📂Electrodynamics

Ampère's Law and Applications

Formulas

The magnetic field $\mathbf{B}$ that arises from the volume current density $\mathbf{J}$ rotates in the direction that satisfies the right-hand rule with the direction of $\mathbf{J}$ as the axis.

$$ \nabla \times \mathbf{B}=\mu_{0} \mathbf{J} $$

Explanation¹

Ampère’s law indicates a special relationship between the current flowing through a conductor and the magnetic field around it. A magnetic field is created around a conductor through which current flows. The direction of this magnetic field follows the ‘right-hand rule’. If you point your right thumb in the direction of the current and curl your fingers, the direction your fingers curl in is the direction of the magnetic field.

Ampère’s law corresponds to Gauss’s law in electrostatics. When there is an equipotential surface, integrating with Gauss’s law simplifies the calculation, allowing the electric field to be easily determined. Similarly, if there is a loop of a constant magnetic field, Ampère’s law simplifies the integration for easily calculating the magnetic field. The rotation of the magnetic field is precisely Ampère’s law in differential form. Integrating both sides over a surface through which the volume current density flows yields

$$ \int (\nabla \times \mathbf{B} ) \cdot d\mathbf{a} = \mu_{0} \int \mathbf{J} \cdot d\mathbf{a} $$

Replacing the left side with Stokes’ theorem yields Ampère’s law in integral form.

$$ \oint \mathbf{B} \cdot d \mathbf{l} = \mu_{0} \int \mathbf{J} \cdot d \mathbf{a} $$

Since the volume current density represents the amount of current passing through a unit area, the integral on the right side represents the total current passing through the said surface. If there is a loop of a constant magnetic field, then $\mathbf{B}$ on the left side remains constant regardless of the integration path, thus allowing the integral to be simplified to an integration over the boundary.

$$ |\mathbf{B}| \oint dl = \mu_{0} I_{in} $$

This is precisely an application of Ampère’s law. Naturally, the sign (direction) of the current and the loop is determined by the right-hand rule. $+$ When the direction of the current is the same as the direction of the right thumb, the direction in which the right hand curls is the direction in which the loop rotates. An important point is that Ampère’s law is only valid for steady currents $I$.

Example

Calculate the magnetic field at a distance $s$ from a long straight conductor carrying a steady current $I$

As shown in the figure, if an Amperian loop is taken, the magnitude of the magnetic field is constant everywhere on the loop. Applying Ampère’s law gives

$$ \oint \mathbf{B} \cdot d \mathbf{l} = B \oint dl = B2\pi s=\mu_{0} I_{in}=\mu_{0} I $$

Therefore,

$$ B=\dfrac{\mu_{0} I}{2 \pi s } $$

Setting the direction of the current flow as direction $\mathbf{z}$ and using the cylindrical coordinate system to also represent the direction of the magnetic field gives

$$ \mathbf{B} =\dfrac{\mu_{0} I}{2 \pi s } \hat{\boldsymbol{\phi}} $$