Motion in electromagnetic fields

Physics · Fields. The idea in plain words, the rules to remember, the mistake to avoid and a worked example, written and checked by Mega Lecture.

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The idea

In a uniform electric field the force on a charge is constant in size and direction. A charge that enters at right angles to the field follows a parabola, like a projectile. A charge accelerated from rest through a potential difference V gains kinetic energy qV.

In a uniform magnetic field the force is always at right angles to the velocity. It changes the direction of motion but not the speed, so a charge moving at right angles to the field moves in a circle. A charge moving parallel to the field feels no force.

In crossed fields the two forces can balance for one speed only. A current-carrying wire in a magnetic field also feels a force.

Rules to remember

Electric force = charge × field strength (F = qE); kinetic energy gained = charge × potential difference (qV)
Magnetic force on a moving charge: F = qvB sin θ, where θ is the angle between the velocity and the field
Circular path: qvB = mv²/r, so radius r = mv ÷ (qB)
Crossed fields, no deflection: qE = qvB, so speed v = E ÷ B, for any charge
Force on a wire: F = BIL sin θ, where θ is the angle between the current and the field
Parallel wires: force per unit length F/L = μ₀I₁I₂ ÷ (2πr); currents in the same direction attract, opposite directions repel
The magnetic force is at right angles to both the field and the velocity (or the current); on a moving charge it does no work, so the speed stays constant

Common mistake

Students say a magnetic field speeds a charge up. The magnetic force is always perpendicular to the velocity, so the speed and kinetic energy stay constant and only the direction changes.

Worked example

An electron moves at 4.0 × 10⁶ m s⁻¹ at right angles to a uniform magnetic field of 2.0 × 10⁻³ T. What is the radius of its path? (m_e = 9.11 × 10⁻³¹ kg, e = 1.6 × 10⁻¹⁹ C)

The magnetic force provides the centripetal force: qvB = mv²/r, so r = mv ÷ (qB).
mv = 9.11 × 10⁻³¹ × 4.0 × 10⁶ = 3.64 × 10⁻²⁴ kg m s⁻¹.
qB = 1.6 × 10⁻¹⁹ × 2.0 × 10⁻³ = 3.2 × 10⁻²².
r = (3.64 × 10⁻²⁴) ÷ (3.2 × 10⁻²²) = 1.1 × 10⁻² m.

Answer: 1.1 × 10⁻² m (about 1.1 cm)

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D.3 Motion in Electromagnetic Fields

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