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JEE Advance - Physics (2018 - Paper 2 Offline - No. 3)

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 center $$O$$ of the spherical shell, as shown in the figure. The permittivity of free space is $${ \in _0}.$$ Which of the following statement is (are) true?

JEE Advanced 2018 Paper 2 Offline Physics - Electrostatics Question 42 English
The electric flux through the shell is $$\sqrt 3 R\lambda /{ \in _0}$$
The $$z$$-component of the electric field is zero at all the points on the surface of the shell
The electric flux through the shell is $$\sqrt 2 R\lambda /{ \in _0}$$
The electric field is normal to the surface of the shell at all points

Explanation

Let Gaussian surface be same as the non-conducting spherical shell. The charge enclosed by this Gaussian surface is the charge on the line segment PQ i.e.,

$${q_{enc}} = PQ\lambda = \sqrt 3 R\lambda $$,

where we used law of cosines in triangle OPQ to get $$PQ = \sqrt {{R^2} + {R^2} - 2(R)(R)\cos 120^\circ } = \sqrt 3 R$$. The electric flux through the shell (Gaussian surface) is given by Gauss's law

$$\phi = {q_{enc}}/{ \in _0} = \sqrt 3 R\lambda /{ \in _0}$$.

The electric field at a point due to the infinitely long line charge is radial i.e., perpendicular to the line PQ. The non-conducting spherical shell does not affect or alter the electric field. Thus, electric field at a point lying on the surface of the shell is perpendicular to the z-axis i.e., its z component is zero. The figure shows electric field on the surface of the shell for a positive line charge. Note that the magnitude of electric field is inversely proportional to the distance of the point from the line charge.

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