Ampere's law

Ampère’s law states that the line integral of the magnetic field over a closed loop (called an Amperian loop) is equal to the net current enclosed by the loop, which pierces the surface bounded by the closed loop.

$${\oint }_C\vec{B}\cdot d\vec{l}={\mu }_0{I}_{\text{enc}}={\mu }_0{\iint }_S\vec{J}\cdot d\vec{S}={\iint }_S(\nabla \times \vec{B})\cdot d\vec{S}$$

This answer is independent of the circumference $s$, because $B$ decreases at the same rate that the circumference increases. We get some of these terms by Stokes’ theorem and the definition of current. The differential form of Ampère’s law is given by:

$$\nabla \times \vec{B}={\mu }_0\vec{J}$$

Stokes’ theorem

One application of Ampère’s law is to use Stokes’ theorem to find the magnetic field from the enclosed current. The Amperian loop corresponds to the closed loop in the left-hand side of Stokes’ theorem.

Ampère’s law is always true in magnetostatic conditions, but not always useful. In general, we are able to consider four different types of symmetry:

• Infinite straight line currents
• Infinite plane currents
• Infinite solenoids (coil around a cylinder)
• Toroids (coil around a donut)

The direction of $\vec{B}$ is given by the right-hand curl rule, with thumb pointing in the $\vec{J}$ direction and fingers curling in the $\vec{B}$ direction.

Similar to Gauss’ law, we have a few core steps:

1. Exploit symmetry. Some symmetry formulations will depend only on some variables. We also need to figure out the direction of the field in our steps.
2. Choose an Amperian loop.
3. Compute the LHS and RHS of Stokes’ theorem.
   1. For the RHS of Stokes’, we should be careful with the correctness of the orientation with respect to the loop.
4. Then we evaluate LHS = RHS.

See also

• Gauss’ law, the electric field equivalent