JEE Advanced · 2015 · 24 May 2015 · Paper 1 · Electrostatics

An infinitely long uniform line charge distribution of charge per unit length

λ

lies parallel to the

y

-axis in the

y - z

plane at

z = \frac{3}{2} a

(see figure). If the magnitude of the flux of the electric field through the rectangular surface

A B C D

lying in the

x - y

plane with its centre at the origin is

\frac{λ L}{n ε _{0}}

(

ε_{0}

=permittivity of free space), then the value of

n

Answer: 6

Solution

The line charge lies parallel to the y-axis at

z = \frac{3}{2} a

. Consider a cylindrical Gaussian surface of radius

a

centered on the line charge. The total flux through the complete cylindrical surface (angle

2 π

) is

\frac{Q _{in}}{ε _{0}} = \frac{λ L}{ε _{0}}

. The rectangular surface

A B C D

in the

x - y

plane subtends an angle of

\frac{2 π}{6} = \frac{π}{3}

at the line charge (since the geometry shows the rectangle spans

60°

of the cylindrical surface). Therefore, the flux through the rectangular surface is

\frac{1}{6}

of the total flux:

Φ = \frac{λ L}{6 ε _{0}}

. Hence

n = 6

JEE Advanced · 2015 · 24 May 2015 · Paper 1 · Integer answer