# Ampere's Law

## Table of Contents

$$ \oint \vec{B} \cdot d\vec{l} = \mu_0 I $$

Where

- The left side is a line integral of the magnetic field \(\vec{B}\) over a closed loop \(\vec{l}\), and
- \(\mu_0\) is the permeability of free space, and
- \(I\) is the current enclosed by the loop.

Ampere’s law is a formula that describes the magnetic field produced by a current-carrying wire. It says that the magneitc field created by an electric current is proportional to the magnitude of the current.

## With displacement current #

Ampere’s law has a more general form that includes displacement current. The displacement current is proportional to the rate of change of the electric flux. This site has a good explanation of why this modification is needed. The modified form is:

$$ \oint \vec{B} \cdot d\vec{l} = \mu_0 I + \mu_0 \epsilon_0 \frac{d\vec{\phi_E}}{dt} $$

Where, in addition to the above definitions,

- \(\epsilon_0\) is the permittivity of free space, and
- \(\vec{\phi_E}\) is the electric flux intercepted by the surface spanned by the integration path of the magnetic field.