Question

(a) Check that Eq. 5.65 is consistent with Eq. 5.63, by applying the divergence.

(b) Check that Eq. 5.65 is consistent with Eq. 5.47, by applying the curl.

(c) Check that Eq. 5.65 is consistent with Eq. 5.64, by applying the Laplacian.

Short answer

(a) The equation 5.65 is consistent with the equation 5.63.

(b) The equation 5.65 is consistent with the equation 5.47.

(c) The equation 5.65 is consistent with the equation 5.64.

Step-by-step solution

Define significance of the curl

The curl inside a vector field mainly describes the tendency of a vector field to swing around. However, the curl mainly describes the rotation inside a particular space.

(a) Check the consistency of the equations 5.65 and 5.63

The equation 5.65 is expressed as:

$A\left(r\right)=\frac{{\mu }_0}{4\pi }\int \frac{J(\pi \text{'})}{\pi }d\tau \text{'}$

Here, A(r) is the vector potential as a function of r and J is the divergence.

The above equation can be written as:

$\nabla ·A=\int \nabla ·\left(\frac{J}{\mathrm{\pi }}\right)d\tau \text{'}$

The equation 5.63 is expressed as:

$\nabla ·A=0$

The above equation can be expressed as:

$\nabla ·\left(\frac{J}{\mathrm{\pi }}\right)=\frac{1}{\mathrm{\pi }}\left(\nabla ·J\right)+J·\nabla \left(\frac{1}{\mathrm{\pi }}\right)$ … (i)

The firm term of this equation is zero as $J\left(\mathrm{\pi }\text{'}\right)$is the source coordinate’s function.

As $\mathrm{\pi }=\mathrm{r}-\mathrm{r}\text{'}$, then $\nabla \left(\frac{1}{\mathrm{\pi }}\right)=-\nabla \text{'}\left(\frac{1}{\mathrm{\pi }}\right)$.

Hence, the equation (i) can be expressed as:

$\nabla ·\left(\frac{J}{\mathrm{\pi }}\right)=-J·\nabla \text{'}\left(\frac{J}{\mathrm{\pi }}\right)$

But $\nabla \text{'}·\left(\frac{J}{\mathrm{\pi }}\right)=\left(\frac{1}{\mathrm{\pi }}\right)\left(J·\nabla \text{'}\right)-J·\nabla \text{'}\left(\frac{1}{\mathrm{\pi }}\right)$and $\nabla \text{'}·J=0$, hence,

$\nabla ·\left(\frac{J}{\mathrm{\pi }}\right)=-\nabla \text{'}\left(\frac{J}{\mathrm{\pi }}\right)$

According to the divergence theorem,

$\begin{array}{rcl}\nabla ·A& =& -\frac{{\mu }_0}{4\pi }\int \nabla \text{'}·\left(\frac{J}{\mathrm{\pi }}\right)d\tau \text{'}\\ & =& -\frac{{\mu }_0}{4\pi }\int \frac{J}{\mathrm{\pi }}·da\text{'}\end{array}$

Hence, as J = 0 in the surface, then $\nabla ·A=0$and $\nabla ·J=0$

Thus, the equation 5.65 is consistent with the equation 5.63.

(b) Check the consistency of the equations 5.65 and 5.47

The equation 5.47 is expressed as:

$B\left(r\right)=\frac{{\mu }_0}{4\pi }\int \frac{J(r\text{'})\times \hat{\mathrm{\pi }}}{{\mathrm{\pi }}^2}d\tau \text{'}$

The above equation can be expressed as:

$\begin{array}{rcl}\nabla \times A& =& \frac{{\mu }_0}{4\pi }\int \nabla \times \left(\frac{J}{\mathrm{\pi }}\right)d\tau \text{'}\\ & =& \frac{{\mu }_0}{4\pi }\int \left[\frac{1}{\mathrm{\pi }}\left(\nabla \times J\right)-J\times \nabla \left(\frac{J}{\mathrm{\pi }}\right)\right]d\tau \text{'}\end{array}$

Substitute$\nabla \times J=0$ and${\nabla }^2\left(\frac{1}{\mathrm{\pi }}\right)=-\frac{\hat{x}}{{\mathrm{\pi }}^2}$ in the above equation.

$\begin{array}{rcl}\nabla \times A& =& \frac{{\mu }_0}{4\pi }\int \frac{J\times \hat{\mathrm{\pi }}}{{\mathrm{\pi }}^2}d\tau \text{'}\\ & =& B\end{array}$

Thus, the equation 5.65 is consistent with the equation 5.47.

(c) Checking the consistency of the equations 5.65 and 5.64

The equation 5.64 is expressed as:

$\nabla 2^A=-{\mu }_{0}J$

The equation 5.65 can be expressed as:

$\begin{array}{rcl}{\nabla }^{2}A& =& \frac{{\mu }_0}{4\pi }\int {\nabla }^2\left(\frac{J}{\mathrm{\pi }}\right)d\tau \text{'}\\ {\nabla }^2\left(\frac{J}{\mathrm{\pi }}\right)& =& J{\nabla }^2\left(\frac{1}{\mathrm{\pi }}\right)\end{array}$

The function J is a constant function as the differentiation is concerned. Hence,

${\nabla }^2\left(\frac{1}{\mathrm{\pi }}\right)=-4{\mathrm{\pi \zeta }}^3\left(\mathrm{\pi }\right)$

The above equation can be expressed as:

$\begin{array}{rcl}{\nabla }^{2}A& =& \frac{{\mu }_0}{4\pi }\int J(r\text{'})\left[-4{\mathrm{\pi \zeta }}^3\left(\mathrm{\pi }\right)\right]d\tau \text{'}\\ & =& -{\mu }_{0}J\left(r\right)\end{array}$

Thus, the equation 5.65 is consistent with the equation 5.64.