Question

In the figure shown, 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 $\lambda$ given by $\lambda =kx$ where $x$ is distance measured along the wire, from the end $A$.

Answer 1

Charge on an element of length dx of wire $AB$

$dq=\lambda ,dx=kxdx$

Total charge on wire $AB$,

$q=\int dq={\int }_0^{1}kxdx=k\left(\frac{x^2}{2}\right){|}_0^1=\frac{1}{2}k{l}^2$

Total flux through$S1=\varphi =\frac{Q}{{\in }_0}$

Total flux through S2

$={\varphi }_2=Q+\frac{q}{{\in }_0}=\frac{Q+\frac{1}{2}k{l}^2}{{\in }_0}$