% vim: tw=50 % 09/02/2023 09AM \begin{center} \includegraphics[width=0.6\linewidth] {images/ac177d40a85a11ed.png} \end{center} Magnetic field lines are the integral curves of $\bf{B}$. Since $\nabla \cdot \bf{B} = \bf{0}$, they are continuous. \subsubsection*{Permanent Magnets} A bar magnet has north and south poles and a dipole moment. THis comes from the superposition of aligned dipoles on the atomic scale. Atoms contain electrons, which are spinning charged particles, with a magnetic dipole moment. \myskip A classical model of a particle is a spinning charged sphere \begin{center} \includegraphics[width=0.3\linewidth] {images/ba203f08a85a11ed.png} \end{center} which is a current loop with a magnetic dipole moment proportional to its charge and spin. As far as we know, there are no magnetic charges (monopoles). \begin{center} \includegraphics[width=0.6\linewidth] {images/d7c63e2ca85a11ed.png} \end{center} \subsubsection*{The Earth as a Magnet} The liquid iron outer core of the Earth is a conducting fluid in convective motion and supports electric currents that generate a magnetic field. At the Earth's structure, this resembles a dipole field. \begin{center} \includegraphics[width=0.6\linewidth] {images/f615d2f2a85a11ed.png} \end{center} \subsection{Magnetic Forces} The Lorentz force on a particle of charge $q_i$ at position $\bf{x}_i(t)$ is \[ q(\bf{E} + \dot{\bf{x}}_i \times \bf{B} \] ($\bf{E}$, $\bf{B}$ evaluated at $\bf{x}_i(t)$). In the limit of continuous charge and current densities, the Lorentz force per unit volume is then \[ \boxed{\rho \bf{E} + \bf{J} \times \bf{B}} \] We can recover the discrete version by substituting \begin{align*} \rho &= \sum_i q_i \delta(\bf{x} - \bf{x}_i(t)) \\ \bf{J} &= \sum_i q_i \dot{\bf{x}}_i(t) \delta(\bf{x} - \bf{x}_i(t)) \end{align*} \subsubsection*{Force between thin wires} Consider two or more thin wires with currents $I_i$ along curves $C_i$. The total magnetic field is $\bf{B} = \sum_i \bf{B}$, where \[ \bf{B}_i(\bf{x}) = \frac{\mu_0 I}{4\pi} \int_{C_i} \frac{\dd \bf{x}_i \times (\bf{x} - \bf{x}_i)}{|\bf{x} - \bf{x}_i|^3} \] is the magnetic field due to wire $i$. The current density is $\bf{J} = \sum_i \bf{J}_i$, where \[ \bf{J}_i(\bf{x}) = I_i \int_{C_i} \delta(\bf{x} - \bf{x}_i) \dd \bf{x}_i \] The total magnetic force acting on a volume $V$ is \[ \bf{F} = \int_V \bf{J} \times \bf{B} \dd V \] The force acting on a volume $V$ is \[ \bf{F} = \int_V \bf{J} \times \bf{B} \dd V .\] The force acting on wire $i$ is \begin{align*} \bf{F}_i &= \int \bf{J}_i(\bf{x}) \times \bf{B}(\bf{X}) \dd^3 \bf{x} \\ &= I_i \int_{C_i} \dd \bf{x}_i \times \bf{B}(\bf{x}_i) \end{align*} Since $\bf{B} = \sum_i \bf{B}_i$, we have \[ \bf{F}_i = \sum_j \bf{F}_{ij} ,\] where \[ \bf{F}_{ij} = I_i \int_{C_i} \dd \bf{x}_i \times \bf{B}_j(\bf{x}_i) \] is the force on wire $i$ due to wire $j$. Using the Biot-Savat Law, \[ \bf{F}_{ij} = \frac{\mu_0 I_i I_j}{4\pi} \int_{C_i} \int_{C_j} \dd \bf{x}_i \times \left( \frac{\dd \bf{x}_j \times (\bf{x}_i - \bf{x}_j)}{|\bf{x}_i - \bf{x}_j|^3} \right) \] This can be rewritten (see Example 2.4) in a manifestly antisymmetric way that shows that \[ \bf{F}_{ji} = -\bf{F}_{ij} \] as expected from Newton's third law. \myskip The self-force $\bf{F}_{ii}$ vanishes, although the thin-wire integral is singular and it is better to treat the case of thick wires. \subsubsection*{Long parallel wires} Consider two infinitely long, parallel, thin wires separated by a distance $r$. Use cylindrical polars centred on wire 2. \begin{center} \includegraphics[width=0.2\linewidth] {images/16efe1e0a85e11ed.png} \end{center} We have $\bf{B}_2 = \frac{\mu_0 I_2}{2\pi r} \bf{e}_\phi$, \[ \bf{F}_{12} = I_1 \int_{-\infty}^\infty \dd z \bf{e}_z \times \bf{B}_2 \] Total force is infinite. Force per unit length is \[ I_1 \bf{e}_z \times \bf{B}_2 = - \frac{\mu_0 I_1 I_2}{2\pi r} \bf{e}_r \] This is directed towards wire 2 if $I_1 I_2 > 0$. So the force is attractive if the currents are aligned and repulsive otherwise. \subsubsection*{Force and torque on a magnetic dipole} Consider a localized current distribution (current loop) confined to a ball $\{V : |\bf{x}| < R\}$. Place this in an external magnetic field $\bf{B}(\bf{x})$ that varies slowly over the length scale $R$. \myskip The magnetic torque (about the origin) on the current loop is \begin{align*} I &= \int_V \bf{x} \times (\bf{J}(\bf{x}) \times \bf{B}(\bf{x})) \dd^3 \bf{x} \\ &= \int_V ((\bf{x} \cdot \bf{B}(\bf{x})) \bf{J}(\bf{x}) - (\bf{x} \cdot \bf{J}(\bf{x})) \bf{B}(\bf{x})) \dd^3 \bf{x} \end{align*} (here $\bf{B}$ is external field -- self-torque of loop vanishes). Within $V$, $\bf{B}(\bf{x})$ can be expanded sa a Taylor series \[ B_i(\bf{x}) = B_i(\bf{0}) + x_j \partial _j B_i(\bf{0}) + \cdots \] Retaining only the zeroth-order term (uniform field), we have \[ T_i \approx B_j(\bf{0}) \in_V x_j J_i \dd^3 \bf{x} - B_i(\bf{0}) \int_V x_j J_j \dd^3 \bf{x} \] Recall the first moments of the current distribution: \[ \int_V x_i J_j \dd^3 \bf{x} = \eps_{ijk} m_k \] Thus $T_i \approx B_j(\bf{0}) \eps_{jik} m_k$. In general, \[ \bf{\tau} \approx \bf{m} \times \bf{B} \] where $\bf{B}$ is evaluated at the dipole's location, and $\bf{\tau}$ is measured about this point. For the force, we need to go to the first order of the Taylor expansion of $\bf{B}$: \begin{align*} \bf{F} &= \int_V \bf{J}(\bf{x}) \times \bf{B}(\bf{x}) \dd^3 \bf{x} \\ F_i &= \int_v \eps_{ijk} J_j(\bf{x}) (B_j(\bf{0}) + x_l \partial_l B_k(\bf{0})) \dd^3 \bf{x} \\ &= \eps_{ijk} B_k(\bf{0}) \int_V J_j \dd^3 \bf{x} + \eps_{ijk} \partial_l B_k(\bf{0}) \int_V x_l J_j \dd^3 \bf{x} \\ &= 0 + \eps_{ijk} \partial_l B_k(\bf{0}) \eps_{ljn} m_n \\ &= \partial_i B_k(\bf{0} m_k - \partial_k B_k(\bf{0}) m_i \\ &= \partial_i(m_k B_k)(\bf{0}) \end{align*} since $\nabla \cdot \bf{B} = 0$. In general, $\bf{F} \approx \nabla (\bf{m} \cdot \bf{B})$. Can also be written as $\bf{F} = -\nabla U$, where \[ \boxed{U = -\bf{m} \cdot \bf{B}} \] is the potential energy of a magnetic dipole in an external field. \myskip As in the electric case, this is minimised when $\bf{m}$ is aligned with $\bf{B}$.