- #1
Mike400
- 59
- 6
Consider a magnetic dipole distribution in space having magnetization ##\mathbf{M}##. The potential at any point is given by:
##\displaystyle\psi=\dfrac{\mu_0}{4 \pi} \int_{V'} \dfrac{ \rho}{|\mathbf{r}-\mathbf{r'}|} dV' + \dfrac{\mu_0}{4 \pi} \oint_{S'} \dfrac{\sigma}{|\mathbf{r}-\mathbf{r'}|} dS'=\psi^{V}+\psi^{S}##
The ##\mathbf{H}## field is:
##\displaystyle\mathbf{H}=\dfrac{\mu_0}{4 \pi} \int_{V'} \rho \dfrac{\mathbf{r}-\mathbf{r'}}{|\mathbf{r}-\mathbf{r'}|^3} dV' + \dfrac{\mu_0}{4 \pi} \oint_{S'} \sigma \dfrac{ \mathbf{r}-\mathbf{r'}}{|\mathbf{r}-\mathbf{r'}|^3} dS'=\mathbf{H}^{V}+\mathbf{H}^{S}##
The ##\mathbf{B}## field is:
##\mathbf{B}=\mathbf{H} + \mu_0 \mathbf{M}=\mathbf{H}^{V} + \mathbf{H}^{S} + \mu_0 \mathbf{M}##
##\mathbf{H}^{V}## has no discontinuity.
##\mathbf{H}^{S}## has discontinuity of ##\mu_0 \mathbf{M} \cdot \hat{n}## at the surface ##S'##
##\mu_0 \mathbf{M}## has discontinuity of ##\mu_0 \mathbf{M}## at the surface ##S'##
From these knowledge, how shall one deduce that ##\mathbf{B}## is continuous at the surface?
My try: (I am getting a contradiction)
We need to show that ##\mu_0 \mathbf{M} \cdot \hat{n}+\mu_0 \mathbf{M}=0##, i.e. ##\mathbf{M} \cdot \hat{n}= -\mathbf{M}##
Since the surface could be oriented at any angle w.r.t. ##\mathbf{M}## at the surface, this is a contradiction. Where am I going wrong?
##\displaystyle\psi=\dfrac{\mu_0}{4 \pi} \int_{V'} \dfrac{ \rho}{|\mathbf{r}-\mathbf{r'}|} dV' + \dfrac{\mu_0}{4 \pi} \oint_{S'} \dfrac{\sigma}{|\mathbf{r}-\mathbf{r'}|} dS'=\psi^{V}+\psi^{S}##
The ##\mathbf{H}## field is:
##\displaystyle\mathbf{H}=\dfrac{\mu_0}{4 \pi} \int_{V'} \rho \dfrac{\mathbf{r}-\mathbf{r'}}{|\mathbf{r}-\mathbf{r'}|^3} dV' + \dfrac{\mu_0}{4 \pi} \oint_{S'} \sigma \dfrac{ \mathbf{r}-\mathbf{r'}}{|\mathbf{r}-\mathbf{r'}|^3} dS'=\mathbf{H}^{V}+\mathbf{H}^{S}##
The ##\mathbf{B}## field is:
##\mathbf{B}=\mathbf{H} + \mu_0 \mathbf{M}=\mathbf{H}^{V} + \mathbf{H}^{S} + \mu_0 \mathbf{M}##
##\mathbf{H}^{V}## has no discontinuity.
##\mathbf{H}^{S}## has discontinuity of ##\mu_0 \mathbf{M} \cdot \hat{n}## at the surface ##S'##
##\mu_0 \mathbf{M}## has discontinuity of ##\mu_0 \mathbf{M}## at the surface ##S'##
From these knowledge, how shall one deduce that ##\mathbf{B}## is continuous at the surface?
My try: (I am getting a contradiction)
We need to show that ##\mu_0 \mathbf{M} \cdot \hat{n}+\mu_0 \mathbf{M}=0##, i.e. ##\mathbf{M} \cdot \hat{n}= -\mathbf{M}##
Since the surface could be oriented at any angle w.r.t. ##\mathbf{M}## at the surface, this is a contradiction. Where am I going wrong?