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 ${120}^o$ at the centre $O$ of the spherical shell, as shown in the figure. The permittivity of free space is ${ϵ}_0$ . Which of the following statements is (are) true?

• Correct answer
  A

  The electric flux through the shell is $\sqrt{3}R\lambda /{ϵ}_0$
• Correct answer
  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 $\sqrt{2}R\lambda /{ϵ}_0$
• D

  The electric field is normal to the surface of the shell at all points

Solution

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

Hence, the correct answer is options: 'A', 'B'.