2.2.7. Magnetization#
2.2.7.1. Single Magnetic Moment (Limit of an Elementary Loop)#
Using the Biot-Savart law, specialized for a conductor carrying current \(i(\vec{r}_0)\)
\[\begin{split}\begin{aligned}
d \vec{b}(\vec{r})
& = - \frac{\mu}{4 \pi} \frac{\vec{r} - \vec{r}_0}{|\vec{r} - \vec{r}_0|^3} \times \vec{j}(\vec{r}_0) d V_0 = \\
& = - \frac{\mu}{4 \pi} i(\vec{r}_0) \frac{\vec{r} - \vec{r}_0}{|\vec{r} - \vec{r}_0|^3} \times \hat{\vec{t}}(\vec{r}_0) d \ell_0 \ ,
\end{aligned}\end{split}\]
we can calculate the magnetic field generated by a loop with path \(\ell_0 = \partial S_0\) using the PSCE
\[\begin{split}\begin{aligned}
\vec{b}(\vec{r})
& = \oint_{\ell_0} d \vec{b}(\vec{r}_0) = \\
& = - \frac{\mu}{4 \pi} i_0 \oint_{\vec{r}_0 \in \ell_0} \frac{\vec{r} - \vec{r}_0}{|\vec{r} - \vec{r}_0|^3} \times \hat{\vec{t}}(\vec{r}_0) = \\
& = \frac{\mu}{4 \pi} i_0 \int_{\vec{r}_0 \in S_0} \hat{\vec{n}}(\vec{r}_0) \cdot \nabla_{\vec{r}_0} \left( \frac{\vec{r} - \vec{r}_0}{|\vec{r} - \vec{r}_0|^3} \right)
\end{aligned}\end{split}\]
The field generated by an elementary loop of surface \(S_0\) with normal \(\hat{\vec{n}}_0\), using the mean value theorem, is
\[\vec{b}(\vec{r}) = \frac{\mu}{4 \pi} i_0 S_0 \hat{\vec{n}}_0 \cdot \nabla_{\vec{r}_0} \left( \frac{\vec{r} - \vec{r}_0}{|\vec{r} - \vec{r}_0|^3} \right) + o(S_0)\]
and as \(i_0 \rightarrow \infty\), \(S_0 \rightarrow 0\) such that \(\vec{M}_0 := i_0 S_0 \hat{\vec{n}}_0\)
\[\begin{split}\begin{aligned}
\vec{b}(\vec{r})
& = \frac{\mu}{4 \pi} \vec{M}_0 \cdot \nabla_{\vec{r}_0} \left( \frac{\vec{r} - \vec{r}_0}{|\vec{r} - \vec{r}_0|^3} \right) \\
& = - \frac{\mu_0}{4\pi} \left[ \frac{(\vec{r}-\vec{r}_0)(\vec{r}-\vec{r}_0)}{|\vec{r}-\vec{r}_0|^5} \cdot \vec{M}_0 - \frac{\vec{M}_0}{|\vec{r}-\vec{r}_0|^3} \right] = \\
& = - \frac{\mu_0}{4\pi} \left[ \frac{(\vec{r}-\vec{r}_0) \otimes (\vec{r}-\vec{r}_0)}{|\vec{r}-\vec{r}_0|^5} - \frac{\mathbb{I}}{|\vec{r}-\vec{r}_0|^3} \right] \cdot \vec{M}_0 \ .
\end{aligned}\end{split}\]
todo Analogy with the electric field produced by a distribution of dipoles.
Details
\[\oint_{\partial S} A \, t_i = \int_S \varepsilon_{ijk} \, n_j \, \partial_k A
\qquad , \qquad
\oint_{\partial S} A \, \hat{\vec{t}} = \int_S \hat{\vec{n}} \times \nabla A\]
\[\begin{split}\begin{aligned}
\oint_{\vec{r}_0 \in \ell_0} \frac{\vec{r} - \vec{r}_0}{|\vec{r} - \vec{r}_0|^3} \times \hat{\vec{t}}(\vec{r}_0) d \ell_0
& = \oint_{\vec{r}_0 \in \ell_0} \varepsilon_{ijk} \frac{r_j - r_{0,j}}{|\vec{r} - \vec{r}_0|^3} t_k = \\
& = \int_{\vec{r}_0 \in S_0} \varepsilon_{krs} n_r \partial^0_s \left( \varepsilon_{ijk} \frac{r_j - r_{0,j}}{|\vec{r} - \vec{r}_0|^3} \right) = \\
& = \int_{\vec{r}_0 \in S_0} \left( \delta_{ir} \delta_{js} - \delta_{is} \delta_{jr} \right) n_r \partial^0_s \left( \frac{r_j - r_{0,j}}{|\vec{r} - \vec{r}_0|^3} \right) = \\
& = \int_{\vec{r}_0 \in S_0} \left\{ n_i \underbrace{\partial^0_j \left( \frac{r_j - r_{0,j}}{|\vec{r} - \vec{r}_0|^3} \right)}_{=0} - n_j \partial^0_i \left( \frac{r_j - r_{0,j}}{|\vec{r} - \vec{r}_0|^3} \right) \right\} = \\
& = - \int_{\vec{r}_0 \in S_0} n_j \partial^0_i \left( \frac{r_j - r_{0,j}}{|\vec{r} - \vec{r}_0|^3} \right) \ .
\end{aligned}\end{split}\]
2.2.7.2. Continuous Distribution of Magnetic Moment#
To calculate the magnetic field generated by a volume distribution of magnetic moment, we can proceed in analogy with what was done to calculate the electric field generated by a distribution of dipoles
\[\begin{split}\begin{aligned}
\vec{b}(\vec{r})
& = \int_{\vec{r}_0 \in V_0} \frac{\mu_0}{4 \pi } \vec{m}(\vec{r}_0) \cdot \nabla_{\vec{r}_0} \left( \frac{\vec{r} - \vec{r}_0}{|\vec{r} - \vec{r}_0|^3} \right) = \\
& = \oint_{\vec{r}_0 \in \partial V_0} \frac{\mu_0}{4 \pi} \frac{\vec{r} - \vec{r}_0}{|\vec{r} - \vec{r}_0|^3} \hat{\vec{n}}(\vec{r}_0) \cdot \vec{m}(\vec{r}_0) + \int_{\vec{r}_0 \in V_0} \frac{\mu_0}{4 \pi} \frac{\vec{r} - \vec{r}_0}{|\vec{r} - \vec{r}_0|^3} \,\left( - \nabla_{\vec{r}_0} \cdot \vec{m}(\vec{r}_0) \right) \ , \\
\end{aligned}\end{split}\]
but without obtaining an analogy with the expression of the Biot-Savart law, which involves the cross product between the term \(\frac{\vec{r}- \vec{r}_0}{|\vec{r} - \vec{r}_0|^3}\) and a current density \(\vec{j}(\vec{r}_0)\).
Details
We can rewrite
\[\begin{split}\begin{aligned}
\oint_{\vec{r}_0 \in \partial V_0} & \frac{\vec{r} - \vec{r}_0}{|\vec{r} - \vec{r}_0|^3} \times \left( \hat{\vec{n}}(\vec{r}_0) \times \vec{m}(\vec{r}_0) \right) \\
& = \oint_{\vec{r}_0 \in \partial V_0} \varepsilon_{ijk} \frac{r_j - r_{0,j}}{|\vec{r}-\vec{r}_0|^3} \varepsilon_{krs} n_r m_s = \\
& = \int_{\vec{r}_0 \in V_0} \left( \delta_{ir} \delta_{js} - \delta_{is} \delta_{jr} \right) \partial^0_r \left( \frac{r_j - r_{0,j}}{|\vec{r}-\vec{r}_0|^3} m_s \right) = \\
& = \int_{\vec{r}_0 \in V_0} \left\{ \partial^0_i \left( \frac{r_j - r_{0,j}}{|\vec{r}-\vec{r}_0|^3} m_j \right) - \partial^0_j \left( \frac{r_j - r_{0,j}}{|\vec{r}-\vec{r}_0|^3} m_i \right) \right\} = \\
& = \int_{\vec{r}_0 \in V_0}
\left\{ \partial^0_i \frac{r_j - r_{0,j}}{|\vec{r}-\vec{r}_0|^3} m_j
+ \frac{r_j - r_{0,j}}{|\vec{r}-\vec{r}_0|^3} \partial^0_i m_j
- \frac{r_j - r_{0,j}}{|\vec{r}-\vec{r}_0|^3} \partial^0_j m_i
- \underbrace{ \partial^0_j \frac{r_j - r_{0,j}}{|\vec{r}-\vec{r}_0|^3}}_{=0} m_i
\right\} = \\
& = \int_{\vec{r}_0 \in V_0}
\left\{ \partial^0_i \frac{r_j - r_{0,j}}{|\vec{r}-\vec{r}_0|^3} m_j
+ \varepsilon_{ijk} \varepsilon_{krs} \frac{r_j - r_{0,j}}{|\vec{r}-\vec{r}_0|^3} \partial^0_r m_s
\right\} = \\
& = \int_{\vec{r}_0 \in V_0}
\left\{ \nabla_{\vec{r}_0} \frac{\vec{r} - \vec{r}_0}{|\vec{r}-\vec{r}_0|^3} \cdot \vec{m}(\vec{r}_0)
+ \frac{\vec{r} - \vec{r}_0}{|\vec{r}-\vec{r}_0|^3} \times \left( \nabla_{\vec{r}_0} \times \vec{m}(\vec{r}_0) \right)
\right\} = \\
\end{aligned}\end{split}\]
using vector calculus identities,
\[\begin{split}\begin{aligned}
\vec{a} \times (\vec{b} \times \vec{c}) & = \varepsilon_{ijk} a_j \varepsilon_{krs} b_r c_s = \\
& = (\delta_{ir} \delta_{js} - \delta_{is} \delta_{jr}) a_j b_r c_s = \\
& = a_j b_i c_j - c_i b_j a_j = \vec{b}(\vec{a} \cdot \vec{c}) - \vec{c} (\vec{a} \cdot \vec{b})
\end{aligned}\end{split}\]
\[\begin{split}\begin{aligned}
a_j \partial_i m_j - a_j \partial_j m_i
& = (\delta_{ir} \delta_{js} - \delta_{is} \delta_{jr}) a_j \partial_r m_s = \\
& = \varepsilon_{ijk} \varepsilon_{krs} a_j \partial_r m_s = \\
& = \vec{a} \times \left( \nabla \times \vec{m} \right)
\end{aligned}\end{split}\]
The magnetic field generated by a distribution of magnetic moment can therefore be rewritten as
\[\begin{split}\begin{aligned}
\vec{b}(\vec{r})
& = \int_{\vec{r}_0 \in V_0} \frac{\mu_0}{4 \pi } \vec{m}(\vec{r}_0) \cdot \nabla_{\vec{r}_0} \left( \frac{\vec{r} - \vec{r}_0}{|\vec{r} - \vec{r}_0|^3} \right) = \\
& = - \frac{\mu_0}{4\pi} \oint_{\vec{r}_0 \in \partial V_0} \frac{\vec{r} - \vec{r}_0}{|\vec{r} - \vec{r}_0|^3} \times \underbrace{ \left( - \hat{\vec{n}}(\vec{r}_0) \times \vec{m}(\vec{r}_0) \right) }_{\vec{j}^s_M}
- \frac{\mu_0}{4 \pi} \int_{\vec{r}_0 \in V_0} \frac{\vec{r} - \vec{r}_0}{|\vec{r}-\vec{r}_0|^3} \times \underbrace{ \left(\nabla_{\vec{r}_0} \times \vec{m}(\vec{r}_0) \right)}_{\vec{j}_M} \ ,
\end{aligned}\end{split}\]
having defined the surface magnetization current density \(\vec{j}^s_M\) and the volume magnetization current density \(\vec{j}_M\) as the intensities of the distributed singularities, in analogy with the expression of the Biot-Savart law.
2.2.7.3. Reformulation of Maxwell’s Equations and Charge Continuity#
The Ampère-Maxwell law can be rewritten
\[\begin{split}\begin{aligned}
& \nabla \times \vec{b} - \mu_0 \varepsilon_0 \partial_t \vec{e} = \mu_0 \vec{j} \\
& \nabla \times \vec{b} - \mu_0 \partial_t \left( \vec{d} - \vec{p} \right) = \mu_0 \left( \vec{j}_f + \vec{j}_P + \vec{j}_M \right) \\
& \nabla \times \underbrace{\left( \vec{b} - \mu_0 \vec{m} \right)}_{=: \mu_0 \vec{h}} - \mu_0 \partial_t \vec{d} + \mu_0 \underbrace{\left( \partial_t \vec{p} - \vec{j}_P \right)}_{= \vec{0}} = \mu_0 \vec{j}_f \\ \\
& \nabla \times \vec{h} - \partial_t \vec{d} = \vec{j}_f
\end{aligned}\end{split}\]
From the continuity equation of electric current,
\[\partial_t \rho + \nabla \cdot \vec{j} = 0 \ ,\]
we derive the continuity equation for magnetization charges
\[\begin{split}\begin{aligned}
0 & = \partial_t \rho_M + \nabla \cdot \vec{j}_M = \\
& = \partial_t \rho_M + \underbrace{ \nabla \cdot \nabla \times \vec{m}}_{ \equiv \vec{0} } \ .
\end{aligned}\end{split}\]