An infinitely long thin non-conducting wire is parallel to the z-axis and carries a uniform line charge density $λ$ . It pierces a thin non-conducting spherical shell of radius R in such a way that the arc PQ subtends an angle 120° 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?

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The electric flux through the shell is $\sqrt{2} R λ / ε_{0}$

The electric flux through the shell is $\sqrt{3} R λ / ε_{0}$

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

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

answer is A, B.

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Detailed Solution

Filed due to straight wire is perpendicular to the wire & radially outward. Hence $E_{z} = 0$ .

Length, $P Q = 2 R \sin 60 = \sqrt{3} R$

According to gauss’s law
Total flux= $\oint \vec{E} . \vec{d s} = \frac{q_{i n}}{\in_{0}} = \frac{λ \sqrt{3} R}{\in_{0}} (∵ L i n e a r c h a r g e d e n s i t y, λ = \frac{q}{l} \Rightarrow q = λ l = λ (P Q))$

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