\varepsilon _0}\pi {r^2}} \over I}{{dv} \over {dt}}$$ .... (From (1))
$$ \Rightarrow d = {{9 	imes {{10}^{ - 12}}} \over {0.15}} 	imes {{22} \over 7} 	imes {1 \over {100}} 	imes 7 	imes 10$$
$$ = {{198} \over {0.15}} 	imes {10^{ - 6}}$$
$$ = 1320 	imes {10^{ - 6}}$$ m
$$ \Rightarrow d = 1320\,\mu m$$

Electrostats & Capacitance

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