Moving Charges and Magnetic Field — Part 2: Forces on moving charges and currents: visual lesson, ISC Class 12 Physics

Lesson

Forces on moving charges and currents

Part 2 — what a magnetic field does to moving charges and to the wires that carry them.

In this lesson you will

Use the Lorentz force, explain why a magnetic force does no work, and describe circular and helical paths
Use F = I(l × B) for the force on a current-carrying conductor
Derive the force between parallel currents and define the ampere and the coulomb

Concept

The Lorentz force

In electric and magnetic fields together, a charge $q$ moving with velocity $\vec v$ feels the Lorentz force $\vec F = q\left(\vec E + \vec v \times \vec B\right)$ For a negative charge the magnetic force is opposite to the direction of $\vec v \times \vec B$ .

Table of the magnetic force on a charge at rest, moving along the field, moving at right angles to it and moving at another angle, and the path that results in each case
Situation	Magnetic force qvB sin θ	Result
Charge at rest (v = 0)	zero	stays at rest
v along B (θ = 0° or 180°)	zero	straight line, constant speed
v perpendicular to B (θ = 90°)	qvB, largest	circle
v at another angle θ	qvB sin θ	helix

Diagram

No work, so constant speed: circular motion

The magnetic force is always perpendicular to $\vec v$ , so it does no work: $P = \vec F \cdot \vec v = 0$ . The speed and kinetic energy stay constant; only the direction changes. If $\vec v \perp \vec B$ , the force provides the centripetal force: $qvB = \dfrac{mv^2}{r} \;\Rightarrow\; r = \dfrac{mv}{qB} = \dfrac{p}{qB},\qquad T = \dfrac{2\pi m}{qB}$

T and the frequency $f = qB/2\pi m$ do not depend on the speed: faster particles move on bigger circles.

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