Chapter 4 · Introduction

Chapter 4 — Moving Charges and Magnetism

Moving charges feel magnetic forces and make magnetic fields — the physics of motors, meters, MRI scanners and particle accelerators.

In the spring of 1820, during a lecture in Copenhagen, Hans Christian Ørsted noticed that a compass needle lying near a wire swung round whenever he switched the current on. Until that moment electricity and magnetism had been thought of as separate subjects; within a few months André-Marie Ampère in Paris had shown that two wires carrying currents push and pull on each other, and Biot and Savart had measured how the field of a current falls off with distance. Electric currents, it turned out, are a source of magnetism — and magnets push on moving charges.

That discovery runs much of the modern world. Every electric motor turns because a coil carrying a current sits in a magnetic field; the scanner in a hospital uses a current-carrying coil to make a strong, uniform field; particle accelerators and mass spectrometers steer charged particles round curves; and an analogue meter measures current by letting a coil twist against a spring. This chapter answers three questions. What force does a magnetic field exert on a moving charge and on a current? What magnetic field does a current produce — for a short element, a circular loop, a long wire and a solenoid? And what happens to a current loop placed in a field — the torque that drives motors and meters?

Lorentz force→Force on a current→Circles & helices→Biot–Savart law→Ampère's law & solenoid→Parallel currents→Torque & galvanometer
🎯 By the end of this chapter you can…
  • use $\mathbf F=q(\mathbf E+\mathbf v\times\mathbf B)$ and $\mathbf F=I\mathbf l\times\mathbf B$ to find magnetic forces on charges and wires, with directions;
  • find the radius, period and pitch of a charged particle moving in a uniform magnetic field;
  • state the Biot–Savart law and use it for the field on the axis of a circular loop and at the centre of an arc;
  • use Ampère's circuital law for a long straight wire (inside and outside) and a long solenoid, and find the force between parallel currents and the definition of the ampere;
  • find the torque $\mathbf m\times\mathbf B$ on a current loop, treat a loop as a magnetic dipole, explain the moving-coil galvanometer and convert it into an ammeter or a voltmeter;
  • solve board, JEE Main, JEE Advanced and NEET problems on all of the above.
Teaser — Ørsted's Compass Needles

A long straight wire runs perpendicular to the page through the middle of a grid of compass needles (grid squares are $2$ cm). Each needle lines up with the total field: a weak uniform background field of $4.0\times10^{-5}$ T pointing up the page (about the size of the Earth's) plus the field of the wire. Switch the current on, reverse it, and drag the wire. Close to the wire its field wins and the needles circle round it.

20
current—
wire's field 5 cm away (T)—
distance where it equals the background—

Sections in this chapter

4.1

The Magnetic Force and Motion of a Charge in a Magnetic Field

The Lorentz force and the tesla, the force on a current-carrying wire, and circular and helical paths.

$\mathbf F=q\mathbf v\times\mathbf B$, $\ r=\dfrac{mv}{qB}$
Open section →4.2

The Biot–Savart Law and the Field of a Circular Loop

The field of a current element, the field on the axis of a loop and at the centre of an arc.

$B=\dfrac{\mu_0IR^2}{2(x^2+R^2)^{3/2}}$
Open section →4.3

Ampère's Circuital Law, the Solenoid and Forces between Currents

Ampère's law, long straight wires, the solenoid, parallel currents and the ampere.

$\oint\mathbf B\cdot d\mathbf l=\mu_0I_\text{enc}$, $\ B=\mu_0nI$
Open section →4.4

Torque on a Current Loop and the Moving-Coil Galvanometer

The torque on a loop, the magnetic dipole moment, the galvanometer, ammeters and voltmeters.

$\boldsymbol\tau=\mathbf m\times\mathbf B$, $\ S=\dfrac{I_gG}{I-I_g}$
Open section →
✅ Before you start
You need the vector (cross) product — its magnitude $ab\sin\theta$, the right-hand rule and the products of $\hat{\mathbf i},\hat{\mathbf j},\hat{\mathbf k}$ — from Class 11 Physics (Chapter 6, torque and angular momentum), and uniform circular motion with $mv^2/r$ from Class 11 Physics, Chapter 3. From this course, Chapter 1 supplies the electric field and the electric dipole (Section 1.2), whose field and torque the current loop imitates, and Gauss's law (Section 1.3), the model for Ampère's law. Chapter 3 (Current Electricity) supplies electric current and drift velocity; the series and parallel rules for resistors (Class 10 Physics) are used for ammeters and voltmeters. Class 10 Physics, "Magnetic Effects of Electric Current", gives a qualitative first look at the field patterns of a wire, a loop and a solenoid.
← 3.4 Kirchhoff's Rules and the Wheatstone Bridge4.1 The Magnetic Force and Motion of a Charge in a Magnetic Field →