A straight wire of length L is bent into a semicircular loop. Use Biot-Savart's to deduce an expression for the magnetic field at its centre due to the curretn I passing through it.

Explanation

As shown in fig. consider a wire of length L bent into a semicular loop of radius `R (R = L/(pi))`. Let a current I flows through it. To calculate magnetic field at centre O of the loop, Let us consider a small curretn Idl. Then as per Biot-Savart.s law
`dB = (mu_0)/(4pi) (I dl sin theta)/(R^2) = (mu_0)/(4 pi) (I dl sin 90^@)/(R^2) = (mu_0 I dl)/(4 pi R^2)`
`:.` Total magnetic field at point O due to entire wire
`B = (mu_0 I)/(4 pi R^2) sum dl = (mu_0 I)/(4 pi R^2) cdot pi R = (mu_0 I)/(4 R) = (mu_0 pi I)/(4L)`