Electrostatics 7 Question 27
29. There are two large parallel metallic plates $S_{1}$ and $S_{2}$ carrying surface charge densities $\sigma_{1}$ and $\sigma_{2}$ respectively $\left(\sigma_{1}>\sigma_{2}\right)$ placed at a distance $d$ apart in vacuum. Find the work done by the electric field in moving a point charge $\mathrm{q}$ a distance $a(a<d)$ from $S_{1}$ towards $S_{2}$ along a line making an angle $\pi / 4$ with the normal to the plates.
(2004, 2M)
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Answer:
Correct Answer: 29. $W=\frac{\left(\sigma_{1}-\sigma_{2}\right) q a}{\sqrt{2} \varepsilon_{0}}$
Solution:
- Electric field near a large metallic plate is given by $E=\sigma / \varepsilon_{0}$. In between the plates the two fields will be in opposite directions. Hence,
$$ E_{\text {net }}=\frac{\sigma_{1}-\sigma_{2}}{\varepsilon_{0}}=E_{0} \text { (say) } $$
Now, $W=(q)$ (potential difference)
$$ \begin{aligned} & =q\left(E_{0} a \cos 45^{\circ}\right)=(q) \frac{\sigma_{1}-\sigma_{2}}{\varepsilon_{0}} \frac{a}{\sqrt{2}} \\ & =\frac{\left(\sigma_{1}-\sigma_{2}\right) q a}{\sqrt{2} \varepsilon_{0}} \end{aligned} $$
