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An infinitely long thin non-conducting wire carrying uniform linear charge density lambda lies parallel to the z-axis. It passes through a thin non-conducting spherical shell of radius R such that the chord (arc PQ) where the wire pierces the shell subtends 120 deg at the centre O. Permittivity of free space is e0. Which statement(s) is/are correct?

1. The electric flux through the shell is sqrt(3)*R*lambda/e0
2. The z-component of the electric field is zero at all points on the surface of the shell
3. The electric flux through the shell is sqrt(2)*R*lambda/e0
4. The electric field is normal to the surface of the shell at all points

Correct answer: The electric flux through the shell is sqrt(3)*R*lambda/e0

Solution

This is a multiple-correct question. The chord length inside the sphere = 2R sin(120/2) = 2R sin60 = sqrt(3) R. Enclosed charge = lambda*sqrt(3)R, so flux = sqrt(3)*R*lambda/e0; the first option is correct, the sqrt(2) one is wrong. An infinite line charge produces a field perpendicular to the wire (in the radial direction away from the z-axis), so its z-component is zero everywhere; that option is correct too. The field is not normal to the spherical surface at all points, so that option is false. Canonical key: the sqrt(3) flux statement.

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