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)

Field due to straight wire is perpendicular to the wire & radially outward. Hence $E_z=0$
Length, $PQ=2R\sin 60=3R$ According to Gauss's law
total flux = $\oint \vec{E}.\overrightarrow{ds}=\frac{q_{in}}{{\in }_0}=\frac{\lambda \sqrt{3}R}{{\in }_0}$