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Standard XII

Physics

Gauss's Law

Answer the fo...

Question

Answer the following Questions.
(i) "The outward electric flux due to charge $+ Q$ is independent of the shape and size of the surface which encloses it." Give two reasons to justify this statement.
(ii) Two identical circular loops $^{'} 1^{'}$ and $^{'} 2^{'}$ of radius $R$ each have linear charge densities $- λ$ and $+ λ C / m$ respectively. The loops are placed coaxially with their centres $R - \sqrt{3}$ distance apart. Find the magnitude and direction of the net electric field at the centre of loop $^{'} 1^{'}$ .

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Solution

Electric field at the centre $O 1$ due to loop $1$ is given by
$| {\to E}_{1} | = 0$ (As $Z = 0$ )
Electric field at a point outside the loop $2$ on the axis passing normally through $O_{2}$ of loop $2$ is
$| {\to E}_{2} | = \frac{λ R}{2 ϵ_{0}} \cdot \frac{Z}{(R^{2} + Z^{2})^{3 / 2}}$
Since $Z = R \sqrt{3}$
$= \frac{λ R}{2 ϵ_{0}} \cdot \frac{R \sqrt{3}}{(R^{2} + 3 R^{2})^{3 / 2}}$
$= \frac{λ \sqrt{3}}{16 ϵ_{0} R}$ towards right (As $λ$ is positive)
So, net electric field at the centre of loop $1$
$\to E = {\to E}_{1} + {\to E}_{2}$
$= 0 + \frac{λ \sqrt{3}}{16 ϵ_{0} R} = \frac{λ \sqrt{3}}{16 ϵ_{0} R}$ .

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