Moving Charges and Magnetism

Lesson

Part 1 — how a magnetic field pushes on moving charges, and how moving charges (currents) create magnetic fields.

In this lesson you will

Use the Lorentz force and describe the motion of a charge in a uniform magnetic field and in crossed fields
State the Biot–Savart law and derive the field on the axis of a circular loop
Apply Ampere’s circuital law to a long straight wire and describe the field of a solenoid

Concept

The Lorentz force

A charge $q$ moving with velocity $\vec v$ in fields $\vec E$ and $\vec B$ feels $\vec F = q\left[\vec E + \vec v \times \vec B\right]$ The magnetic part has size $F = qvB\sin\theta$ and is perpendicular to both $\vec v$ and $\vec B$ . Unit of $B$ : tesla, $1\ \text{T} = 1\ \text{N s C}^{-1}\,\text{m}^{-1}$ .

No magnetic force on a charge at rest, or moving parallel to $\vec B$ .
The magnetic force does no work: it changes the direction of motion, not the speed.
For a negative charge the force is opposite to $\vec v \times \vec B$ .

Table of the magnetic force for a charge at rest, moving parallel, perpendicular and at an angle to the field
Situation	Magnetic force
Charge at rest	0
v parallel to B	0
v perpendicular to B	qvB (largest)
v at angle θ to B	qvB sin θ

Diagram

Circular motion in a uniform field

If $\vec v \perp \vec B$ , the force $qvB$ is always towards a fixed centre, so the charge moves in a circle: $qvB = \dfrac{mv^2}{r} \Rightarrow r = \dfrac{mv}{qB},\qquad T = \dfrac{2\pi m}{qB},\qquad f = \dfrac{qB}{2\pi m}$ The period does not depend on the speed. With a velocity component along $\vec B$ the path is a helix.

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