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 = $\oint \vec{E} . \overrightarrow{ds} = \frac{q_{in}}{\in_0} = \frac{ \lambda \sqrt{3} R}{\in_0}$