Ampere-Maxwell Law

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Here is the Ampere-Maxwell law again $$ \oint_C \vec{B} \circ d\vec{l} = \mu_0 \bigg(I + \varepsilon_0 \frac{d}{dt} \int_S \vec{E} \circ \hat{n} dA \bigg) $$ We can convert the Ampere-Maxwell law from integral form to differential form with our definition of curl. $$ (\vec{\nabla} \times \vec{B}) \circ \hat{n} \equiv \lim_{\Delta A \rightarrow 0} \frac{1}{\Delta A} \oint _C \vec{B} \circ d\vec{l}$$ If we substitute it into our definition of curl we get the following $$ (\vec{\nabla} \times \vec{B}) \circ \hat{n} = \lim_{\Delta A \rightarrow 0} \frac{1}{\Delta A} \bigg( \mu_0 I + \mu_0 \varepsilon_0 \frac{d}{dt} \int_S \vec{E} \circ \hat{n} dA \bigg) $$ $$ = \mu_0 \lim_{\Delta A \rightarrow 0} \frac{I}{\Delta A} + \mu_0 \varepsilon_0 \lim_{\Delta A \rightarrow 0} \frac{1}{\Delta A} \frac{d}{dt} \int_S \vec{E} \circ \vec{n} dA $$

The quantity $ \frac{I}{\Delta A} $ in the first limit is the current density across the surface $ \Delta A $ in the $ \hat{n} $ direction. In the limit that the area shrinks to a point it becomes the current density at that point. In the second limit, we can say the area $ \Delta A $ is small enough to make the approximation that $ \vec{E} $ is constant across it,

$$ (\vec{\nabla} \times \vec{B}) \circ \hat{n} = \mu_0 J_n + \mu_0 \varepsilon_0 \lim_{\Delta A \rightarrow 0} \frac{1}{\Delta A} \frac{d}{dt} (\vec{E}\circ \hat{n} \Delta A) $$ $$ = \mu_0 J_n + \mu_0 \varepsilon_0 \frac{d}{dt} \vec{E} \circ \hat{n} $$ If the surface normal $ \hat{n} $ is constant in time, we can bring the differential operator $ \frac{d}{dt} $ inside the dot product $$ (\vec{\nabla} \times \vec{B}) \circ \hat{n} = \mu_0 J_n + \mu_0 \varepsilon_0 \frac{d\vec{E}}{dt} \circ \hat{n} $$ The vector field $ \vec{E} $ is a function of position too, so the derivative becomes a partial derivative. We can also undo the dot product with $ \hat{n} $ giving $$ \vec{\nabla} \times \vec{B} = \mu_0 \bigg(\vec{J} + \varepsilon_0 \frac{\partial \vec{E}}{\partial t} \bigg) $$

Dotting a vector with a unit vector brings out the component that's parallel to the unit vector, i.e. $ \vec{J} \circ \hat{n} = J_n $. We have just reversed this process.

The Ampere-Maxwell law in differential form states that the curl of the magnetic field $ \vec{B} $ at a location is related to the sum of the current density $ \vec{J} $ and time derivative of the electric field $ \vec{E} $ at that location.