Moving Charges and Magnetism: Class 12 Physics · Chapter 4

4.1 Introduction

Every electric motor, loudspeaker, hospital MRI scanner and particle accelerator depends on a link that stayed hidden until 1820: an electric current produces a magnetic field. Before then the two subjects had separate histories and separate toolkits. Electricity meant rubbed amber, sparks and Leyden jars; magnetism meant lodestones and the compass needles that sailors steered by.

Hans Christian Oersted, a professor in Copenhagen, came across it in 1820 while showing an experiment to his students, and it is easy to see for yourself (Fig. 4.1). Lay a wire across the top of a compass, parallel to its needle, and connect it for a moment to a cell. The needle kicks sideways and stays turned for as long as the current flows. Swap the connections and it kicks the other way. The kick grows with the current, and shrinks as the wire is lifted away from the needle.

Where exactly is the needle trying to point? Stand a current-carrying wire upright through a card, and surround it with a ring of small compasses or a sprinkling of iron filings. The needles and the filings line up along circles centred on the wire. So the field of a straight current does not point away from its source, the way the electric field of a charge does; it wraps round it. A current, then, does more than warm a wire or light a bulb: it fills the space around it with a magnetic field. That discovery is the starting point of this chapter.

Two questions run through the chapter. How does a magnetic field push on a moving charge, or on a wire carrying a current? And how does a current create a magnetic field in the first place? The answers explain why charged particles circle and spiral in a field (the idea behind the cyclotron), why neighbouring wires pull on each other, and how a small coil turning in a magnet becomes the moving coil galvanometer used to measure currents and voltages.

Many diagrams in this chapter show currents and fields that point straight into or out of the page, and we draw them with two symbols borrowed from archery (Fig. 4.2). A circle with a dot marks something coming out of the page, towards you, like the point of an arrow flying straight at you. A circle with a cross marks something going into the page, away from you, like the crossed flights at the tail of an arrow that is flying away.

4.2 Magnetic Force

4.2.1 Sources and fields

In Chapter 1 we split the electric interaction into two steps. A charge $Q$ first fills the space around it with an electric field; a second charge $q$ then responds to whatever field it finds at its own position. For a point charge at the origin, the field at position $𝐫$ is

and the force on a charge $q$ placed there is

Chapter 1 left us two ideas worth keeping. The first is superposition: the field of many sources is the vector sum of the fields they would make one at a time. The second is that a field is more than bookkeeping. When charges jiggle in a mobile-phone tower, the charges in your phone respond only after the disturbance has crossed the gap at the speed of light, carrying energy and momentum with it. Magnetic fields share both properties. Here, though, every current is steady, so every field stays the same from moment to moment and no delays arise.

Magnetism works in the same two steps. Moving charges, and so currents, set up a magnetic field $𝐁(𝐫)$ (चुंबकीय क्षेत्र), a vector with a size and direction at every point in space; any other moving charge then responds to the field it finds at its own position. What is new is how it responds, and that is where we begin.

4.2.2 Magnetic field, Lorentz force

Fire a charged particle through a region that contains both an electric field $𝐄$ and a magnetic field $𝐁$. The electric force is the familiar $q𝐄$. The magnetic force turns out to depend on the particle's velocity $𝐯$ as well, through a cross product. The complete force on a point charge $q$ at position $𝐫$ is

This expression, named after the Dutch physicist Hendrik Lorentz, summarises many experiments by Ampère and his successors. Its magnetic part, $q𝐯\times 𝐁$, is strikingly unlike the electric part:

• A charge at rest feels no magnetic force at all, however strong the field. The force exists only while the charge moves.
• The force is always at right angles to the velocity and to the field, in the direction of $𝐯\times 𝐁$ for a positive charge (Fig. 4.3).
• Its size, $qvB\sin \theta$, depends on the angle $\theta$ between $𝐯$ and $𝐁$. Motion along the field lines, forwards or backwards, gives no force.
• Changing the sign of the charge reverses the force: an electron and a proton moving side by side are pushed in opposite directions.

To find the direction of $𝐯\times 𝐁$, point the fingers of your right hand along $𝐯$ and curl them towards $𝐁$ through the smaller angle between them. Your thumb then points along $𝐯\times 𝐁$. For a positive charge this is the direction of the magnetic force; for a negative charge the force is the opposite way. In Cartesian axes the cyclic products $\widehat{𝐢}\times \widehat{𝐣}=\widehat{𝐤}$, $\widehat{𝐣}\times \widehat{𝐤}=\widehat{𝐢}$ and $\widehat{𝐤}\times \widehat{𝐢}=\widehat{𝐣}$ do the same job.

The formula also tells us how to measure $B$. Rearranged, $B=F\mathrm{/}(qv\sin \theta )$, so the SI unit of magnetic field is a newton per coulomb per (metre per second). This unit is the tesla (T), named after the engineer Nikola Tesla: a $1\mathrm{T}$ field pushes with a force of $1\mathrm{N}$ on a $1\mathrm{C}$ charge crossing it at right angles at $1\mathrm{m}\mathrm{/}\mathrm{s}$. In base units,

Everyday fields are tiny in these units. The Earth's field, the one that turns a compass, is only a few times ${10}^{-5}\mathrm{T}$, so for weak fields people often use the older, non-SI gauss ($1\mathrm{G}={10}^{-4}\mathrm{T}$). Table 4.2 shows how enormous the range of magnetic fields is.

4.2.3 Magnetic force on a current-carrying conductor

A wire carrying a current is full of moving charges, so a magnetic field must push on the wire as a whole. Take a straight rod, length $l$ and cross-section $A$, lying in a uniform external field $𝐁$ (Fig. 4.4). Suppose, as in a metal, that it has one kind of mobile carrier, with charge $q$ and number density $n$.

Two cautions go with (4.4). First, $𝐁$ is the external field, not the field that the rod's own current creates: a wire cannot push itself along. Second, the formula is for a straight piece of wire in a uniform field. A wire of any shape can be broken into short, nearly straight pieces $d{𝐥}_j$, and the forces on them added as vectors,

a sum that becomes an integral when the pieces are made very short.

4.3 Motion in a Magnetic Field

The rate at which a force does work on a moving particle is the power $P=𝐅\cdot 𝐯$. For the magnetic force, $𝐅\cdot 𝐯=q(𝐯\times 𝐁)\cdot 𝐯=0$, because a cross product is perpendicular to each of the vectors that make it. So a magnetic field never does work on a charged particle and never changes its kinetic energy. It steers the particle, changing the direction of its momentum, but it cannot speed it up or slow it down. To give a particle energy you need an electric field, whose force $q𝐄$ can point along the motion; that is how every particle accelerator adds energy.

Velocity at right angles to the field

Let a particle of mass $m$ and charge $q$ move with speed $v$ in a uniform field $𝐁$, with $𝐯$ perpendicular to $𝐁$. The magnetic force has a constant size $qvB$ and always points sideways, at right angles to the motion. A force of constant size that stays perpendicular to the velocity is exactly what uniform circular motion needs: the magnetic force supplies the centripetal force, and the particle moves in a circle in the plane perpendicular to $𝐁$ (Fig. 4.5).

(4.5) says that a particle with more momentum $mv$ moves on a larger circle, and that a stronger field bends it into a tighter one. (4.6(a)) holds a surprise: the speed has cancelled out. A fast particle runs round a big circle and a slow one round a small circle, but both take exactly the same time. The frequency $\nu =qB\mathrm{/}2\pi m$ is called the cyclotron frequency (साइक्लोट्रॉन आवृत्ति); it depends only on the particle's charge-to-mass ratio and on the field.

Velocity at an angle: the helix

Suppose instead that $𝐯$ makes an angle $\theta$ with $𝐁$. Split the velocity into a component along the field, $v_{\parallel }=v\cos \theta$, and a component across it, $v_{\perp }=v\sin \theta$. The magnetic force depends only on $v_{\perp }$ (the cross product of $𝐁$ with a vector along $𝐁$ is zero), so $v_{\parallel }$ never changes. The perpendicular motion is a circle of radius $m{v}_{\perp }\mathrm{/}qB$, just as before. Put together, the particle drifts steadily along the field while it circles round it: its path is a helix (कुंडलिनी), like the thread of a screw.

Two numbers describe the helix. Its radius, $r=m{v}_{\perp }\mathrm{/}qB$, is the radius of the circle you would see looking straight along the field. Its pitch $p$ (चूड़ी अंतराल) is how far the particle advances along the field in one turn, like the spacing between the threads of a screw:

4.4 Magnetic Field due to a Current Element, Biot–Savart Law

So far the magnetic field has simply been given. Now we ask how a current produces it. Only two sources of magnetic field are known: electric currents, that is, moving charges, and the intrinsic magnetism carried by particles such as electrons. This section deals with the first. The rule for currents was found in 1820 by Jean-Baptiste Biot and Félix Savart, who timed the swings of a magnetised needle at different distances from current-carrying wires.

Picture a wire carrying a steady current $I$, cut up in imagination into tiny pieces. Choose one piece of length $dl$, and let $d𝐥$ be a vector of that length pointing along the current. The product $Id𝐥$ is called a current element. We want the field $d𝐁$ that this element produces at a point P, where $𝐫$ is the displacement from the element to P and $\theta$ is the angle between $d𝐥$ and $𝐫$ (Fig. 4.7).

In symbols,

Because $\mathrm{\mid }d𝐥\times 𝐫\mathrm{\mid }=rdl\sin \theta$, the magnitude of the field is

The constant ${\mu }_0$ is called the permeability of free space (निर्वात की चुंबकशीलता). In SI units its value is

so ${\mu }_0=4\pi \times {10}^{-7}\mathrm{T}\mathrm{m}\mathrm{/}\mathrm{A}$.11For decades this value was exact by definition. Since the 2019 revision of the SI, ${\mu }_0$ is a measured quantity, but it still equals $4\pi \times {10}^{-7}\mathrm{T}\mathrm{m}\mathrm{/}\mathrm{A}$ to better than one part in a billion. For the direction, use the right-hand rule for a cross product: fingers along $d𝐥$, bent towards $𝐫$, and the thumb gives $d𝐁$. In Fig. 4.7 this puts the field into the page at points to the right of an upward current element and out of the page at points to its left.

It is worth setting the Biot–Savart law beside Coulomb's law, which gives the electric field of a point charge (Table 4.3).

The two constants of electricity and magnetism are tied together by the speed of light $c$ in vacuum:

The match is no accident. Light is an electromagnetic wave, and Chapter 8 shows that its speed is fixed by these two constants, $c=1\mathrm{/}\sqrt{{\mu }_0{\epsilon }_0}$. Since $c$ is known exactly, ${\mu }_0$ and ${\epsilon }_0$ carry the same information: a laboratory that pins down one of them has, in effect, measured the other as well.

4.5 Magnetic Field on the Axis of a Circular Current Loop

The Biot–Savart law gives the field of any current once the contributions of all its elements are added up. The simplest shape for which the sum can be done exactly is a circular loop, and the simplest place to do it is a point on the loop's axis.

Take a loop of radius $R$ carrying a steady current $I$ in vacuum, with its centre O at the origin, its plane in the $yz$-plane and its axis along the $x$-axis (Fig. 4.8). We want the field at a point P on the axis, a distance $x$ from O.

At the centre of the loop, $x=0$, and the field is

A flat coil of $N$ closely wound turns acts like $N$ loops on top of one another, so both results are multiplied by $N$. Far out along the axis, where $x\gg R$, the field falls off as $1\mathrm{/}{x}^3$; we come back to this in Section 4.9.2.

The field lines of a current loop are closed curves that thread through the loop (Fig. 4.9). Their direction follows a right-hand thumb rule: curl the fingers of your right hand round the loop in the direction of the current, and your thumb points along the field inside the loop. Seen from the face where the field lines come out, the current flows anticlockwise. That face behaves like the north pole of a magnet, and the opposite face, where the current looks clockwise, like a south pole.

4.6 Ampère's Circuital Law

Adding up current elements with the Biot–Savart law always works, but the integrals can be hard. Ampère's circuital law packages the same physics differently, and whenever a current arrangement is symmetric enough it hands us the field almost for free. It does for magnetism what Gauss's law does for electrostatics.

The law is about a quantity called the circulation of $𝐁$ round a closed path C. Imagine carrying a probe once round C, keeping a running total of how much the field points along your direction of travel: for each short step $d𝐥$ you add $𝐁\cdot d𝐥$, which is positive where the field points the way you are going, negative where it points back, and zero where it points across your path. The total for one complete trip is written ${\oint }_C𝐁\cdot d𝐥$ (Fig. 4.11). Ampère's law says that this total depends on nothing except the current that threads the path.

Which currents count as positive? Look at C from the side from which your walk round it appears anticlockwise. Currents coming towards you count as positive and currents going away from you as negative. (This is the right-hand rule in disguise: fingers along the walk, thumb along a positive current.) Currents that do not thread C do not count at all, however close to it they pass; they change $𝐁$ along the path, but their contributions to the circulation add up to zero.

The circulation is easy to work out when the path follows the symmetry of the field. On a stretch where $𝐁$ runs along the path with a constant size $B$, the contribution is simply $B$ times the length of that stretch. On stretches where $𝐁$ crosses the path at right angles, or is zero, the contribution is zero. A path chosen so that it is made only of such stretches is called an Amperian loop (ऐम्पियरी पाश). If the stretches of the first kind have total length $L$ and the loop encloses a net current $I_e$, the law becomes

Field of a long straight wire

An infinitely long straight wire carrying a current $I$ has cylindrical symmetry. Nothing changes if we turn the wire about its own axis or slide it along its length. So the field can depend only on the distance $r$ from the wire, and it circles round the wire. Choose as the Amperian loop a circle of radius $r$ centred on the wire, in a plane at right angles to it. $𝐁$ is tangential to this circle with the same size all the way round, so $L=2\pi r$ and $I_e=I$:

Three quick checks make this result believable. Distance: the field falls off as $1\mathrm{/}r$, more slowly than the $1\mathrm{/}{r}^2$ of a single current element, because a long wire has elements at every distance from you. Infinity: the wire never ends, yet the field is finite everywhere off the wire; its far-off parts are distant and seen almost end-on, where $\sin \theta$ is small, so they add less and less. Size: a current of $10\mathrm{A}$ in a single straight wire gives $2\times {10}^{-5}\mathrm{T}$ at $10\mathrm{c}\mathrm{m}$. That is about half the Earth's field, weak by laboratory standards, yet enough to pull a compass held that close about ${30}^{\circ }$ off north.

Because the field has the same size all round any circle centred on the wire and points along that circle, the field lines are rings round the wire, which is exactly the pattern Oersted's compasses traced out. Like every magnetic field line they close on themselves, with no beginning or end; electric field lines, by contrast, start and stop on charges. The right-hand grip rule gives their sense: wrap your right hand round the wire with the thumb pointing along the current, and the curl of your fingers shows which way the field lines run.

When should you use which law? The Biot–Savart law always gives the answer, though it may need a hard integral. Ampère's law is always true, but it is useful for finding $𝐁$ only when symmetry lets us take $B$ outside the integral: long straight wires, thick wires, solenoids, toroids. For a single circular loop no path has a constant field along it, so the field at the centre had to come from the Biot–Savart law in Section 4.5. The two laws are not rivals: Ampère's law can be proved from the Biot–Savart law, and for steady currents they contain the same physics.