Curl Operator/Examples/Magnetic Field of Conductor

Example of Curl Operator

Consider a conductor of electricity $C$.

Let $C$ be carrying a steady current $I$.

Let $P$ be an arbitrary point in the magnetic field $\mathbf H$ induced by $I$.

Let a small plane surface be placed at $P$, turned into a position so that the contour integral of the magnetic force taken around its boundary is the greatest possible.

This value per unit area is the curl of $\mathbf H$.

This is the magneto-motive force per unit area at $P$.

If $P$ is within $C$ at the point where current density is $\mathbf J$, this will be the total current passing normally through the closed curve when the contour integral is greatest.

We have from the Ampère-Maxwell Law that:

$\curl \mathbf H = \mathbf J$

For a point in the magnetic field outside the conductor there is no current density and so:

$\curl \mathbf H = 0$

Sources

• 1951: B. Hague: An Introduction to Vector Analysis (5th ed.) ... (previous) ... (next): Chapter $\text {IV}$: The Operator $\nabla$ and its Uses: $5$. Simple Examples of Curl