Magnetic Field and Electric Currents

Electric current is the ordered motion of electric charges in a conductor. It is measured in amperes: $1\,\mathrm{A} = 1\,\mathrm{C/s}$. The current intensity is defined as the charge crossing a transverse section of the conductor per unit time: $$I = \frac{dq}{dt}$$

Current density: to describe the charge flow locally, the vector $\vec{J}$ is introduced: $$\vec{J} = nq\vec{v}_d$$ where $n$ is the number density of charge carriers, $q$ their charge, and $\vec{v}_d$ the drift velocity. The drift velocity is typically very small ($\sim \mathrm{mm/s}$), but the electrical signal propagates at nearly the speed of light in the medium ($\sim 2\times10^8\,\mathrm{m/s}$ in cables).

Microscopic Ohm's law: in many conducting materials, the current density is proportional to the applied electric field: $$\vec{J} = \sigma\vec{E}$$ where $\sigma$ is the electrical conductivity of the material (unit: $\Omega^{-1}\mathrm{m}^{-1}$). The resistivity $\rho = 1/\sigma$ measures the material's opposition to current flow.

Resistance and macroscopic Ohm's law: for a conductor of length $L$ and cross-section $A$: $$R = \rho\frac{L}{A}, \qquad V = RI$$ The resistance $R$ is measured in ohms ($\Omega$). The voltage $V$ across the conductor is proportional to the current $I$.

Joule effect: current flowing through a resistor dissipates energy as heat: $$P = I^2R = \frac{V^2}{R} = IV$$ The dissipated power is proportional to the square of the current: this is why thicker cables (lower $R$) heat up less.

Applications:

• Electrical circuits — lighting, motors, household appliances.
• Resistive heating — electric heaters, ovens, kettles.
• Fuses — protect circuits by breaking the current when it exceeds a threshold (they melt due to Joule heating).
• Thermistors — temperature sensors based on the variation of resistivity with $T$.

Magnetic force acts on electric charges in motion. It is described by the Lorentz force: $$\vec{F} = q\vec{v}\times\vec{B}$$ where $\vec{v}$ is the velocity of the charge $q$ and $\vec{B}$ is the magnetic field. The force is always perpendicular to both $\vec{v}$ and $\vec{B}$ (right-hand rule).

A key feature: the magnetic force does no work ($\vec{F}\cdot\vec{v}=0$). It only changes the direction of the velocity, not its magnitude, so kinetic energy remains constant.

Full Lorentz force: in the presence of both electric and magnetic fields: $$\vec{F} = q(\vec{E} + \vec{v}\times\vec{B})$$

Motion in a uniform magnetic field: if $\vec{v} \perp \vec{B}$, the trajectory is circular. The radius (Larmor radius) and angular frequency (cyclotron frequency) are: $$r = \frac{mv}{qB}, \qquad \omega_c = \frac{qB}{m}$$ If $\vec{v}$ has a component parallel to $\vec{B}$, the trajectory is helical with constant pitch.

Force on a current-carrying conductor: a current element $I\,d\vec{l}$ in a field $\vec{B}$ experiences the force: $$d\vec{F} = I\,d\vec{l}\times\vec{B}$$ Integrating over the whole conductor gives the total force (e.g., a current loop in a magnetic field experiences a torque — principle of the electric motor).

Hall effect: when a current-carrying conductor is placed in a perpendicular magnetic field, a transverse voltage develops: $$V_H = \frac{IB}{nqt}$$ This effect allows measuring $B$ (Hall probe) or determining the sign of charge carriers in semiconductors.

Applications:

• Electric motors — torque on current loops in a magnetic field.
• Mass spectrometers — separate ions by $m/q$ ratio by measuring the curvature radius.
• Particle accelerators (cyclotrons, synchrotrons).
• Hall sensors — position, speed, and current detection.
• Magnetic confinement (tokamaks) — confine plasma with magnetic fields.

The Biot-Savart law describes the magnetic field generated by a current-carrying circuit. An infinitesimal element $d\vec{l}$ of the circuit produces an infinitesimal magnetic field at point $P$: $$d\vec{B} = \frac{\mu_0}{4\pi}\frac{I\,d\vec{l}\times\hat{r}}{r^2}$$ where $\mu_0 = 4\pi\times10^{-7}\,\mathrm{T\cdot m/A}$ is the magnetic permeability of free space, $\hat{r}$ is the unit vector from $d\vec{l}$ to $P$, and $r$ is the distance. The total field is obtained by integrating over the entire circuit.

Key fields:

• Infinite straight wire carrying current $I$: $$B = \frac{\mu_0 I}{2\pi r}$$ The field lines are concentric circles around the wire. The direction is given by the right-hand rule (thumb in the direction of $I$, fingers curl in the direction of $\vec{B}$).
• Circular loop of radius $R$ on its axis (at centre): $$B = \frac{\mu_0 I}{2R}$$ The field is perpendicular to the plane of the loop. Along the axis at distance $z$ from the centre: $B = \dfrac{\mu_0 I R^2}{2(R^2+z^2)^{3/2}}$.
• Ideal solenoid with $n$ turns per metre: $$B = \mu_0 n I$$ The interior field is uniform and parallel to the axis; outside, the field is negligible. Solenoids are the magnetic equivalent of parallel-plate capacitors.

Applications:

• Electromagnets — solenoids with ferromagnetic cores to amplify $B$.
• Helmholtz coils — two loops generate a uniform field in a central region (useful for laboratory experiments).
• Magnetic resonance imaging (MRI) — uses powerful superconducting solenoids to generate fields of several tesla.
• Transformers and inductors in electrical circuits.

Ampère's law is the magnetic analogue of Gauss's law for electricity. It states that the circulation of the magnetic field around a closed path is proportional to the enclosed current (the current passing through the surface bounded by the path): $$\oint_C \vec{B}\cdot d\vec{l} = \mu_0 I_{enc}$$ where $I_{enc}$ is the net current crossing the surface bounded by curve $C$.

Usefulness: like Gauss's law, Ampère's law is especially powerful in the presence of symmetries (cylindrical, planar, toroidal). It allows computing $B$ without integrating the Biot-Savart law.

Differential form: applying Stokes' theorem: $$\nabla\times\vec{B} = \mu_0\vec{J}$$ The curl of $\vec{B}$ at a point is proportional to the local current density.

Gauss for the magnetic field: $$\nabla\cdot\vec{B} = 0$$ This equation states that magnetic monopoles do not exist: $\vec{B}$ field lines are always closed (there is no magnetic analogue of electric charge). In integral form: $\oint_S \vec{B}\cdot d\vec{A} = 0$, the flux through any closed surface is zero.

Magnetic vector potential: since $\nabla\cdot\vec{B}=0$, $\vec{B}$ can be expressed as the curl of a vector potential $\vec{A}$: $$\vec{B} = \nabla\times\vec{A}$$ This potential is useful in electrodynamics and quantum mechanics (Aharonov-Bohm effect).

Applications:

• Computing $B$ in solenoids and toroids — Ampère's law gives immediate results.
• Plasma confinement — in tokamaks, toroidal currents generate confining fields.
• Magnet design — electromagnets, MRI magnets.

Magnetic materials respond to an external magnetic field by developing a magnetisation $\vec{M}$, defined as the magnetic dipole moment per unit volume. To describe the field inside materials, the $\vec{H}$ field is introduced: $$\vec{H} = \frac{\vec{B}}{\mu_0} - \vec{M}$$ The circulation of $\vec{H}$ depends only on free currents (not on magnetisation currents): $$\nabla\times\vec{H} = \vec{J}_{free}$$
Linear materials: for many materials, $\vec{M}$ is proportional to $\vec{H}$: $$\vec{B} = \mu_0\mu_r\vec{H}$$ where $\mu_r$ is the relative magnetic permeability.

Classification of magnetic materials:

• Diamagnetic ($\mu_r < 1$, typically $\mu_r \approx 0.9999$): have no permanent atomic magnetic moments. The magnetic field induces an opposing moment (Lenz's law at the atomic level). Examples: bismuth, copper, water, carbon. They are weakly repelled by magnets.
• Paramagnetic ($\mu_r > 1$, typically $\mu_r \approx 1.00001$): have permanent atomic magnetic moments that partially align with the external field. Alignment is opposed by thermal agitation (Curie's law: $M = C B/T$). Examples: aluminium, platinum, liquid oxygen.
• Ferromagnetic ($\mu_r \gg 1$, up to $10^5$): exhibit spontaneous ordering of magnetic moments within microscopic domains. In an external field, domains align producing strong magnetisation. They show hysteresis (magnetisation does not reversibly follow the applied field) and a Curie temperature ($T_C$) above which they become paramagnetic. Examples: iron ($T_C \approx 770^\circ\mathrm{C}$), nickel, cobalt, gadolinium.

Superconductors: completely expel the magnetic field from their interior (Meissner effect), behaving as perfect diamagnets ($\mu_r = 0$). This enables magnetic levitation.

Applications:

• Hard disk drives — magnetic recording on ferromagnetic layers.
• Transformers and motors — ferrite or silicon-steel cores to concentrate magnetic flux.
• Magnetic resonance imaging (MRI) — intense fields from superconducting magnets.
• Maglev trains — use repulsion between superconducting magnets and rails.
• Magnetic memories — tapes, floppy disks, magnetic cards.