Magnetic Field and Ampere's Law

Magnetism is electricity’s velocity-dependent twin: it acts only on moving charges, always at right angles to their motion. A force that can never do work, only steer — which is why magnetic fields curl particle paths into circles and why a compass needle twists rather than launches.

Moving charges create and experience magnetic fields. The magnetic force on a charge is:

$$\vec{F} = q\vec{v}\times\vec{B}$$

This force is always perpendicular to $\vec{v}$ — it changes direction but not speed.

Biot-Savart law — Field from a current element:

$$d\vec{B} = \frac{\mu_0}{4\pi}\frac{I\,d\vec{l}\times\hat{r}}{r^2}$$

Ampere's law — For steady currents with symmetry:

$$\oint_C \vec{B}\cdot d\vec{l} = \mu_0 I_{\text{enc}}$$

Key results from Ampere's law

Force between parallel wires carrying currents $I_1$ and $I_2$:

$$\frac{F}{L} = \frac{\mu_0 I_1 I_2}{2\pi d}$$

Parallel currents attract; antiparallel currents repel.

Physics link: A charged particle in a uniform $\vec{B}$ moves in a circle of radius $r = mv/(qB)$. This is the principle behind cyclotrons and mass spectrometers.

Common pitfall: Because $\vec{F} = q\vec{v}\times\vec{B}$ is perpendicular to $\vec{v}$, a magnetic field can never change a particle’s speed or kinetic energy — only its direction. If a problem needs energy added, some electric field must be doing it.