GATE 2022 PH – Question 31
Electric field is measured along the axis of a uniformly charged disc of radius 25 cm. At a distance d from the centre, the field differs by 10% from that of an infinite plane having the same charge density. The value of d is _______cm. (Round off to one decimal place)
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Correct answer: 2.49 to 2.51
Explanation
**Field on the axis of a uniformly charged disc** of radius $R$ and surface charge density $\sigma$, at a distance $d$ from its centre:
$$E=\frac{\sigma}{2\epsilon_0}\left(1-\frac{d}{\sqrt{d^2+R^2}}\right).$$
**Infinite plane:** $E_\infty=\dfrac{\sigma}{2\epsilon_0}$.
**The disc field differs by 10% from the plane field,** so the correction term is 10%:
$$\frac{d}{\sqrt{d^2+R^2}}=0.1\;\Rightarrow\;d^2=0.01\,(d^2+R^2)\;\Rightarrow\;d^2\,(0.99)=0.01R^2 .$$
$$d=\frac{R}{\sqrt{99}}=\frac{25\text{ cm}}{9.950}=\mathbf{2.5\ cm}.$$