Derive the expression for the magnetic field due to a current carrying coil of radius r at a distance x from the center along the X-axis.

Consider a conduct element dl of the loop. This is shown above . The magnitude dB of the magnetic field due to dl is given the Biot savant law dB=μ04πI|dl×r|r3r2=x2+R2al→+r→So dB=M_04π I dl r^2μ04π×Idlx2+R2dBy component will cancel each other and all the dBx component will add.sodBx=sdBcosθdBhed=dBx=μ04π×Idl(x2+R2)×R(x2+R2)12B=Bx↑=∫(dBx)=μ0I(2πR)×R4π×(x2+R2)32B=μ0IR22(x2+R2)32↑If we need to find field at centerX=0B=μ0I2R↑