Ampere's circuital law states that the line integral of the magnetic field B\mathbf{B}B around any closed loop is equal to the permeability of free space μ0\mu_0 μ0 times the total electric current III enclosed by the loop. Mathematically, it is expressed as:
∮B⋅dl=μ0I\oint \mathbf{B}\cdot d\mathbf{l}=\mu_0 I∮B⋅dl=μ0I
This law links magnetic fields and electric currents, showing that electric current acts as a source of magnetic fields. The magnetic field around the loop depends directly on the amount of current enclosed. Ampere's law is especially useful for calculating magnetic fields in cases with high symmetry, such as long straight wires, solenoids, and toroids. It applies to steady currents and is a fundamental part of magnetostatics.
Key points about Ampere's circuital law:
- The path of integration (Amperian loop) is a closed loop around the current.
- Only currents passing through the area enclosed by the loop are considered.
- The direction of current and magnetic field matters, with the right-hand rule used for direction.
- It is conceptually similar to Gauss’s law for electric fields but applies to magnetism.
- The law is valid strictly for steady currents; when time-varying electric fields are present, Maxwell added a displacement current correction.
This law provides a practical method to find magnetic fields generated by currents in extremely symmetrical situations by converting the magnetic field calculation into a line integral problem.
