A long straight metal rod has a very long hole of radius `'a'` drilled parallel to the rod axis as shown in the figure. If the rod carries a current `I`, find the magnetic field on axis of hole. Given `C` is the centre of the hole and `OC=c`.

Options

A. `(mu_0Ir)/(4pi(r_2^2-r_1^2))`
B. `(mu_0Ir)/(2pi(r_2^2-r_1^2)`
C. `(mu_0Ir)/(pi(r_2^2-r_1^2))`
D. `(mu_0Ir)/(6pi(r_2^2-r_1^2))`

Explanation

Draw a cylindrical gaussian surface whose axis lies at the axis of hole as shown in figure.

Let r be the radius of this gaussian surface.
Area of crosssection of this gaussian surface `=pir^2`
Current threading the gaussian surface
`I^'=(Ixxpir^2)/(pi(r_2^2-r_1^2))=(Ir^2)/((r_2^2-r_1^2))`
Magnetic field induction at point C is
`B=(mu_0)/(4pi)(2I^')/(r)=(mu_0)/(4pi)2/r(Ir^2)/((r_2^2-r_1^2))=(mu_0Ir)/(2pi(r_2^2-r_1^2))`