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Q.

In the fig. calculate the total flux of the electrostatic field through the spheres $S_{1}$ and $S_{2}$ The wire AB shown here has a linear charge density $λ$ given by $λ = k x$ , where the x is the distance measured along the wire from end A.

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a

$\frac{Q - k l^{2}}{\in_{0}}$

b

$\frac{Q + k l^{2}}{\in_{0}}$

c

$\frac{Q + \frac{1}{2} k l^{2}}{\in_{0}}$

d

$\frac{Q - \frac{1}{2} k l^{2}}{\in_{0}}$

answer is A.

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Detailed Solution

Charge an element of length $d x$ of wire AB $d q = λ d x = k x d x$
Total charge on wire AB $q = \int d q = \int_{0}^{l} k x d x = \frac{1}{2} k l^{2}$
Total flux through $S_{1} = ϕ_{1} = \frac{Q}{\in_{0}}$
Total flux through $S_{2} = ϕ_{2} = \frac{Q + q}{\in_{0}} = \frac{Q + \frac{1}{2} k l^{2}}{\in_{0}}$

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