# Ampere's Circuital law in non-linear and dispersive medium

For a linear and non-dispersive medium $$\vec{B} = \mu \vec{H}$$.

So amperes circuital law in integral form (without Maxwell's correction term) can be written in two ways $$\oint_C \vec{B}\cdot\vec{dl} = \mu \int_S \vec{J}\cdot \hat{n}dS$$

and

$$\oint_C \vec{H}\cdot\vec{dl} = \int_S \vec{J}\cdot \hat{n}dS$$

Both of them work in linear and non-dispersive medium and we can get one equation from the other using this relation $$\vec{B} = \mu \vec{H}$$.

However, if we're dealing with non-linear and dispersive medium, we can not get one equation from the other. So one of them has to be correct and the other one wrong. In this case which one of them should we use?

• What you mean by non linear? – Jack Rod Aug 3 '20 at 14:26

Ampère's law follows from Maxwell's curl $$\vec{H}$$ equation, so you should use $$\vec{H}$$ in Ampère's law.