Magnetic Force on Wires and Moving Charges
A magnetic field exerts a force on a current-carrying wire, F = BIL sin θ, and on a moving charge, F = qvB sin θ, always perpendicular to both the field and the velocity (or current), so it changes the direction of motion but not the speed, making charged particles move in circles of radius r = mv ÷ qB. In brief: the direction of the force is given by Fleming’s left-hand rule (first finger = field, second finger = current, thumb = force/motion) or the right-hand rule for the cross product. The force is zero when the current or the velocity is parallel to the field. The unit tesla is 1 T = 1 N ÷ (A·m).
Electric motors, loudspeakers, mass spectrometers, particle accelerators and the auroras all depend on magnetic forces. This guide gives the formulas and directions for the force on wires and charges, describes circular motion in a field, compares electric and magnetic forces in a table, works through numerical examples and corrects common mistakes.
What is the force on a current-carrying wire?
A wire of length L carrying a current I in a uniform field B, at an angle θ between the wire and the field, experiences a force F = BIL sin θ. The force is greatest (F = BIL) when the wire is perpendicular to the field, and zero when parallel. The direction is given by Fleming’s left-hand rule: hold the thumb, first finger and second finger of the left hand at right angles. The first finger points along the field (north to south), the second finger along the conventional current, and the thumb shows the direction of the force. This is the principle of the electric motor: the force on the sides of a current-carrying coil in a field produces a torque. In a loudspeaker, the force on a coil in the field of a magnet makes the cone move.
What is the force on a moving charge?
Current is moving charge, so a single charge q moving with velocity v in a field B experiences F = qvB sin θ, where θ is the angle between v and B. The force is perpendicular to both v and B (for a positive charge use the left-hand rule with the second finger along v; for a negative charge, the force is reversed). Because the force is perpendicular to the velocity, it does no work and cannot change the speed or kinetic energy of the particle: it only changes its direction. A charge at rest feels no magnetic force. If both electric and magnetic fields are present, the total force is the Lorentz force F = q(E + v × B). In a velocity selector, perpendicular E and B fields let only particles with v = E ÷ B pass straight through.
Why do charges move in circles?
When a charged particle enters a uniform magnetic field perpendicular to its velocity, the magnetic force is perpendicular to the velocity at each moment and has constant magnitude, so it acts as a centripetal force: qvB = mv2 ÷ r. The path is a circle of radius r = mv ÷ qB. The period T = 2πm ÷ qB does not depend on the speed, which is the basis of the cyclotron. If the velocity has a component along the field, the path is a helix. Applications: mass spectrometers (ions of different mass follow circles with different radii, which allows isotopes to be separated), cyclotrons and synchrotrons (accelerating particles on circular paths), cathode-ray tubes (older TVs), and bubble chamber tracks in which the curvature of the tracks shows the momentum and the sign of charge. The Earth’s field traps charged particles in the Van Allen belts and guides them to the poles, producing auroras.
Magnetic and electric forces compared
| Feature | Electric field | Magnetic field |
|---|---|---|
| Force on a charge | F = qE | F = qvB sin θ |
| Acts on charge at rest? | Yes | No |
| Direction | Along the field (or opposite) | Perpendicular to v and B |
| Does work? | Yes | No |
| Path of a charge entering perpendicular | Parabola | Circle |
Worked example: forces and orbits
- Wire in a field. A wire of length 0.20 m carries 4.0 A at right angles to a field of 0.50 T. F = BIL = 0.50 × 4.0 × 0.20 = 0.40 N. At 30° to the field: F = 0.40 × sin 30° = 0.20 N.
- Electron. An electron (q = 1.60 × 10−19 C, m = 9.11 × 10−31 kg) moves at 1.0 × 106 m/s perpendicular to a field of 0.010 T. F = qvB = 1.60 × 10−19 × 1.0 × 106 × 0.010 = 1.6 × 10−15 N.
- Radius of the circle: r = mv ÷ qB = (9.11 × 10−31 × 1.0 × 106) ÷ (1.60 × 10−19 × 0.010) = 9.11 × 10−25 ÷ 1.60 × 10−21 = 5.7 × 10−4 m = 0.57 mm.
- Proton. A proton (m = 1.67 × 10−27 kg) at 1.0 × 107 m/s in a 1.0 T field: r = 1.67 × 10−27 × 1.0 × 107 ÷ (1.60 × 10−19 × 1.0) = 0.10 m. The period T = 2πr ÷ v = 2π × 0.104 ÷ 1.0 × 107 = 6.6 × 10−8 s.
- Velocity selector. With E = 2.0 × 104 V/m and B = 0.10 T, particles with v = E ÷ B = 2.0 × 105 m/s pass without deflection.
How were magnetic forces explored?
André-Marie Ampère studied the forces between current-carrying wires in 1820. Hendrik Lorentz gave the expression for the force on a moving charge in the 1890s. J. J. Thomson used electric and magnetic fields in 1897 to deflect cathode rays and measure the charge-to-mass ratio of the electron. Ernest Lawrence built the first cyclotron in 1931. The mass spectrometer was developed by Francis Aston in the 1910s and 1920s to measure isotopes, for which he received the Nobel Prize in Chemistry in 1922.
Which mistakes are common?
Mistake 1: Thinking that the magnetic force is along the field
It is perpendicular to both the field and the velocity or current.
Mistake 2: Saying that a magnetic field speeds up a particle
The force does no work. It only changes the direction of the velocity.
Mistake 3: Using the left-hand rule with electron flow
Fleming’s rule uses conventional current. For electrons, reverse the direction.
Mistake 4: Expecting a force on a stationary charge
A static charge in a magnetic field experiences no magnetic force.