4.1 Lorentz Force, Biot–Savart's Law and the Vector Potential

Moving charges feel a sideways force in a magnetic field and, as currents, produce one: the Lorentz force says what the field does, Biot–Savart's law says where it comes from, and from the two follow the vector potential, Ampère's law and the fields of wires, loops and coils.

39 solved examples and exercises6 interactive tools

In this section you will learn

what a magnetic field does: the Lorentz force $F = q (E + v \times B)$ on a moving charge, and its wire form $F = I l \times B$ ,
how a current is described by the current density $J$ $[A/ m^{2}]$ , and why a moving charge is a current,
where a magnetic field comes from: Biot–Savart's law, the magnetic counterpart of Coulomb's law, and the two field vectors $B$ $[T]$ and $H$ $[A/m]$ ,
what the vector potential $A$ is, why $B = \nabla \times A$ , and the two fundamental equations of magnetostatics, $\nabla \times H = J$ and $\nabla \cdot B = 0$ ,
what magnetic flux is, and why Ampère's law $\oint_{C} H \cdot d l = I_{enc}$ turns symmetric problems into one-line calculations, and
how all this explains Hall sensors, DC motors, the field under a power line, clamp meters and flat charging coils.

Chapter 3 dealt with charges at rest. Everything there grew out of one experimental fact, Coulomb's law, and one mathematical tool, superposition. In this chapter the charges move. A moving charge is a current, and currents do something that resting charges never do: they push and pull on other moving charges with a force that is not the Coulomb force. A compass needle turns next to a wire that carries a current (Ørsted, 1820), two parallel wires attract each other when their currents run the same way (Ampère, 1820), and a beam of electrons bends in a curve when it passes a magnet. None of this is explained by $E$ alone.

We shall follow the same road as in electrostatics. First we say what the new force does: this is the Lorentz force, which introduces the magnetic flux density $B$ . Then we say where the field comes from: this is Biot–Savart's law, which plays the role of Coulomb's law. From the two we derive, exactly as Gauss's law and $\nabla \times E = 0$ followed from Coulomb's law, the two fundamental equations of magnetostatics. One of them, Ampère's law, is the magnetic twin of Gauss's law and will do most of the work in the problems. The section is the largest of the chapter because it contains both the theory (Lecture 8) and its exercise session (Lecture 9).

Throughout, as in Chapter 3, we take the engineering point of view: the field intensities $E$ and $H$ are the quantities we work with, and $D = ε_{0} E$ and $B = μ_{0} H$ are derived from them, with

μ_{0} = 4 π \times 1 0^{- 7} H/m (4.1.1)

the permeability of free space. The combination $μ_{0} /2 π = 2 \times 1 0^{- 7}$ H/m will appear in almost every numerical example of this section.

4.1 Lorentz Force, Biot–Savart's Law and the Vector Potential

Contents of Section 4.1

Problems · 8

Home exercises · 7

Further exercises · 20

Worked examples · 4