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 $7 \times 10^8 \mathrm{~V} / \mathrm{s}$ then the integer value of the distance between the parallel plates is

$\left(\right.$ Take, $\left.\epsilon_0=9 \times 10^{-12} \frac{\mathrm{~F}}{\mathrm{~m}}, \pi=\frac{22}{7}\right)$ ____________ $\mu \mathrm{m}$.

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