Electromagnetic Fields and Waves: Notes, Solved Problems and Interactive Lab

What the experiments show

Coulomb's law of Section 3.1 tells us the force on a charge $q$ that stands still in the field $E$ of some source: $F = q E$ . Now let the test charge move, with velocity $v$ , near a magnet or near a wire that carries a current. Experiments show three things:

the force on the moving charge is different from $q E$ ,
the difference $F - q E$ is proportional to the speed of the charge (a charge at rest feels only $q E$ , and doubling $v$ doubles the extra force), and
the difference is perpendicular to the velocity $v$ , whatever the direction of $v$ .

A force that is always perpendicular to the velocity and proportional to it has the form of a vector product with some fixed vector. This observation is turned into the basic assumption of magnetism.

Definition · Assumption: the Lorentz law

Let the force acting on a point charge $q$ which moves in an electric field $E$ with a certain velocity $v$ be $F$ . Then the difference $F - q E$ is both perpendicular to and linearly dependent on the product $q v$ . The vector that expresses this linear dependence is called the magnetic induction or magnetic flux density $B$ :

F = q (E + v \times B) [N], B in [T] = [Wb/ m^{2}] . (4.1.2)

The unit of $B$ is the tesla, $1 T = 1 N/ (A m) = 1 Wb/ m^{2}$ (the weber, Wb, is the unit of magnetic flux, Section 4.1.6). The Earth's field at the surface is $25$ to $65$ μT, a strong permanent magnet gives a few tenths of a tesla at its surface, and the magnet of a hospital MRI scanner $1.5$ to $3$ T. The older unit gauss is $1 0^{- 4}$ T.

Figure 4.1.1 shows the magnetic part of the force. A positive charge moving to the right through a field pointing into the page is pushed upward, at right angles both to $v$ and to $B$ . The three vectors $v$ , $B$ , $F$ form a right-handed triad: point the fingers of the right hand along $v$ , curl them towards $B$ , and the thumb gives $v \times B$ . For a negative charge the force is reversed. The magnitude is

∣ F_{mag} ∣ = ∣ q ∣ v B sin θ, (4.1.3)

where $θ$ is the angle between $v$ and $B$ : largest when the charge moves across the field, zero when it moves along it.

The magnetic force on a moving charge. Left: a positive charge moving to the right in a field \vec B pointing into the page (\otimes) is pushed upward, perpendicular to both \vec v and \vec B. Because the force stays perpendicular to the velocity, it only turns the velocity and the charge runs on a circle (dashed). Right: \vec v, \vec B and \vec F=q\vec v\times\vec B form a right-handed triad. — **Figure 4.1.1.** The magnetic force on a moving charge. Left: a positive charge moving to the right in a field $B$ pointing into the page ( $\otimes$ ) is pushed upward, perpendicular to both $v$ and $B$ . Because the force stays perpendicular to the velocity, it only turns the velocity and the charge runs on a circle (dashed). Right: $v$ , $B$ and $F = q v \times B$ form a right-handed triad.

Drag the velocity arrow or change $B$ : the charge runs on a circle whose radius grows with $v$ , while the period stays the same.

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In plain words

An electric field pushes a charge along the field, whether the charge moves or not. A magnetic field pushes a charge sideways, at right angles to its motion, and only while it moves. The faster the charge, the harder the sideways push.

The magnetic force does no work

Because $F_{mag} = q v \times B$ is perpendicular to $v$ , the power it delivers is $F_{mag} \cdot v = 0$ . A magnetic field can bend the path of a charge, but it can never speed it up or slow it down: the kinetic energy of a particle in a pure magnetic field is constant. This is why a charge in a uniform $B$ moves on a circle at constant speed (Home exercise 4.1.4), and why particle accelerators use electric fields to accelerate and magnetic fields to steer.

Components, and why B is an axial vector

Writing all four vectors in Cartesian components,

F v E B = F_{x} a_{x} + F_{y} a_{y} + F_{z} a_{z}, = v_{x} a_{x} + v_{y} a_{y} + v_{z} a_{z}, = E_{x} a_{x} + E_{y} a_{y} + E_{z} a_{z}, = B_{x} a_{x} + B_{y} a_{y} + B_{z} a_{z},

and expanding the vector product as in Chapter 2, the Lorentz force reads

F_{x} F_{y} F_{z} = q (E_{x} + v_{y} B_{z} - v_{z} B_{y}), = q (E_{y} + v_{z} B_{x} - v_{x} B_{z}), = q (E_{z} + v_{x} B_{y} - v_{y} B_{x}) . (4.1.4)

These three equations are what one programs when tracing a particle through a given field, and they are used in Exercise 4.1.1.

Pause the motion and compare the arrows $q E$ , $q v \times B$ and $F$ : the magnetic part turns with $v$ and never changes the speed.

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Quick check

A proton moves along $+ a_{x}$ at $1 0^{6}$ m/s through $B = 0.1 a_{z}$ T. What are the direction and magnitude of the magnetic force? What if it were an electron?

$v \times B = v B (a_{x} \times a_{z}) = - v B a_{y}$ , so $F = - e v B a_{y} = - 1.602 \times 1 0^{- 19} \times 1 0^{6} \times 0.1 a_{y} = - 1.6 \times 1 0^{- 14} a_{y}$ N. For an electron $q = - e$ and the force is $+ 1.6 \times 1 0^{- 14} a_{y}$ N: same size, opposite direction.

The force on a current-carrying wire

In a wire the moving charges are the carriers, and the Lorentz force on each of them adds up to a force on the wire. Consider a straight piece of wire of length $l$ and cross-section $A$ carrying the current $I$ in a uniform field $B$ (Figure 4.1.2). If the wire contains $n$ carriers per unit volume, each of charge $q$ and drift velocity $v$ , the current is $I = n q v A$ (this relation is derived in Section 4.1.2). The piece contains $n A l$ carriers, each feeling $q v \times B$ , so

F = n A l q v \times B = (n q v A) l a_{v} \times B,

where $a_{v}$ is the unit vector along the drift. Writing $l = l a_{v}$ for the piece of wire taken in the direction of the current,

Force on a straight current-carrying wire

F = I l \times B [N], and for a curved wire F = I \int_{C} d l \times B . (4.1.5)

The force on a wire is the Lorentz force on its carriers added up. Each carrier, drifting with \vec v along the wire, feels q\vec v\times\vec B (short green arrows). The nAl carriers of a piece of length l together give I\,\vec l\times\vec B, where \vec l points along the current. The carriers cannot leave the wire, so the force is handed over to the metal. — **Figure 4.1.2.** The force on a wire is the Lorentz force on its carriers added up. Each carrier, drifting with $v$ along the wire, feels $q v \times B$ (short green arrows). The $n A l$ carriers of a piece of length $l$ together give $I l \times B$ , where $l$ points along the current. The carriers cannot leave the wire, so the force is handed over to the metal.

Note that the result contains the current and not the sign of the carriers: in copper the carriers are electrons ( $q < 0$ ) moving against $I$ , and the two sign changes cancel. The direction of the force is given by the right hand with the fingers along the current. A wire of $1$ m carrying $10$ A across a field of $0.5$ T feels $F = I l B = 5$ N, about the weight of half a kilogram. This is the force that drives every electric motor (Application 4.1.2).

Common mistake · Using the carrier velocity instead of the current

Trying to find the direction of the force on a copper wire by thinking about the electrons, and forgetting that they carry a negative charge. The force on the wire is $I l \times B$ with $l$ along the conventional current. A positive carrier moving with $I$ and a negative carrier moving against $I$ are pushed to the same side, which is exactly why the Hall effect (Application 4.1.1) is needed to tell them apart.

Quick check

A horizontal wire runs from west to east carrying $20$ A. The Earth's field at the place has the horizontal component $20$ μT pointing north. What is the force on $10$ m of the wire?

$l$ east, $B$ north, so $l \times B$ points up (east $\times$ north $=$ up). $F = I l B = 20 \times 10 \times 20 \times 1 0^{- 6} = 4$ mN, a vertical force far too small to notice against the weight of the wire.

4.1.1 The Lorentz force

What the experiments show

Components, and why B is an axial vector

The force on a current-carrying wire