91Ó°ÊÓ

Magnetostatics is described as the subfield of electromagnetics that describes a static field of the magnet. Moreover, magnetostatics also appears around the magnetized bodies’ surface.

The equation 5.78 is expressed as:

$\frac{∂A_{above}}{∂n}-\frac{∂A_{below}}{∂n}=-{\mathrm{μ}}_{0}K$

Here, $\frac{∂A_{above}}{∂n}$ is the derivative of the potential of $A_{above}$, $\frac{∂A_{below}}{∂n}$ is the derivative of the potential of $A_{below}$, ${\mathrm{μ}}_0$ is the permeability and K is the constant.

The equation 5.77 is expressed as:

$A_{above}=A_{below}$

The equation 5.76 is expressed as:

$B_{above}-B_{below}={\mathrm{μ}}_0(K×\hat{n})$ …(¾±)

Here, $B_{above}$ is one component of the magnetic field, $\hat{n}$ is the position vector and role="math" localid="1657534284873" $B_{below}$ is another component of the magnetic field.

The equation 5.63 is expressed as:

$∇â‹\dots A=0$

Here, $∇$ is the curl.

As $A_{above}=A_{below}$, then the value of $\frac{∂a}{∂y}$ and $\frac{∂a}{∂x}$ are also same in below and also in above.

The equation of the magnetic field can be expressed as:

$B_{above}-B_{below}=\left(-\frac{∂{\mathrm{A}}_{\mathrm{y}\mathrm{above}}}{∂\mathrm{z}}+\frac{∂{\mathrm{A}}_{\mathrm{y}\mathrm{below}}}{∂\mathrm{z}}\right)\hat{x}+\left(-\frac{∂{\mathrm{A}}_{\mathrm{x}\mathrm{above}}}{∂\mathrm{z}}+\frac{∂{\mathrm{A}}_{\mathrm{x}\mathrm{below}}}{∂\mathrm{z}}\right)\hat{\mathrm{y}}$ …(¾±¾±¾±)

Comparing the equation (ii) and (iii).

role="math" localid="1657533984204" $\left(-\frac{∂{\mathrm{A}}_{\mathrm{y}\mathrm{above}}}{∂\mathrm{z}}+\frac{∂{\mathrm{A}}_{\mathrm{y}\mathrm{below}}}{∂\mathrm{z}}\right)\hat{\mathrm{x}}+\left(-\frac{∂{\mathrm{A}}_{\mathrm{x}\mathrm{above}}}{∂\mathrm{z}}+\frac{∂{\mathrm{A}}_{\mathrm{x}\mathrm{below}}}{∂\mathrm{z}}\right)\hat{\mathrm{y}}={\mathrm{μ}}_0(K×\hat{n})$

As the equation 5.76 is along the axis, then the above equation can be expressed as:

role="math" localid="1657534042142" $\left(-\frac{∂{\mathrm{A}}_{\mathrm{y}\mathrm{above}}}{∂\mathrm{z}}+\frac{∂{\mathrm{A}}_{\mathrm{y}\mathrm{below}}}{∂\mathrm{z}}\hat{\mathrm{x}}\right)+\left(-\frac{∂{\mathrm{A}}_{\mathrm{x}\mathrm{above}}}{∂\mathrm{z}}+\frac{∂{\mathrm{A}}_{\mathrm{x}\mathrm{below}}}{∂\mathrm{z}}\right)\hat{\mathrm{y}}={\mathrm{μ}}_0\mathrm{K}(-\hat{\mathrm{y}})$

Due to the x and the y components, the above equation can be reduced to two values such as:

$$\begin{gathered} \left(-\frac{∂A_{yabove}}{∂z}-\frac{∂{A_y}_{below}}{∂z}\right)\hat{x}=\hat{x} \\ \frac{∂A_{yabove}}{∂z}-\frac{∂{A_y}_{below}}{∂z} \\ \left(-\frac{∂A_{xabove}}{∂z}-\frac{∂{A_x}_{below}}{∂z}\right)\hat{y}=-{\mathrm{μ}}_{0}K\hat{y} \\ \left(\frac{∂A_{xabove}}{∂z}-\frac{∂A_{x_{}below}}{∂z}\right)=-{\mathrm{μ}}_{0}K \end{gathered}$$

According to the equation 5.63, the normal derivative of the A component is parallel to the product of the permeability and constant K .

Hence, the above equation can be expressed as:

$\frac{∂A_{above}}{∂n}-\frac{∂A_{below}}{∂n}=-{\mathrm{μ}}_{0}K$

Thus, the equation 5.78 is proved.