91影视

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.

The equation 5.65 is expressed as:

$A\left(r\right)=\frac{{渭}_0}{4蟺}鈭�\frac{J(蟺\text{'})}{蟺}d蟿\text{'}$

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

The above equation can be written as:

$鈭�路A=鈭�鈭�路\left(\frac{J}{\mathrm{蟺}}\right)d蟿\text{'}$

The equation 5.63 is expressed as:

$鈭�路A=0$

The above equation can be expressed as:

$鈭�路\left(\frac{J}{\mathrm{蟺}}\right)=\frac{1}{\mathrm{蟺}}\left(鈭�路J\right)+J路鈭�\left(\frac{1}{\mathrm{蟺}}\right)$ 鈥� (i)

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

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

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

$鈭�路\left(\frac{J}{\mathrm{蟺}}\right)=-J路鈭�\text{'}\left(\frac{J}{\mathrm{蟺}}\right)$

But $鈭�\text{'}路\left(\frac{J}{\mathrm{蟺}}\right)=\left(\frac{1}{\mathrm{蟺}}\right)\left(J路鈭�\text{'}\right)-J路鈭�\text{'}\left(\frac{1}{\mathrm{蟺}}\right)$and $鈭�\text{'}路J=0$, hence,

$鈭�路\left(\frac{J}{\mathrm{蟺}}\right)=-鈭�\text{'}\left(\frac{J}{\mathrm{蟺}}\right)$

According to the divergence theorem,

$\begin{array}{rcl}鈭�路A& =& -\frac{{渭}_0}{4蟺}鈭�鈭�\text{'}路\left(\frac{J}{\mathrm{蟺}}\right)d蟿\text{'}\\ & =& -\frac{{渭}_0}{4蟺}鈭�\frac{J}{\mathrm{蟺}}路da\text{'}\end{array}$

Hence, as J = 0 in the surface, then $鈭�路A=0$and $鈭�路J=0$

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

The equation 5.47 is expressed as:

$B\left(r\right)=\frac{{渭}_0}{4蟺}鈭�\frac{J(r\text{'})脳\widehat{\mathrm{蟺}}}{{\mathrm{蟺}}^2}d蟿\text{'}$

The above equation can be expressed as:

$\begin{array}{rcl}鈭�脳A& =& \frac{{渭}_0}{4蟺}鈭�鈭�脳\left(\frac{J}{\mathrm{蟺}}\right)d蟿\text{'}\\ & =& \frac{{渭}_0}{4蟺}鈭�\left[\frac{1}{\mathrm{蟺}}\left(鈭�脳J\right)-J脳鈭�\left(\frac{J}{\mathrm{蟺}}\right)\right]d蟿\text{'}\end{array}$

Substitute$鈭�脳J=0$ and${鈭�}^2\left(\frac{1}{\mathrm{蟺}}\right)=-\frac{\hat{x}}{{\mathrm{蟺}}^2}$ in the above equation.

$\begin{array}{rcl}鈭�脳A& =& \frac{{渭}_0}{4蟺}鈭�\frac{J脳\widehat{\mathrm{蟺}}}{{\mathrm{蟺}}^2}d蟿\text{'}\\ & =& B\end{array}$

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

The equation 5.64 is expressed as:

$鈭�2^A=-{渭}_{0}J$

The equation 5.65 can be expressed as:

$\begin{array}{rcl}{鈭�}^{2}A& =& \frac{{渭}_0}{4蟺}鈭�{鈭�}^2\left(\frac{J}{\mathrm{蟺}}\right)d蟿\text{'}\\ {鈭�}^2\left(\frac{J}{\mathrm{蟺}}\right)& =& J{鈭�}^2\left(\frac{1}{\mathrm{蟺}}\right)\end{array}$

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

${鈭�}^2\left(\frac{1}{\mathrm{蟺}}\right)=-4{\mathrm{蟺味}}^3\left(\mathrm{蟺}\right)$

The above equation can be expressed as:

$\begin{array}{rcl}{鈭�}^{2}A& =& \frac{{渭}_0}{4蟺}鈭�J(r\text{'})\left[-4{\mathrm{蟺味}}^3\left(\mathrm{蟺}\right)\right]d蟿\text{'}\\ & =& -{渭}_{0}J\left(r\right)\end{array}$

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