Question 824: Electrostatics & Capacitance

Question

Six infinitely large and thin non-conducting sheets are fixed in configurations I and II. As shown in the figure, the sheets carry uniform surface charge densities which are indicated in terms of $\sigma_0$. The separation between any two consecutive sheets is $1~\mu 	ext{m}$. The various regions between the sheets are denoted as 1, 2, 3, 4 and 5. If $\sigma_0 = 9~\mu	ext{C/m}^2$, then which of the following statements is/are correct:
(Take permittivity of free space $\epsilon_0 = 9 	imes 10^{-12}$ F/m)

Accepted Answer

$\begin{aligned}\left(E_4ight)_I= & \frac{\sigma_0}{2 \epsilon_0}[1-1+1-1-1+1]=0 \ \left(V_{	ext {First }}ight)_I & =\frac{-\sigma_0}{2 \epsilon_0}[-1+2-3+4-5] \mathrm{d} \ & =\frac{-\sigma_0}{2 \epsilon_0}[-3] \mathrm{d}=\frac{\sigma_0 3 \mathrm{~d}}{2 \epsilon_0}\end{aligned}$
$\begin{aligned} & \begin{aligned}\left(V_{	ext {Last }}ight)_I & =\frac{-\sigma_0}{2 \epsilon_0}[1-2+3-4+5] \ & =\frac{\sigma_0}{2 \epsilon_0}[-3 \mathrm{~d}]\end{aligned} \ & \begin{aligned}\left(V_{	ext {First }}-V_{	ext {Last }}ight)_I & =\frac{3 \sigma_0 \mathrm{~d}}{\epsilon_0} \ & =\frac{3 	imes 9 	imes 10^{-6} 	imes 1 	imes 10^{-6}}{9 	imes 10^{-12}}=3 \mathrm{volt}\end{aligned}\end{aligned}$
$\begin{aligned} & \begin{aligned}\left(\mathrm{E}_3ight)_{	ext {II }}= & \frac{\sigma_0}{2 \epsilon_0}\left[\frac{1}{2}-1+1+1-1+\frac{1}{2}ight]=\frac{\sigma_0}{2 \epsilon_0} \ & =\frac{-\sigma_0}{2 \epsilon_0}\left[-1+2-3+4-\frac{5}{2}ight] \mathrm{d} \ & =\frac{-\sigma_0}{2 \epsilon_0}[2-2.5] \mathrm{d}=\frac{\sigma_0 \mathrm{~d}}{4 \epsilon_0}\end{aligned} \ & \left(\mathrm{~V}_{	ext {Last }}ight)_{	ext {II }}=\frac{-\sigma_0}{2 \epsilon_0}\left[1-2+3-4+\frac{5}{2}ight] \mathrm{d} \ & \ & \quad=\frac{-\sigma_0}{2 \epsilon_0}[6.5-6] \mathrm{d}=\frac{-\sigma_0 \mathrm{~d}}{4 \epsilon_0} \ & \left(\mathrm{~V}_{	ext {First }}-\mathrm{V}_{	ext {Last }}ight)_{	ext {II }}=\frac{\sigma_0 \mathrm{~d}}{2 \epsilon_0} 
eq 0\end{aligned}$

Answer

In region 4 of the configuration I, the magnitude of the electric field is zero.

Answer

In region 3 of the configuration II, the magnitude of the electric field is $\dfrac{\sigma_0}{\epsilon_0}$.

Answer

Potential difference between the first and the last sheets of the configuration I is 5 V.

Answer

Potential difference between the first and the last sheets of the configuration II is zero.

Electrostatics & Capacitance

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