JEE Advance - Physics (2015 - Paper 1 Offline - No. 9)
An infinitely long uniform line charge distribution of charge per unit length $$\lambda$$ lies parallel to the y-axis in the y-z plane at $$z = {{\sqrt 3 } \over 2}$$a (see figure). If the magnitude of the flux of the electric field through the rectangular surface ABCD lying in the x-y plane with its centre at the origin is $${{\lambda L} \over {n{\varepsilon _0}}}$$ ($${{\varepsilon _0}}$$ = permittivity of free space), then the value of n is
Answer
6
Explanation
ANBP is cross-section of a cylinder of length L. The line charge passes through the centre O and perpendicular to paper.
$$AM = {a \over 2}$$, $$MO = {{\sqrt 3 a} \over 2}$$
$$\therefore$$ $$\angle AOM = {\tan ^{ - 1}}\left( {{{AM} \over {OM}}} \right)$$
$$ = {\tan ^{ - 1}}\left( {{1 \over {\sqrt 3 }}} \right) = 30^\circ $$
Electric flux passing from the whole cylinder
$${\phi _1} = {{{q_{in}}} \over {{\varepsilon _0}}} = {{\lambda L} \over {{\varepsilon _0}}}$$
$$\therefore$$ Electric flux passing through ABCD plane surface (shown only AB) = Electric flux passing through cylindrical surface ANB
$$ = \left( {{{60^\circ } \over {360^\circ }}} \right)({\phi _1}) = {{\lambda L} \over {6{\varepsilon _0}}}$$
$$\therefore$$ n = 6
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