Poisson's Differential Equation for Rotational and Solenoidal Field

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Theorem

Let $R$ be a region of ordinary space.

Let $\mathbf V$ be a vector field over $R$.

Let $\mathbf V$ be both rotational and solenoidal.


Let $\mathbf A$ be a vector field such that $\mathbf V = \curl \mathbf A$.


Then $\mathbf V$ satisfies this version of Poisson's differential equation:

$\curl \mathbf V = -\nabla^2 \mathbf A \ne \bszero$


Proof

As $\mathbf V$ is rotational it is not conservative.

Hence from Vector Field is Expressible as Gradient of Scalar Field iff Conservative $\mathbf V$ cannot be the gradient of some scalar field.

However, by definition of rotational vector field:

$\curl \mathbf V \ne \bszero$

As $\mathbf V$ is solenoidal:

$\operatorname {div} \mathbf V = 0$

Hence from Divergence of Curl is Zero, for some vector field $\mathbf A$ over $R$:

$\operatorname {div} \mathbf V = \operatorname {div} \curl \mathbf A = 0$

and so:

$\mathbf V = \curl \mathbf A$

Then we have:

\(\ds \curl \mathbf V\) \(=\) \(\ds \curl \curl \mathbf A\)
\(\ds \) \(=\) \(\ds \grad \operatorname {div} \mathbf A - \nabla^2 \mathbf A\) Curl of Curl is Gradient of Divergence minus Laplacian
\(\ds \) \(=\) \(\ds -\nabla^2 \mathbf A\) Definition of Solenoidal Vector Field: setting $\operatorname {div} \mathbf A = 0$

$\blacksquare$


Examples

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:

We have from Ampère's Law that:

$\curl \mathbf H = \mathbf J$

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.


Sources