Question

Prove the following uniqueness theorem: If the current density J isspecified throughout a volume V ,and eitherthe potential A orthe magnetic field B is specified on the surface Sbounding V,then the magnetic field itself is uniquely determined throughout V.[Hint:First use the divergence theorem to show that

$\mathbf{\int }\mathbf{[}\left(\stackrel{\rightharpoonup }{\nabla }\times \stackrel{\rightharpoonup }{U}\right)\mathbf{.}\left(\stackrel{\rightharpoonup }{\nabla }\times \stackrel{\rightharpoonup }{V}\right)\mathbf{-}\mathbf{U}\mathbf{.}\left(\stackrel{\rightharpoonup }{\nabla }\times \stackrel{\rightharpoonup }{\nabla }\times \stackrel{\rightharpoonup }{\nabla }\right)\mathbf{]}\mathit{d}\mathit{r}\mathbf{=}\mathbf{\oint }\mathbf{}\mathbf{(}\stackrel{\mathbf{\rightharpoonup }}{\mathbf{U}}\mathbf{\times }\stackrel{\mathbf{\rightharpoonup }}{\mathbf{}\mathbf{\nabla }}\mathbf{\times }\stackrel{\mathbf{\rightharpoonup }}{\mathbf{V}}\mathbf{)}\mathit{d}\stackrel{\mathbf{\rightharpoonup }}{\mathbf{a}}$

for arbitrary vector functions $\stackrel{\rightharpoonup }{\mathit{U}}$and $\stackrel{\mathbf{\rightharpoonup }}{\mathbf{V}}$ ]

Short answer

It is proved If the current density is specified throughout a volume and eitherthe magnetic vector potential orthe magnetic field is specified on the surface bounding the volumethen the magnetic field itself is uniquely determined throughout throughout the volume.

Step-by-step solution

Given data

The specified current density is $\stackrel{\rightharpoonup }{J}$.

To prove,

$\int \left[\left(\stackrel{\rightharpoonup }{\nabla }\times \stackrel{\rightharpoonup }{U}\right).\left(\stackrel{\rightharpoonup }{\nabla }\times \stackrel{\rightharpoonup }{V}\right)-\stackrel{\rightharpoonup }{U}.\left(\stackrel{\rightharpoonup }{\nabla }\times \stackrel{\rightharpoonup }{\nabla }\times \stackrel{\rightharpoonup }{V}\right)\right]d\tau =\oint \left(\stackrel{\rightharpoonup }{U}\times \stackrel{\rightharpoonup }{\nabla }\times \stackrel{\rightharpoonup }{V}\right).d\stackrel{\rightharpoonup }{a}$

Vector product, divergence theorem and curl of magnetic field

Vector product rule of two arbitrary vector functions $\stackrel{\rightharpoonup }{U}$and $\stackrel{\rightharpoonup }{V}$is

localid="1658559820207" $\stackrel{\mathbf{\rightharpoonup }}{\mathbf{\nabla }}\mathbf{.}\mathbf{(}\stackrel{\mathbf{\rightharpoonup }}{\mathbf{U}}\mathbf{\times }\stackrel{\mathbf{\rightharpoonup }}{\mathbf{\nabla }}\mathbf{\times }\stackrel{\mathbf{\rightharpoonup }}{\mathbf{V}}\mathbf{)}\mathbf{=}\mathbf{(}\stackrel{\mathbf{\rightharpoonup }}{\mathbf{\nabla }}\mathbf{\times }\stackrel{\mathbf{\rightharpoonup }}{\mathbf{V}}\mathbf{)}\mathbf{.}\mathbf{(}\stackrel{\mathbf{\rightharpoonup }}{\mathbf{\nabla }}\mathbf{\times }\stackrel{\mathbf{\rightharpoonup }}{\mathbf{U}}\mathbf{)}\mathbf{.}\mathit{U}\mathbf{.}\mathbf{(}\stackrel{\mathbf{\rightharpoonup }}{\mathbf{\nabla }}\mathbf{\times }\stackrel{\mathbf{\rightharpoonup }}{\mathbf{\nabla }}\mathbf{\times }\stackrel{\mathbf{\rightharpoonup }}{\mathbf{V}}\mathbf{)}$

According to divergence theorem, the volume integral of the divergence of a vector function is

localid="1658559830383" $\mathbf{\int }\mathbf{\nabla }\mathbf{.}\stackrel{\mathbf{\rightharpoonup }}{\mathbf{U}}\mathit{d}\mathit{\tau }\mathbf{=}\mathbf{\oint }\stackrel{\mathbf{\rightharpoonup }}{\mathbf{U}}\mathbf{.}\mathit{d}\stackrel{\mathbf{\rightharpoonup }}{\mathbf{a}}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{\left(}\mathbf{3}\mathbf{\right)}$

The curl of the magnetic field is

localid="1658559839318" $\stackrel{\mathbf{\rightharpoonup }}{\mathbf{\nabla }}\mathbf{\times }\stackrel{\mathbf{\rightharpoonup }}{\mathbf{B}}\mathbf{=}{\mathit{\mu }}_{\mathbf{0}}\stackrel{\mathbf{\rightharpoonup }}{\mathbf{J}}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{\left(}\mathbf{4}\mathbf{\right)}$

Here, ${\mu }_0$is the permeability of free space.

Proof of continuity equation

Take volume integral of both sides of equation (2) and use equation (3) on the left hand side to get,

$\oint \left(\stackrel{\rightharpoonup }{U}\times \stackrel{\rightharpoonup }{\nabla }\times \stackrel{\rightharpoonup }{V}\right).d\stackrel{\rightharpoonup }{a}=\int \left(\stackrel{\rightharpoonup }{\nabla }\times \stackrel{\rightharpoonup }{V}\right).\left(\stackrel{\rightharpoonup }{\nabla }\times \stackrel{\rightharpoonup }{U}\right)-\stackrel{\rightharpoonup }{U}\left(\stackrel{\rightharpoonup }{\nabla }\times \stackrel{\rightharpoonup }{\nabla }\times \stackrel{\rightharpoonup }{V}\right)d\tau$

Assume that there are two values of magnetic fields ${\stackrel{\rightharpoonup }{B}}_1$ and role="math" localid="1657773504274" ${\stackrel{\rightharpoonup }{\mathrm{B}}}_2$ and two corresponding magnetic vector potentials ${\stackrel{\rightharpoonup }{A}}_1$ and ${\stackrel{\rightharpoonup }{A}}_2$. The difference between the values is defined as

$$\begin{gathered} {\stackrel{\rightharpoonup }{B}}_3={\stackrel{\rightharpoonup }{B}}_1-{\stackrel{\rightharpoonup }{B}}_2 \\ {\stackrel{\rightharpoonup }{A}}_3={\stackrel{\rightharpoonup }{A}}_1-{\stackrel{\rightharpoonup }{A}}_2 \\ {\stackrel{\rightharpoonup }{B}}_3=\stackrel{\rightharpoonup }{\nabla }\times {\stackrel{\rightharpoonup }{A}}_3 \end{gathered}$$

Since the current density is uniquely specified, from equation (4),

$$\begin{gathered} \stackrel{\rightharpoonup }{\nabla }\times {\stackrel{\rightharpoonup }{B}}_3=\stackrel{\rightharpoonup }{\nabla }\times {\stackrel{\rightharpoonup }{B}}_1-\stackrel{\rightharpoonup }{\nabla }\times {\stackrel{\rightharpoonup }{B}}_2 \\ ={\mu }_0\stackrel{\rightharpoonup }{J}-{\mu }_0\stackrel{\rightharpoonup }{J} \\ =0 \end{gathered}$$

Set $\stackrel{\rightharpoonup }{U},\stackrel{\rightharpoonup }{V}=\stackrel{\rightharpoonup }{A_3}$in equation (1), to get,

$$\begin{gathered} \int \left[\left(\stackrel{\rightharpoonup }{\nabla }\times {\stackrel{\rightharpoonup }{A}}_3\right).\left(\stackrel{\rightharpoonup }{\nabla }\times {\stackrel{\rightharpoonup }{A}}_3\right)-{\stackrel{\rightharpoonup }{A}}_3\left(\stackrel{\rightharpoonup }{\nabla }\times \overrightarrow{\nabla }\times {\stackrel{\rightharpoonup }{A}}_3\right)\right]d\tau =\oint \left({\stackrel{\rightharpoonup }{A}}_3\times \overrightarrow{\nabla }\times {\stackrel{\rightharpoonup }{A}}_3\right).d\overrightarrow{a} \\ \int {\stackrel{\rightharpoonup }{B}}_3.{\stackrel{\rightharpoonup }{B}}_3-{\stackrel{\rightharpoonup }{A}}_3.\left(\overrightarrow{\nabla }\times {\stackrel{\rightharpoonup }{B}}_3\right)d\tau =\oint \left({\stackrel{\rightharpoonup }{A}}_3\times {\stackrel{\rightharpoonup }{B}}_3\right).da \end{gathered}$$

Use equation (5) to get,

$\int B_3^{2}d\tau =\oint \left({\stackrel{\rightharpoonup }{A}}_3\times {\stackrel{\rightharpoonup }{B}}_3\right).d\stackrel{\rightharpoonup }{a}$

If either magnetic field or the potential is uniquely specified on the surface then either ${\stackrel{\rightharpoonup }{A}}_3=0$ or ${\stackrel{\rightharpoonup }{B}}_3=0$. In both cases the right hand side of the previous equation becomes zero. Hence $B_3=0$ in the volume.

Thus, the magnetic field is uniquely determined in the volume.