Question

Calculate current density J in the region where (x²+y²)<<d₂

Where the vector potential A is given as

Hol %3D A = - Td2 ((1/a)y² – ax?) nd2

Answer 1

The relation between vector potential and the current density is given by

                                 

Ã= -HoJ

                        or,    $\triangledown (\triangledown . \vec{A}) - \triangledown \times (\triangledown \times \vec{A})= - \mu _{0} \vec{J}$

Putting the value of A in left-hand side of the equation, we get

                               $L.H.S =\triangledown (\triangledown . \vec{A}) - \triangledown \times (\triangledown \times \vec{A})$

going to calculate all the terms separately:

  $\triangledown . \vec{A} = \left ( \hat{x} .\frac{\partial }{\partial x} + \hat{y} .\frac{\partial }{\partial y} + \hat{z} .\frac{\partial }{\partial z}\right ).\left ( \hat{z} \frac{\mu _{0} I}{\pi d^{2}} ((1/\alpha) y^{2} - \alpha x^{2}) \right ) = 0$

and

                 $\triangledown \times \vec{A} =\begin{vmatrix} \hat{x} & \hat{y} &\hat{z} \\ \frac{\partial }{\partial x}&\frac{\partial }{\partial y} &\frac{\partial }{\partial z} \\ 0 &0 & \frac{\mu _{0} I}{\pi d^{2}} ((1/\alpha) y^{2} - \alpha x^{2}) \end{vmatrix} = \frac{2 \mu _{0} I}{\pi d^{2}} \left ( \hat{x} . \frac{y}{\alpha} +\hat{y} . \alpha x \right )$

and

                  $\triangledown \times\left ( \triangledown \times \vec{A} \right )=\begin{vmatrix} \hat{x} & \hat{y} &\hat{z} \\ \frac{\partial }{\partial x}&\frac{\partial }{\partial y} &\frac{\partial }{\partial z} \\ \frac{2y \mu _{0} I}{\pi \alpha d^{2}} & \frac{2x \alpha \mu _{0} I}{\pi d^{2}} &0 \end{vmatrix} =\hat{z}. \frac{2 \mu _{0} I}{\pi d^{2}} \left ( \alpha - \frac{1}{\alpha} \right )$

So,

              $\small L.H.S =\triangledown (\triangledown . \vec{A}) - \triangledown \times (\triangledown \times \vec{A}) = 0 - \hat{z} \frac{2 \mu_{0}I}{\pi d^{2}}\left ( \alpha - \frac{1}{\alpha} \right )= \hat{z} \frac{2 \mu_{0}I}{\pi d^{2}}\left ( \frac{1}{\alpha} - \alpha \right )$

So, the current density will be

                 $\small \vec{J }= -\frac{L.H.S}{\mu_{0}} = \hat{z} \frac{2 \mu_{0}I}{\pi d^{2}}\left ( \alpha - \frac{1}{\alpha} \right )$