# Ampere's Circuital law proof for closed circular loop

How do we prove the magnetic field for a circular loop is $$\frac{\mu_0I}{2r}$$ using Ampere's Circuital law

I proved it using Biot-Savart Law but i am getting $$\mu_0I/2\pi r$$ instead of $$\frac{\mu_0I}{2r}$$ while using Ampere's law.

Please help me where I am doing wrong

## 1 Answer

You probably misapplied Ampere's law. This law is usually used to find magnetic field only in special cases when the contour integral can be found as a function of single field value based on symmetry.

Magnetic field of a circular current loop is not so simple and Ampere's law cannot be easily used to find it. In such cases, the method of choice is to use the Biot-Savart law (integrate the contributions to the field due to elements of the circuit) or find vector potential as a function of position and then derive magnetic field from it.

• I understand that the magnetic field is complicated but i would still like a rigorous mathematical proof Nov 16, 2018 at 15:07
• I'll try brother Nov 16, 2018 at 15:44