Magnetic Force — F = qvB, Direction and Circular Motion

Only moving charges feel it

Electric fields push charges along the field lines; magnetic fields play a different game. A magnetic field B exerts a force only on charges that are moving, and the force is perpendicular to both the velocity and the field:

$$F = qvB\sin\theta$$

where $\theta$ is the angle between the velocity v and the field B. Three consequences fall out immediately:

• $v = 0$ → no force. Stationary charges ignore magnetic fields entirely.
• $\theta = 0°$ (moving along the field) → no force. The field grabs only the perpendicular part of the motion.
• $\theta = 90°$ → maximum force $F = qvB$.

The tesla (T) is the SI unit of B — a strong refrigerator magnet is about $10^{-2}\ \mathrm{T}$; an MRI machine runs at 1.5–3 T.

The right-hand rule

Because three directions are involved, we need a convention — the right-hand rule (for positive charges):

1. Point your fingers along the velocity v.
2. Curl them toward the magnetic field B.
3. Your thumb points along the force F.

For electrons and other negative charges, the force points the opposite way (equivalently, use your left hand). Test it: a positive charge moving right in a field pointing into the page feels an upward force — fingers right, curl into the page, thumb up.

Getting the direction right is a pure practice skill. Sketch the three arrows every time; guessing from memory is where most sign errors are born.

Force on a current-carrying wire

A wire carrying current $I$ is just many charges moving together, so the field pushes the whole wire. For a straight wire of length $L$ in a uniform field:

$$F = ILB\sin\theta$$

where $\theta$ is the angle between the wire (current direction) and the field. Maximum force when the wire is perpendicular to the field; zero when parallel.

This is the working principle of every electric motor: current in the coil's wires feels sideways forces on opposite sides of the coil, and those paired forces form a torque that spins it. Loudspeakers do the same thing in miniature — a coil glued to the speaker cone jerks back and forth in a magnet's field, pushing air.

Magnetic forces do no work

Here is the strangest fact about magnetic forces: they never do work. Work needs a force component along the displacement ($W = Fd\cos\phi$), but the magnetic force is always at 90° to the velocity — and therefore 90° to the displacement.

Consequences:

• A charge in a uniform magnetic field keeps its speed forever; the field only bends its path into a circle (or helix, if part of the velocity is along the field).
• Magnets cannot be a free energy source: no work, no energy gain, no matter how clever the arrangement.
• To accelerate particles you need electric fields; magnetic fields are for steering. Particle accelerators use both: electric fields for speed, magnetic fields for bending the beam around the ring.

The circular radius follows from $qvB = mv^2/r$: $r = mv/(qB)$.

Worked example: electron in a field

An electron moves at $3.0 \times 10^6\ \mathrm{m/s}$ perpendicular to a uniform $0.50\ \mathrm{T}$ field (into the page). Find the force's magnitude, direction, and the radius of its path.

Magnitude: $F = qvB = (1.602 \times 10^{-19})(3.0 \times 10^6)(0.50) = 2.4 \times 10^{-13}\ \mathrm{N}$.

Direction: the electron is negative, so the force is opposite the right-hand-rule result. With velocity right and B into the page, a positive charge would feel an upward force; the electron feels a downward force of $2.4 \times 10^{-13}\ \mathrm{N}$.

Radius: $r = mv/(qB) = (9.11 \times 10^{-31})(3.0 \times 10^6) / (2.4 \times 10^{-13}) \approx 1.1 \times 10^{-2}\ \mathrm{m}$ — about 1.1 cm. The electron circles at constant speed, bent into a tight loop by the field.

Variables and units

Common mistakes

Real-world applications

• Electric motors spin because current-carrying coils feel magnetic forces that twist them toward alignment with the field.

• Particle accelerators and mass spectrometers steer and sort charged particles with precisely shaped magnetic fields.

• Cathode-ray tubes and electron microscopes focus electron beams using magnetic lenses.

• Hall-effect sensors measure current and detect position (in cars, phones, brushless motors) by reading the sideways magnetic force on moving charges.