Poisson's Differential Equation for Rotational and Solenoidal Field/Examples

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

Magnetic Field in Conductor carrying Steady Current

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:

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

Rotational Motion of Incompressible Fluid

Let $B$ be a body of incompressible fluid.

Let $B$ be undergoing rotational motion.

Let $\mathbf V$ be the vector field which describes the rotational motion of $B$.

Then $\mathbf V$ satisfies Poisson's Differential Equation for Rotational and Solenoidal Field.