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An infinitely long thin non-conducting wire is parallel to the $z$-axis and carries a uniform line charge density $\lambda$. It pierces a thin non-conducting spherical shell of radius $R$ in such a way that the arc $PQ$ subtends an angle $120^{\circ}$ at the centre $O$ of the spherical shell, as shown in the figure. The permittivity of free space is $\epsilon_0$. Which of the following statements is (are) true?
$(A)$ The electric flux through the shell is $\sqrt{3} R \lambda / \epsilon_0$
$(B)$ The z-component of the electric field is zero at all the points on the surface of the shell
$(C)$ The electric flux through the shell is $\sqrt{2} R \lambda / \epsilon_0$
$(D)$ The electric field is normal to the surface of the shell at all points
$A,C$
$A,B$
$A,D$
$A,B,C$
Solution
Gauss law
$\phi=\oint \overrightarrow{ E } \cdot d \overrightarrow{ A }=\frac{ Q }{\epsilon_0}$
where $Q =\lambda L$
$L = PQ =2 R \sin 60^{\circ}$
$L = R \sqrt{3}$
$\phi=\frac{\lambda R \sqrt{3}}{\epsilon_0}$
Electric field lines are radially outwards, perpendicular to length of wire. Hence component of $E.F.$ is zero along $z$-axis.
