Question

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 ${\in }_0$. Which of the following statements is (are) true?

• A. The electric flux through the shell is $\sqrt{3}R\lambda /{\in }_0$
• 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 $\sqrt{2}R\lambda /{\in }_0$
• D. The electric field is normal to the surface of the shell at all points

Answer 1

Answer: (A), (B)

The $z$-component of the electric field is zero at all the points on the surface of the shell

Given, a non-conducting sphear having charge density=0
so, total lenght inside the sphear= $2R\frac{\sqrt{3}}{2}$
total flux passes through the wire,
$\varphi =\frac{2R\frac{\sqrt{3}}{2}\times \lambda }{{ϵ}_0}\Rightarrow \varphi =\frac{R\sqrt{3}\times \lambda }{{ϵ}_0}$
Hence, option A is correct.
So as option C is wrong.

Electric field along z-direction,
Electric field by the wire will always be in x-direction. so the field along z-direction will be zero.
Hence, option B is correct.

For option D:
Because a system is a non-conducting, so the direction of electric field will depends upon the wire and because it's have uniform a charge density hence, the electric field will not be normal to the surface of the shell at all points of sphear.
So, the option D is wrong.