EM-15 · Magnetostatics field

The Magnetic Force and the Lorentz Force

A magnetic field exerts force on a moving charge: F = qv × B, perpendicular to both the velocity and the field. Change the angle between v and B to see the force vary as sinθ, and learn that the magnetic force does no work, so the charge moves in a circle.

Change the velocity angle to see the force

Rotate the charge's velocity in a uniform magnetic field. How big is the force when the velocity is parallel to the field? And when perpendicular? Which way does the force point?

Angle θ between v and Bθ = 55°

Drag to orbit. The slider sets the angle θ between velocity and field.

The magnetic force on this charge

F = qv × B = qvB sinθ · perpendicular to both v and B

F ⊥ v, so the magnetic force does no work · speed unchanged, only direction bends

Force magnitude (vs maximum) · 82%

Slanted

The field exerts a force

EM-13 and EM-14 were about current making a magnetic field. Now the reverse: a magnetic field exerts force on a charge passing through it. But a charge at rest feels no magnetic force; only when it moves does a force arise, tied to its velocity. This is the magnetic force. Where the electric force acted on the amount of charge, the magnetic force acts on its motion.

A cross product, three perpendiculars

The magnetic force is written as a cross product: F = qv × B. So the force is perpendicular to the velocity v and to the field B alike; the three vectors are mutually at right angles. Its magnitude is F = qvB sinθ, where θ is the angle between velocity and field. When the velocity is parallel to the field (θ=0), sinθ=0 and there is no force; when perpendicular (θ=90°) it is strongest. The direction follows the right-hand rule: curl your fingers from v to B and the thumb gives the force on a positive charge.

ObserveF = qvB sinθ

The force scales with sinθ; parallel gives zero.

ChooseF = q ?

The force is the cross product of velocity and field.

It does no work

The strangest thing about the magnetic force is that it does no work. Work is the dot product of force with displacement (the velocity direction), and the magnetic force is always perpendicular to velocity, so that dot product is zero. The magnetic force can therefore never change a charge's speed; its kinetic energy is untouched. All it can do is bend the direction. This is why a magnet in an accelerator can steer a particle's path but not speed it up, and why doing electrical work needs an electric field.

Fill inF ⊥ v → W = ?

Perpendicular to velocity, the work done is zero.

Circular motion, Lorentz, current

When the velocity is perpendicular to the field, the force stays at right angles to it and the charge traces a circle at constant speed. This force serves as the centripetal force, fixing the radius r = mv/(qB); a stronger field makes a smaller circle. If the velocity has a component along the field, there is no force that way so it drifts on, and the whole path becomes a helix. Combining the electric and magnetic forces gives the Lorentz force F = q(E + v × B). Since a current is a stream of moving charges, the force on a wire in a field is F = IL × B.

On your ownF = ?

It is the force on a current-carrying wire in a field.

Back to the first screen

On the first screen, aligning the velocity with the field made the force vanish to zero; turning it perpendicular made it largest. Its size followed sinθ. And the force always pointed out of the plane of velocity and field, at right angles to both. That is the cross product v × B at work. Because it is perpendicular to the velocity, this force cannot change the speed, only bend the direction, so a charge in a field traces a circle or a helix. F = qv × B.

A magnetic field exerts force only on a moving charge: F = qv × B, perpendicular to both velocity and field (a cross product), of magnitude qvB sinθ. It is zero when the velocity is parallel to B and largest when perpendicular. Because it is always perpendicular to the velocity it does no work, so the speed is unchanged and only the direction bends — the charge traces a circle or a helix. With the electric force added it is the Lorentz force F = q(E + v × B); the force on a current-carrying wire is F = IL × B.

What comes next

Ampère's law (EM-14) was the integral form ∮B·dl = μ₀I. The curl and Stokes' theorem in the next unit (EM-16) turn this loop integral into the pointwise differential form ∇ × B = μ₀J. What the divergence theorem (EM-07) did for Gauss's law, Stokes' theorem does for Ampère's. The curl is the tool that measures how much a field swirls at a point.