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Electromagnetism overview

Topic 5 of 7

Magnetic force on a charged particle

Magnetic force depends on a particle's velocity as well as its charge. Determine the force for positive-charge motion first, then reverse that force for a negative particle moving the same way.

For charge Q moving at speed v through magnetic flux density B, with angle θ between velocity and field, the force magnitude is:

F = B|Q|v sin θ

The form F = BQv sin θ uses Q as the charge magnitude when calculating a magnitude. If Q is signed, handle the direction separately rather than reporting a negative magnitude. At perpendicular entry, F = B|Q|v. A stationary charge, a neutral particle, or motion exactly parallel or antiparallel to B has zero magnetic force.

Keep velocity and charge sign separate

Use x right, y up and z out of the page. For a positive charge, its motion has the conventional-current direction, so Fleming's left-hand rule gives the force. For an electron with the same stated velocity, reverse that force once. Do not reverse the velocity as well and then reverse the force a second time.

Same velocity and field; opposite charge signs

Same velocity and field; opposite charge signsTwo distinct selected-particle cases have velocity right and the same magnetic field into the page. The positive charge on the left has upward force; the electron on the right has downward force. Blue arrows show their rightward velocities and purple arrows show force. The teal circle-and-cross above them is the common magnetic-field direction key, not a current. The arrow lengths are schematic, not a claim of equal charge magnitudes or accelerations.B into the page+q-eFFvvPositive chargeElectron

Apply the positive-charge direction first, then reverse it for an electron. The actual electron velocity remains rightward; do not reverse both velocity and charge.

Velocity up and field right

Velocity up and field rightThe two selected particles both move upward in a magnetic field directed right. For the positive charge, v cross B points into the page, shown by a purple circle-and-cross at the left particle location. For the electron, force is out of the page, shown by a purple circle-and-dot at the right particle location. The charge labels below the symbols identify the bodies; the purple symbols represent force, not magnetic field or current.B rightvv+q-eForce into pageForce out of page

Here purple cross means force into the page and purple dot means force out. A stationary charge, or one travelling along B, would instead have zero magnetic force.

Each comparison holds the stated velocity and field directions fixed while changing charge sign. The first view gives upward or downward forces; the second gives forces into or out of the page. Dot and cross labels identify which quantity points through the page.
  • Velocity right, B into the page: positive-charge force is up; electron force is down.
  • Velocity up, B right: positive-charge force is into the page; electron force is out of the page, towards the observer.

Reversing only velocity, only B or only charge sign reverses the force for a nonzero-force case. Reversing both velocity and B restores the original force for the same charge. The force is perpendicular to both velocity and B; it does not point along either.

Calculate a magnitude

An electron with |Q| = 1.60 × 10-19 C moves right at 2.00 × 107 m/s through a 1.00 mT field into the page. Convert 1.00 mT to 1.00 × 10-3 T:

F = B|Q|v
= (1.00 × 10-3)(1.60 × 10-19)(2.00 × 107)
= 3.20 × 10-15 N downward

At a different angle, retain sin θ. The angle is between the actual velocity vector and B, not between force and either of them.

Why the magnetic force does no work

The magnetic force is perpendicular to the particle's instantaneous velocity and therefore to its small displacement at that instant. It does no work on the particle. With magnetic force alone, speed and kinetic energy remain constant while velocity direction can change.

A uniform B does not mean a fixed force direction: as the velocity turns, the perpendicular force turns too. For perpendicular entry this leads to the circular path. Any velocity component parallel to B remains unaffected by the magnetic force, so not every entry direction gives a circle in the page.

Optional check An electron moves upward in the page through a uniform magnetic field directed right. What is the direction of its magnetic force?
An electron moves upward in the page through a uniform magnetic field directed right. What is the direction of its magnetic force?