If current I passes through a square-shaped conducting loop of side a, what is the value of the magnetic field at the point of intersection of its two diagonals ?

Explanation

Distacne of the point of intersection of the two diagonals, (O) from each arm of the square [Fig 1.88]. Is `r=(a)/(2)`.
For two extreme point of each side, angle subtended at O, `theta_(1)=theta_(2)=45^(@)`
So, magnetic field at O due to current I through each arm,
`B=(mu_(0))/(4pi)*(I)/(r) (sin theta_(1)+ sin theta_(2))`
`=(mu_(0))/(4pi)*(I)/((a)/(2))(sin 45^(@)+ sin 45^(@))`
`=(mu_(0))/(4pi)*(2I)/(a)*2xx (1)/(sqrt(2)) =(mu_(0))/(4pi) (2sqrt(2)I)/(a)`

The direction of the magnetic field at the point O due to current I in each arm is the same.
Hene the resultant magnetic field at the point O
`=4B=(mu_(0))/(4pi)*(8 sqrt(2I))/(a)`