Two identical wires `A and B` , each of length 'l', carry the same current `I`. Wire A is bent into a circle of radius `R and wire B` is bent to form a square of side 'a' . If ` B_(A) and B_(B)` are the values of magnetic field at the centres of the circle and square respectively , then the ratio `(B_(A))/(B_(B))` is :

Options

A. `(pi^(2))/8`
B. `(pi^(2))/(16sqrt(2))`
C. `(pi^(2))/16`
D. `(pi^(2))/(8sqrt(2))`

Answer

D

Explanation

Wire A is bent into a circle of radius R,
`1=2piRimpliesR=1/(2pi)`
`B_(A)=(mu_(0)I)/(2R)=(mu_(0)I)/(2xx(1/(2pi)))(mu_(0)piI)/1`
Wire B is bent into a square of side a
`1=4aimpliesa=1/4`
`B_(B)=4xx[(mu_(0)I)/(4pi(a//2))(sin 45^(@)+sin 45^(@))]=(2mu_(0)I)/(pia)xx2/(sqrt(2))=(16 mu_(0)I)/(sqrt(2)pi1)`
`:.(B_(A))/(B_(B))=(mu_(0)piI/1)/(16 mu_(0)I//sqrt(2)pi1)=(pi^(2))/(4sqrt(2))`