Poisson's Differential Equation for Rotational and Solenoidal Field/Examples/Magnetic Field in Conductor carrying Steady Current

Example of Poisson's Differential Equation for Rotational and Solenoidal Field

Consider a conductor of electricity $C$.

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

From Curl Operator: Magnetic Field of Conductor, the curl of the magneto-motive force per unit area $\mathbf H$ is given by:

$\curl \mathbf H = \mathbf J$

Hence this satisfies Poisson's Differential Equation for Rotational and Solenoidal Field:

$\curl \mathbf J = -\nabla^2 \mathbf J$

Sources

• 1951: B. Hague: An Introduction to Vector Analysis (5th ed.) ... (previous) ... (next): Chapter $\text {V}$: Further Applications of the Operator $\nabla$: $7$. The Classification of Vector Fields: $\text {(iii)}$