Ampere’s Law allows physicists to determine the magnitude and direction of the magnetic field induced by an electric current. Understanding Ampere’s Law. Electric Currents, Induced Magnetic Fields, & Current in Closed Loop.

Ampere’s law is a mathematical relationship between the amount of magnetic field around a closed loop and the total amount of electric current enclosed by the loop. It is somewhat analogous to **Gauss**‘s law, relating the total electric flux over a closed surface to the total charge enclosed by the surface.

## Origin of Ampere’s Law

In 1820 **Oersted** **discovered** that **electric currents** can induce **magnetic fields**. Several weeks later **Andre Ampere** went to a talk in Paris reporting Oersted’s discovery. **Ampere** began doing detailed experiments to investigate the nature of these induced magnetic fields and their relationship to the electric currents.

## Right Hand Rule and Magnetic Field Direction

Ampere discovered that the induced magnetic field pointed in a direction perpendicular to the electric current. The magnetic field lines formed circles around the wire carrying the electric current. To find the direction of the magnetic field lines use the right hand. Point the thumb along the wire in the direction that the electric current is traveling. Curl the fingers in a circle around the wire. If the thumb points in the direction of the electric current, the fingers will point in the direction of the magnetic field around the wire.

The fact that the magnetic field lines are counter intuitively perpendicular to the inducing current may have contributed to the difficulty 19th century physicists had in discovering the connection between electric and magnetic phenomena.

## Mathematics of Ampere’s Law

If we consider any arbitrarily shaped closed loop then the line integral of the parallel component of the magnetic field over the entire loop equals the magnetic permeability multiplied by the total electric current enclosed by the closed loop. The equation is given in the figure.

To apply Ampere’s law without calculus we need to divide the closed loop into discrete sections over which the magnetic field is constant. Then find the sum for all sections of the component of the magnetic field parallel to the section multiplied by the length of the section. This sum then equals the magnetic permeability multiplied by the total electric current enclosed by the loop.

## Meaning of Ampere’s Law

Ampere’s law is especially useful when symmetries in the problem make it unnecessary to actually do the integral. The magnetic field depends only on the amount of current enclosed by the loop and not on how that current is distributed. For example if an electric current is flowing in a wire, the magnetic field outside the wire will depend on the amount of current but not on the diameter of the wire.

If there is an electric current flowing through a hollow pipe, it will induce a magnetic field outside the pipe. However the magnetic field inside the pipe will be zero because a closed loop just inside the pipe will not have any current flowing through it. This situation is similar to a Faraday cage where the electric field inside a hollow conducting shell is zero.

Ampere’s law tells us that an electric current will induce a magnetic field and provides a mathematical formula for calculating the magnetic field from the total electric current flowing through a closed loop.