EM-15 · Magnetostatics field

The Magnetic Force and the Lorentz Force

A magnetic field exerts a force on moving charges. With F = qv × B, that force is perpendicular to both the velocity and the field. Changing the angle between velocity and field, you see the force vary as sinθ, and confirm that because the magnetic force does no work, the charge moves in a circle.

Change the angle of the velocity and see the force

The cyan arrows are a uniform magnetic field pointing up, and the gold ball is a positive charge. Changing the angle the velocity arrow makes with the field using the slider changes the length of the force arrow on the charge. When the velocity is parallel to the field, the force vanishes; when it is perpendicular, the force is largest, and then the faint circle shows the orbit the charge will follow. The force always points at right angles to both the velocity and the field.

Angle θ between v and Bθ = 55°

Drag to rotate; the slider changes the angle θ between the velocity and the 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 currents making magnetic fields. This time it is the reverse: a magnetic field exerts a force on charges passing through it. A charge at rest feels no magnetic force; only when a charge moves does a force matched to its velocity appear. This is the magnetic force. If the electric force acts on the amount of charge, the magnetic force acts on the motion of charge.

A cross product, three perpendiculars

The magnetic force is written with a cross product. F = qv × B. So the force is perpendicular to the velocity and to the field, and the three vectors stand at right angles to each other. 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 so there is no force, and it is strongest when perpendicular (θ=90°). The direction is set by the right-hand rule: curl your fingers from the velocity toward the field, and your thumb gives the direction of the force on a positive charge. For a negative charge the force points the other way.

ObserveF = qvB sinθ

The magnetic force is proportional to sinθ and is 0 when parallel.

ChooseF = q ?

The force is the charge times 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 and displacement, and since the magnetic force is always perpendicular to the velocity, that dot product is 0. So the magnetic force cannot change a charge's speed and leaves its kinetic energy unchanged. All it can do is bend the direction. That is why particle accelerators use magnets to bend particles' paths and leave the job of raising their speed to electric fields.

Fill inF ⊥ v → W = ?

The force is perpendicular to the velocity, so the work done is 0.

Circular motion, Lorentz, current

When the velocity is perpendicular to the field, the force is always at right angles to the velocity, so the charge moves in a circle at constant speed. This force acts as the centripetal force and sets the radius r = mv/(qB); the stronger the field, the smaller the circle. The time for one revolution depends only on the field, the mass, and the charge, not on the speed, and the cyclotron makes use of this. If the velocity has a component parallel to the field, there is no force in that direction and the charge keeps going, so the whole path becomes a helix. Combining the electric and magnetic forces gives the Lorentz force F = q(E + v × B), and since a current is a collection of moving charges, the force on a wire in a magnetic field is F = IL × B.

On your ownF = ?

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

Back to the first screen

On the first screen, setting the velocity parallel to the field made the force vanish to 0, and 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, as a result of the cross product v × B. Because this force is perpendicular to the velocity, it cannot change the speed and only bends the direction, so a charge in a magnetic field traces a circle or a helix. F = qv × B.

A magnetic field exerts a force only on moving charges. With F = qv × B, the force is perpendicular to both velocity and field (cross product), with magnitude qvB sinθ. It is 0 when the velocity is parallel to B and greatest when perpendicular. Because the force is always perpendicular to the velocity, it does no work, so the speed stays the same and only the direction bends, and the charge traces a circle or helix. Adding the electric force gives the Lorentz force F = q(E + v × B), and 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. In the next unit, curl and Stokes' theorem (EM-16) turn this loop integral into the differential form that holds at a point, ∇ × B = μ₀J. What the divergence theorem (EM-07) did for Gauss's law, Stokes' theorem does for Ampère's law. Curl is a tool that measures how much a field swirls at a point.