2024-02-29 Magnetic Materials

Diamagnetism Paramagnetism ferromagnetism

The magnetic vector potential $\vec{A}$ of a magnetized object is given by

$$\vec{A}(\vec{r})=\frac{{\mu }_0}{4\pi }{\int }_V\frac{{\vec{j}}_b({\vec{r}}^{\prime })}{|\vec{r}-{\vec{r}}^{\prime }}d{V}^{\prime }+\frac{{\mu }_0}{4\pi }\oint \frac{k_b({\vec{r}}^{\prime })}{|\vec{r}-{\vec{r}}^{\prime }|}d{a}^{\prime }$$

$$\mathbf{B}=\nabla \times \mathbf{A}$$ $$\nabla \times \mathbf{B}={\mu }_0\mathbf{J}$$ $$\nabla \times \mathbf{A}=\left[\frac{1}{s}\frac{\partial A_z}{\partial \varphi }-\frac{\partial A_{\varphi }}{\partial z}\right]\hat{s}+\left[\frac{\partial A_s}{\partial z}-\frac{\partial A_z}{\partial s}\right]\hat{\varphi }+\frac{1}{s}\left[\frac{\partial }{\partial s}(s{A}_{\varphi })-\frac{\partial A_s}{\partial \varphi }\right]\hat{z}$$ $$k=A_{\varphi }$$ $$\nabla \times \mathbf{A}=\left[\frac{1}{s}(0)-0\right]\hat{s}+0+\frac{1}{s}\left[\frac{\partial }{\partial s}(sk)-0\right]\hat{z}=\frac{k}{s}\hat{z}$$ $$\mathbf{B}=\frac{k}{s}\hat{z}={\mathbf{B}}_z$$ $$\nabla \times \mathbf{B}={\mu }_0\mathbf{J}$$ $$\nabla \times \mathbf{B}=\cdots +\left[\frac{\partial {\mathbf{B}}_{\mathbf{s}}}{\partial z}-\frac{\partial {\mathbf{B}}_{\mathbf{z}}}{\partial s}\right]\hat{\varphi }+\cdots =\left[0+\frac{k}{s^2}\right]\hat{\varphi }=\frac{k}{s^2}\hat{\varphi }=\mathbf{J}{\mu }_0$$ $$\mathbf{J}=\frac{k}{s^2{\mu }_0}\hat{\varphi }$$