Find the magnetic field B at the centre of a rectangular loop of length l and width b, carrying a current i.

Explanation

B at P due to AB, `=(mu_oi)/(4pi((b)/(2))). (2l)/(sqrt(l^2 + 4((b)/(2)))^2) = (mu_oi)/(pi b) . (l)/(sqrtl^2 +b^2)`
B at due to BC,
`=(mu_oi)/(4pi((l)/(2))). (2b)/(sqrt(b^2 + l^2)) = (mu_oi)/(pil) . (l)/(sqrtl^2 +b^2)`
B at P due to CD, `=(mu_oi)/(4pi((b)/(2))). (2l)/(sqrt(l^2 + b^2)) = (mu_oi)/(pib) . (b)/(sqrtl^2 +b^2) `
B at P due to DA,
`=(mu_oi)/(4pi((b)/(2))). (2l)/(sqrt(l^2 + b^2)) = (mu_oi)/(pib) . (b)/(sqrtl^2 +b^2)`
So, net B = `=2[(mu_oi)/(pib). (l)/(sqrt(l^2 + b^2)) = (mu_oi)/(pib) . (b)/(sqrtl^2 +b^2)]`
`=2 (mu_oi)/pisqrt(l^2 + b^2) [(l)/(b) + (b)/(l)]`
`=(2mu_oi)/(pisqrtl^2 + b^2) (b^2+ l^2)/(bl)`
`=(2mu_oi sqrtb^2 + l^2)/(pibl).`