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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

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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.

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  • $\begingroup$ I understand that the magnetic field is complicated but i would still like a rigorous mathematical proof $\endgroup$
    – Sayak Adak
    Nov 16, 2018 at 15:07
  • $\begingroup$ I'll try brother $\endgroup$
    – Aditya Garg
    Nov 16, 2018 at 15:44

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