Magnetic Forces, Materials, and Devices

A charge \(Q\) moving with velocity \(\vec{u}\) through fields \(\vec{E}\) and \(\vec{B}\) feels the Lorentz force — the union of the electric push and the magnetic deflection:

The magnetic part is always perpendicular to the velocity (it is a cross product), so it can change a charge's direction but never its speed. A purely magnetic force therefore does no work — it bends a charged particle into a circle without speeding it up, the principle behind cyclotrons, mass spectrometers, and the beam steering in many pulsed-power and accelerator systems.

A current is moving charge, so a current-carrying wire in a field feels a force — the sum of the Lorentz forces on its drifting charges. For a straight segment \(\vec{L}\) carrying current \(I\) in a uniform field:

Each wire sits in the field of the other, so two parallel currents exert forces on each other. Using the wire field \(B=\mu_0 I/2\pi\rho\) from Chapter 11, the force per unit length between wires a distance \(d\) apart is:

Currents in the same direction attract; opposite currents repel — the reverse of the electrostatic rule for charges. This force once defined the ampere itself, and it is the mechanical stress that high-current busbars and pulsed-power conductors must be braced against.

A current loop in a uniform field feels no net force — but it can feel a torque that tries to rotate it. Defining the loop's magnetic dipole moment \(\vec{m}=I\vec{S}\) (current times the vector area, normal by the right-hand rule), the torque is a clean cross product:

The torque is greatest when the loop lies parallel to the field and zero when its moment aligns with \(\vec{B}\) — the loop wants to swing until \(\vec{m}\parallel\vec{B}\). This is the operating principle of every electric motor and moving-coil meter. Note the exact parallel with the electric dipole of Chapter 6, where \(\vec{T}=\vec{p}\times\vec{E}\).

Inside matter, the orbiting and spinning electrons act as countless atomic current loops, each a tiny dipole. Their alignment per unit volume is the magnetization \(\vec{M}\) (A/m) — the magnetic twin of polarization \(\vec{P}\). It modifies the relation between \(\vec{B}\) and \(\vec{H}\), and collapses (for linear materials) into a single permeability:

The relative permeability \(\mu_r=1+\chi_m\) sorts materials into three classes — and ferromagnets break the linear picture entirely, retaining magnetization after the field is removed:

In a ferromagnet \(\vec{B}\) depends on the material's history, tracing a hysteresis loop as \(H\) cycles. The loop's width gives the remanence (field kept at \(H=0\)) and coercivity (reverse field needed to erase it); its enclosed area is the energy dissipated as heat each cycle — the core loss that matters in every transformer and inductor.

Like the electric field, the magnetic field bends at a material interface. The rules mirror Chapter 7: the normal \(\vec{B}\) is continuous (from \(\nabla\cdot\vec{B}=0\)), and the tangential \(\vec{H}\) is continuous when no free surface current flows:

When a field is guided around a core of high \(\mu\) — a transformer or inductor — the flux behaves like current in a wire, and we analyse it as a magnetic circuit. The magnetomotive force \(\mathcal{F}=NI\) drives flux \(\Psi_m\) against reluctance \(\mathcal{R}\), in a direct echo of Ohm's law:

For each item, identify whether it is a force, torque, material, or circuit problem, then apply the matching relation. Difficulty rises down the list.