Sections#

Review & Summary#

Gauss’ Law for Magnetic Fields#

The simplest magnetic structures are magnetic dipoles. Magnetic monopoles do not exist (as far as we know). Gauss’ law for magnetic fields states that the net magnetic flux through any closed Gaussian surface is zero:

(204)#\[ \oint \vec{B} \cdot d\vec{A} = 0 \]

It implies that magnetic monopoles do not exist.

Induced Magnetic Fields#

A changing electric flux \(\Phi_E\) through a loop induces a magnetic field along the loop. This is Maxwell’s addition to Ampère’s law.

Displacement Current#

The displacement current due to a changing electric field is defined as \(i_d = \varepsilon_0 \, d\Phi_E/dt\). The Ampère–Maxwell law combines conduction current and displacement current:

(205)#\[ \oint \vec{B} \cdot d\vec{s} = \mu_0 i_{d,\mathrm{enc}} + \mu_0 i_{\mathrm{enc}} \]

where \(i_{d,\mathrm{enc}}\) is the displacement current encircled by the integration loop. Displacement current is not a transfer of charge but allows continuity of “current” through a capacitor.

Maxwell’s Equations#

Maxwell’s equations summarize electromagnetism and form its foundation, including optics. They unify electric and magnetic phenomena and predict electromagnetic waves.

Spin Magnetic Dipole Moment#

An electron has an intrinsic spin angular momentum \(\vec{S}\) and an associated spin magnetic dipole moment \(\vec{\mu}_s = -(e/m_e)\vec{S}\). For a measurement along a \(z\) axis, \(S_z\) can have only the values \(\pm \hbar/2\).

Orbital Magnetic Dipole Moment#

An electron in an atom has orbital angular momentum \(\vec{L}_{\mathrm{orb}}\) and an associated orbital magnetic dipole moment \(\vec{\mu}_{\mathrm{orb}} = -(e/2m_e)\vec{L}_{\mathrm{orb}}\). Orbital angular momentum is quantized: \(L_z = m_\ell \hbar\) with \(m_\ell = 0, \pm 1, \ldots, \pm \ell\). The energy of the dipole in an external field \(\vec{B}_{\mathrm{ext}}\) is \(U = -\vec{\mu}_{\mathrm{orb}} \cdot \vec{B}_{\mathrm{ext}}\).

Diamagnetism#

Diamagnetic materials exhibit magnetism only when placed in an external field; they form magnetic dipoles directed opposite the external field. In a nonuniform field, they are repelled from the region of greater magnetic field.

Paramagnetism#

Paramagnetic materials have atoms with permanent magnetic dipole moments that are randomly oriented unless in an external field \(\vec{B}_{\mathrm{ext}}\), where the dipoles tend to align. The magnetization \(\vec{M}\) (dipole moment per unit volume) is \(M = C B_{\mathrm{ext}}/T\) at low \(B_{\mathrm{ext}}/T\), where \(T\) is temperature and \(C\) is the Curie constant. In a nonuniform field, paramagnetic materials are attracted to the region of greater field.

Ferromagnetism#

In ferromagnetic materials, magnetic dipole moments can be aligned by an external field and remain partially aligned in domains after the field is removed. Alignment is eliminated above the Curie temperature \(T_C\). Hysteresis in the \(B\)–\(H\) curve allows permanent magnets.