Question

In the current carrying conductor (AOCDEFG) as shown, the magnetic induction at point O is ______.

(R₁ and R₂ are radii of CD and EF respectively. l = current in the loop, μ₀ = permeability of free space)

 

Options

• `(mu_0l)/8((R_1 + R_2)/(R_1R_2))`

• `(mu_0l)/8((R_1 - R_2)/(R_1R_2))`

• `(mu_0l)/8((R_1R_2)/(R_1 - R_2))`

• `(mu_0l)/8((R_1R_2)/(R_1 + R_2))`

Answer 1

In the current carrying conductor (AOCDEFG) as shown, the magnetic induction at point O is `underlinebb((mu_0l)/8((R_1 + R_2)/(R_1R_2)))`.

Explanation:

The magnetic induction due to an arc of angle θ is given by

B = `mu_0/(4pi)(l/a)theta`

For wires AO, DE and FG, angle θ is 0°, so they do not contribute to magnetic field induction. The arc CD of radius R₁ and arc EF of radius R₂, both subtend an angle of 90° the centre O. So, total magnetic induction at O,

B = `(mu_0l)/(4piR_1). pi/2 + (mu_0l)/(4piR_2). pi/2 = (mu_0l)/(4pi) pi/2(1/R_1 + 1/R_2) = (mu_0l)/8((R_1 + R_2)/(R_1R_2))`