Two indentical wires A and B, each of length I, carryi the same current I. Wire 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, thne the ratio `B_(A)//B_(B)` is

Options

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

Explanation

We have, `I=2pi R or, R =(l)/(2pi)`
Again, `l=4a or, a=(l)/(4)`
Magnetic field a the centre of the circle,
`B_(A)=(mu_(0)I)/(2R)=(mu_(0)piI)/(l)`
Magnetic field at the centre of the square,
`B_(B)=4xx (mu_(0)I)/(4pixx(a)/(2))(sin45^(@)+sin45^(@))`
`=(mu_(0)I)/((pil)/(8))xx(2)/(sqrt(2))=(16mu_(0)I)/(pi sqrt(2)l)`
`:. (B_(A))/(B_(B))=(mu_(0)piI)/(l)xx(pi sqrt(2)l)/(16 mu_(0)l)=(pi^(2))/(8 sqrt(2))`
The option D is correct.