In the loop shown , the magnetic induction at the point 'O' is

Options

A. `(mu_(0) I)/(8) ((R_(1) - R_(2))/(R_(1) R_(2)))`
B. `(mu_(0) I)/(8) ((R_1 + R_2)/(R_1 R_2))`
C. `(mu_(0) I)/(8) ((R_(1) R_(2))/(R_(1) + R_(2)))`
D. zero

Answer

B

Explanation

Since the point O lies on the lines of the straight wires , they contribute no magnetic induction at O . The magnetic induction at O due to arcs only . The magnetic induction at O due to arc of radius `R_(1)` is `B_(1) = (mu_(0))/(4 pi) (I)/(R_(1)) (pi)/(2) = (mu_(0) I)/(8 R_(1)) ox`
The magnetic filed at O due to arc of radius `R_(2)` is `B_(2) = (mu_(0))/(4pi) (I)/(R_(2)) (pi)/(2) = (mu_(0) I)/(8 R_(2)) ox`
The net magnetic field at O is `B = B_(1) + B_(2)`
`B = (mu_(0) I)/( 8 R_(1)) + (mu_(0) I)/(8 R_(2))`
`= (mu_(0)I)/(8) ((1)/(R_(1)) + (1)/(R_(2))) = (mu_(0))/(8) ((R_(2) + R_(1))/(R_(1) R_(2)))`