EM , magnetic field at centre of a carrying circular loop

The question is to find the magnetic field at the centre of a current carrying circular loop of radius R, where the current = I

Okay so I’m trying to do this by both Amp’s Law and Biot Savarts Law, and I can’t get my answers to agree.

First method – Biot Savarts Law:

B=[itex]\frac{I\mu_{0}}{4\pi}[/itex][itex]\int[/itex][itex]\frac{dl X \hat{n}}{n^{2}}[/itex]


dl[itex]X[/itex][itex]\hat{n}[/itex]=dl (as |n|=R is always perpendicular to a given line element dl)

=> B=[itex]\frac{I\mu_{0}}{4\pi}[/itex][itex]\int[/itex][itex]\frac{dl X \hat{n}}{n^{2}}[/itex]=[itex]\frac{I\mu_{0}}{4R^{2}\pi}[/itex][itex]\int[/itex]dl=[itex]\frac{I\mu_{0}}{4R^{2}\pi}[/itex]2R[itex]\pi[/itex]=[itex]\frac{\mu_{0}I}{2R}[/itex]

Second method – Ampere’s Law:


So B[itex]\oint[/itex]dl=[itex]\mu_{0}[/itex]I


=> B=[itex]\frac{\mu_{0}I}{2πR}[/itex]

Thanks in advance.


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