${\displaystyle I\,}$

${\displaystyle \mathbf {B} \,}$

${\displaystyle \oint _{\partial S}\mathbf {B} \cdot d\mathbf {l} =\mu _{0}\int _{S}(\mathbf {J} +\epsilon _{0}{\frac {\partial \mathbf {E} }{\partial t}})\cdot d\mathbf {S} \,}$

The equation says that the integral of the magnetic field ${\displaystyle \mathbf {B} \,}$ around a loop ${\displaystyle \partial S\,}$ is equal to the current ${\displaystyle \mathbf {J} \,}$ through any surface spanning the loop, plus a term depending on the rate of change of the electric field ${\displaystyle \mathbf {E} \,}$ through the surface. This term, the second term on the right, is the displacement current. For applications with no time varying electric fields (unchanging charge density) it is zero and is ignored. However in applications with time varying fields, such as circuits with capacitors, it is needed, as shown below. Any surface intersecting the wire, such as ${\displaystyle S_{1}\,}$, has current ${\displaystyle I\,}$ passing through it so Ampere's law gives the correct magnetic field:

${\displaystyle B={\frac {\mu _{0}I}{2\pi r}}\,}$

But surface ${\displaystyle S_{2}\,}$ spanning the same loop that passes between the capacitor's plates has no current flowing through it, so without the displacement current term Ampere's law gives:

${\displaystyle B=0\,}$

So without the displacement current term Ampere's law fails; it gives different results depending on which surface is used, which is inconsistent. The 'displacement current' term provides a second source for the magnetic field besides current; the rate of change of the electric field ${\displaystyle \mathbf {E} \,}$. Between the capacitor's plates, the electric field is increasing, so the rate of change of electric field through the surface ${\displaystyle S_{2}\,}$ is positive, and its magnitude gives the correct value for the field field ${\displaystyle \mathbf {B} \,}$ found above.

James Clerk Maxwell added the displacement current term to Ampere's law around 1861.