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Standard XII
Physics
Gauss Law
Question

Let a total charge $$2Q$$ be distributed in a sphere of radius $$R$$, with the charge density given by $$\rho(r) = kr$$, where $$r$$ is the distance from the centre. Two charges $$A$$ and $$B$$, of $$-Q$$ each, are placed on diametrically opposite points, at equal distance, a form the centre. If $$A$$ and $$B$$ do not experience any force, then:

A
$$a = \dfrac{3R}{2^{1/4}}$$
B
$$a = 8^{-1/4} R$$
C
$$a = 2^{-1/4} R$$
D
$$a = \dfrac{R}{\sqrt{3}}$$
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Solution
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Correct option is C. $$a = 8^{-1/4} R$$
$$E \,4\pi a^2 = \dfrac{\int^a_0 kr \,4\pi r^2 \,dr}{\varepsilon_0}$$
$$E = \dfrac{k \,4\pi a^4}{4 \times 4\pi \varepsilon_0}$$
$$2Q = \int^{R}_{0} kr \,4\pi r^2 \,dr$$
$$k = \dfrac{2Q}{\pi R^4}$$ --------(1)
$$QE = \dfrac{1}{4\pi \varepsilon_0} \dfrac{QQ}{(2a)^2}$$--------(2)
sum. (1) and (2)
$$R = a8^{1/4}$$

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