4 Magnetism

Fields, lines, and poles

A magnetic field describes the magnetic influence at each point in space. It is represented by the vector $\mathbf{B}$, and its SI unit is the tesla ($\text{T}$). A compass needle turns to align with the local field. Magnetic field lines visualize the field: the field is tangent to each line, and closer-spaced lines indicate a stronger field.

Field lines form continuous loops. A bar magnet has north and south poles, but cutting it produces smaller magnets with both poles rather than isolated magnetic poles. A current loop also behaves like a magnetic dipole, with a north-like face and a south-like face.

Takeaway: Field lines show direction and relative strength; they do not represent separate magnetic charges.

Magnetic forces on charges and wires

A moving charge in a magnetic field experiences the magnetic force

where $q$ is charge, $\mathbf{v}$ is velocity, and $\theta$ is the angle between the velocity and the field. The force is perpendicular to both vectors. A stationary charge, or one moving parallel to the field, has no magnetic force.

Use the right-hand rule for a positive charge: point your fingers along its velocity and curl them toward the field; your thumb gives the force direction. For a negative charge, reverse that direction.

For example, a positive charge of $2.0\times10^{-6}\,\text{C}$ moving at $3.0\times10^4\,\text{m/s}$ perpendicular to a $0.50\,\text{T}$ field experiences a force of magnitude

Because the magnetic force is perpendicular to the particle’s instantaneous motion, it does no work. It can change the direction of the velocity, but not the particle’s speed or kinetic energy.

A current-carrying wire also experiences a force in a magnetic field. For a straight segment in a uniform field,

where $I$ is conventional current and $\mathbf{L}$ points along the current. This force is the operating principle behind electric motors.

Takeaway: Magnetic forces redirect moving charges and currents; they do not increase a charged particle’s speed.

Charged-particle paths

In a uniform magnetic field, a charged particle moving perpendicular to the field follows a circular path. The magnetic force provides the centripetal force, so the radius is

where $m$ is the particle’s mass. The circular-motion period is

A greater speed or mass gives a wider circle, while a stronger field or larger charge magnitude gives a tighter circle.

If velocity has both perpendicular and parallel components, the perpendicular component produces circular motion while the parallel component remains unchanged. Together, these motions create a helix around the field direction. This behavior helps explain charged-particle motion in Earth’s magnetic field and is used in devices such as mass spectrometers.

Takeaway: Separate velocity into components parallel and perpendicular to the field to predict the particle’s path.

Magnetic fields from electric currents

Moving electric charges produce magnetic fields. The Biot–Savart law gives the contribution from a small current segment:

Here, $d\mathbf{l}$ points along the current, and $\hat{\mathbf{r}}$ points from the segment toward the observation point. The total field is the vector sum of contributions from all segments. In vacuum, $\mu_0\approx4\pi\times10^{-7}\,\text{T·m/A}$.

For a long, straight wire, the field forms circles around the wire, with magnitude

where $r$ is the distance from the wire. Use the right-hand grip rule: point your right thumb along the conventional current; your curled fingers show the field direction.

A circular current loop produces a field resembling that of a bar magnet. A long coil, or solenoid, produces a nearly uniform field inside it. For an ideal long solenoid,

where $n$ is the number of turns per unit length.

Ampère’s law relates the field around a closed path to the current enclosed by that path:

It is especially useful for symmetric arrangements, such as a long straight wire or an ideal solenoid.

Takeaway: Use the Biot–Savart law to add field contributions and Ampère’s law when symmetry makes the field around a path easy to describe.

Magnetic moments and materials

Electrons have magnetic moments associated with their intrinsic spin and orbital motion. In many materials, these moments largely cancel. In ferromagnetic materials, groups of moments can align in magnetic domains. When many domains align, the material becomes strongly magnetized.

This microscopic alignment helps explain permanent magnets. An electromagnet, by contrast, produces its field chiefly through current in a coil. Both illustrate how magnetic fields can arise from aligned microscopic moments or from moving electric charge.

Takeaway: Magnetic materials and current-carrying coils produce fields through different mechanisms, but both are rooted in magnetic moments or moving charges.