Due to flow of current I through a square shaped conducting loop, magnetic field generated at its centre is B. the magnetic field generated at the centre of a circular conducting loop having the same perimeter as that of the square and for the flow of the same current is B'. determine the ratio of B to B'.

Explanation

Magnetic field at the centre of the circular loop of radius r,
`B'=(mu_(0)I)/(2r),` circumference of the circle `2pir`
So, the perimeter of the given, square loop `2pi=r` and hence the lengh of the each side,
`a=(2pir)/(4)=(pir)/(2)`
Magnetic field at the centre of the square loop,
`B=(mu_(0))/(4pi)*(8 sqrt(2)I)/(a)`
`=(mu_(0))/(4pi)*(8 sqrt(2)I)/((pir)/(2))=(4 sqrt(2) mu_(0)I)/(pi^(2)r)= (8sqrt(2))/(pi^(2))*(mu_(0)I)/(2r)=(8 sqrt(2))/(pi^(2))B'`
So, `(B)/(B')=(8 sqrt(2))/(pi^(2))`