91Ó°ÊÓ

The curl is mainly used for quantifying the circulation of a particular electric field. It mainly measures the tendency of a particular fluid that swirls around a point.

The equation of the dot product of the curl and the function A is expressed as:

$∇.A=-\frac{1}{2}∇.\left(\stackrel{→}{r}×\stackrel{→}{B}\right)$

Here,$∇$is the curl, is the function,$\stackrel{→}{r}$is the radius of the magnetic field and $\stackrel{→}{B}$is the magnetic field.

Calculating the above equation

$$\begin{gathered} ∇.A=-\frac{1}{2}\left[\stackrel{→}{B}.\left(∇×\stackrel{→}{r}\right)-\stackrel{→}{r}.\left(∇×\stackrel{→}{B}\right)\right] \\ =0 \end{gathered}$$

The equation of the dot product of the curl and the function A is expressed as:

$∇×A=-\frac{1}{2}∇×\left(\stackrel{→}{r}×\stackrel{→}{B}\right)$

Here,$∇$is the curl,A is the function,$\stackrel{→}{r}$is the radius of the magnetic field and $\stackrel{→}{B}$is the magnetic field.

Calculating the above equation

$∇×A=-\frac{1}{2}\left[\left(\stackrel{→}{B}.∇\right)\stackrel{→}{r}-\left(\stackrel{→}{r}.∇\right)\stackrel{→}{B}+\stackrel{→}{r}\left(∇.\stackrel{→}{B}\right)-\stackrel{→}{B}\left(∇.\stackrel{→}{r}\right)\right]$

Substitute 3 for$∇.\stackrel{→}{r}$and $\stackrel{→}{B}$for$\left(\stackrel{→}{B}.∇\right)\stackrel{→}{r}$ in the above equation.

$$\begin{gathered} ∇×A=-\frac{1}{2}\left[\stackrel{→}{B}-0+0-3\stackrel{→}{B}\right] \\ =\stackrel{→}{B} \end{gathered}$$

The above solution shows that the vector potential does not produce a uniform magnetic field.

The solution is not a unique solution as the vector $\stackrel{→}{r}$can be replaced with$\stackrel{→}{r}+\stackrel{→}{d}$where$\stackrel{→}{d}$is a constant vector and the same result will come.

Thus, $A\left(r\right)=-\frac{1}{2}\left(r×B\right)$works. The solution is not a unique solution as the vector $\stackrel{→}{r}$can be replaced with $\stackrel{→}{r}+\stackrel{→}{d}$where$\stackrel{→}{d}$ is a constant vector and the same result will come.