Moving Charges and Magnetism

1. Check this before you revise anything

The "Additional Exercises" section has been removed from this chapter, as from all 14 chapters of the current Class 12 Physics book. The questions run contiguously from 4.1 to 4.13 with no gaps, and a full-text search of every chapter finds zero occurrences of the phrase.

The cyclotron is promised but never taught. Older editions carried a section on motion in combined electric and magnetic fields, containing the velocity selector and the cyclotron. That section is gone: searching this chapter for "velocity selector" returns zero hits, and no cyclotron is ever described.

Yet the Introduction still tells you "We shall see how particles can be accelerated to very high energies in a cyclotron," and Summary point 3 still says the cyclotron frequency "is exploited in a machine, the cyclotron, which is used to accelerate charged particles."

So the chapter advertises a machine it no longer explains. What survives is the frequency formula itself, ${\nu }_c=qB/2\pi m$, developed in section 4.3 as a property of circular motion. That formula is examinable; the machine built on it is not part of this syllabus.

The toroid has been removed too. Section 4.7 is titled "The Solenoid" alone, where older editions read "The Solenoid and the Toroid." The word "toroid" does not occur anywhere in the chapter.

A symbol trap if you work from an extracted PDF. The permeability constant ${\mu }_0$ extracts as a plain "m0", the same micro-to-m substitution seen in Chapters 1 to 3. In this chapter it also hits the gauss values in Exercise 4.11.

2. The Lorentz Force (Textbook 4.2)

Oersted's observation that a current deflects a compass needle is the starting point: moving charges produce magnetic fields, and magnetic fields exert forces on moving charges.

The force on a single charge. A charge $q$ moving with velocity $\vec{v}$ in a magnetic field $\vec{B}$ experiences:

$\vec{F}=q(\vec{v}\times \vec{B})$

Combined with the electric force, this gives the full Lorentz force:

$\vec{F}=q(\vec{E}+\vec{v}\times \vec{B})$

Three consequences follow from the cross product, and every one of them is examined:

• The force vanishes when $\vec{v}$ is parallel or antiparallel to $\vec{B}$, since the cross product of parallel vectors is zero.
• The force vanishes for a stationary charge. A magnetic field does nothing to a charge at rest.
• The force is always perpendicular to the velocity, so it does no work and cannot change the speed.

That last point is the whole content of Exercise 4.11. The magnetic force changes only the direction of motion, never the kinetic energy.

The force on a current-carrying conductor (4.2.3). A wire is a stream of moving charges, so summing the force over all of them gives:

$\vec{F}=I(\vec{l}\times \vec{B})$

with magnitude $F=BIl\sin \theta$. The direction follows the right-hand rule or Fleming's left-hand rule.

3. Circular and Helical Motion (Textbook 4.3)

When a charge enters a field perpendicular to it, the force stays perpendicular to the velocity at every instant, which is exactly the condition for uniform circular motion. Equating the magnetic force to the centripetal requirement:

$qvB=\frac{m{v}^2}{r}⟹r=\frac{mv}{qB}$

The radius grows with momentum and shrinks with field strength.

The frequency is independent of speed. From $\omega =v/r$:

$\omega =\frac{qB}{m},{\nu }_c=\frac{qB}{2\pi m}$

Neither expression contains $v$. A faster particle traces a proportionally larger circle and completes it in the same time.

This is the result Exercise 4.12 asks you to interpret, and it is the principle the cyclotron was built on — though, as noted above, the machine itself is no longer in the chapter.

Helical motion. If the velocity has a component parallel to $\vec{B}$, that component feels no force and continues unchanged. The perpendicular component still circles, so the path is a helix. The distance advanced along the field in one revolution is the pitch:

$p=v_{\parallel }T=\frac{2\pi m{v}_{\parallel }}{qB}$

4. The Biot-Savart Law and Its Consequences (Textbook 4.4 to 4.5)

The Biot-Savart law is to magnetism what Coulomb's law is to electrostatics: it gives the field from an infinitesimal source, to be integrated over the whole configuration.

The law. A current element $Id\vec{l}$ produces at displacement $\vec{r}$ the field:

$d\vec{B}=\frac{{\mu }_0}{4\pi }\frac{Id\vec{l}\times \hat{r}}{r^2}$

with ${\mu }_0=4\pi \times 10^{-7}$ T m A${}^{-1}$. Note the inverse-square dependence and the cross product, which makes $d\vec{B}$ perpendicular to both the element and the line joining it to the point.

Straight infinite wire. Integrating along the wire gives:

$B=\frac{{\mu }_{0}I}{2\pi a}$

at perpendicular distance $a$. The field lines are concentric circles around the wire, with direction given by the right-hand thumb rule. This single formula answers Exercises 4.2, 4.3 and 4.4, where the only real work is deciding the direction.

On the axis of a circular loop (4.5). For a loop of radius $R$ carrying current $I$, at axial distance $x$:

$B=\frac{{\mu }_{0}I{R}^2}{2(R^2+x^2{)}^{3/2}}$

Setting $x=0$ gives the field at the centre, and multiplying by $N$ turns:

$B_{\text{centre}}=\frac{{\mu }_{0}NI}{2R}$

which is what Exercise 4.1 needs.

5. Ampere's Circuital Law and the Solenoid (Textbook 4.6 to 4.7)

Where the Biot-Savart law always works but often demands a hard integral, Ampere's law trades generality for speed in symmetric situations — exactly as Gauss's law does in electrostatics.

The law. For any closed loop:

$\oint \vec{B}\cdot d\vec{l}={\mu }_0{I}_{\text{enclosed}}$

Only the current threading the loop counts. Currents outside contribute nothing to the integral, though they do contribute to $\vec{B}$ at individual points.

Choosing the loop is the whole skill. The Amperian loop must be chosen so that $\vec{B}$ is either constant along it or perpendicular to it. For a straight wire, a circle of radius $a$ makes $\vec{B}$ constant and parallel to $d\vec{l}$, so the integral is simply $B(2\pi a)$, recovering the straight-wire result in one line.

The solenoid (4.7). A long, closely wound solenoid has a nearly uniform field inside and a negligible field outside. A rectangular Amperian loop straddling the winding gives:

$B={\mu }_{0}nI$

where $n$ is the number of turns per unit length — not the total number of turns. Converting the total turns and the length into $n$ is the step Exercises 4.6 and 4.8 are really testing.

For the multi-layer solenoid of Exercise 4.8, all five layers count: $n$ is the total turns across every layer divided by the length. The diameter is given but never needed, since $B$ inside does not depend on the radius.

6. Force Between Parallel Currents (Textbook 4.8)

Each wire sits in the magnetic field of the other, so each feels a force. Combining the straight-wire field with the force on a conductor gives the force per unit length:

$\frac{F}{l}=\frac{{\mu }_0{I}_1{I}_2}{2\pi d}$

The direction rule is worth memorising exactly, because it runs opposite to the electrostatic case that students meet first:

• Parallel currents, flowing the same way, attract.
• Antiparallel currents, flowing opposite ways, repel.

Two like charges repel; two like currents attract. Exercise 4.7 turns on this distinction, and Exercise 4.5 uses the same formula in its simplest form.

This relation defines the ampere. Historically, one ampere was the current which, flowing in two infinitely long parallel wires one metre apart in vacuum, produces a force of $2\times 10^{-7}$ N per metre of length. The 2019 SI revision redefined the ampere through a fixed value of the elementary charge, but this force relation remains how the unit is taught here.

7. Torque on a Current Loop, and the Galvanometer (Textbook 4.9 to 4.10)

A current loop is a magnetic dipole. In a uniform field, the forces on opposite sides of a loop are equal and opposite, so the net force is zero — but they do not act along the same line, so they produce a torque.

Defining the magnetic moment of a loop of $N$ turns, area $A$, carrying current $I$:

$\vec{m}=NI\vec{A},\vec{\tau }=\vec{m}\times \vec{B}$

with magnitude $\tau =NIAB\sin \theta$, where $\theta$ is the angle between the normal to the loop and the field.

Two facts fall straight out of this, and Exercise 4.13 asks about both:

• The torque is maximum when the plane of the coil is parallel to the field, and zero when the plane is perpendicular to it.
• The torque depends only on the enclosed area, not on the shape. A circular, square or irregular loop of equal area experiences the same torque.

A circular loop as a magnetic dipole (4.9.2). Comparing the axial field of a loop at large distances with the field of an electric dipole shows the same $1/x^3$ falloff, which is what justifies calling the loop a dipole at all. This is the bridge into Chapter 5.

The moving coil galvanometer (4.10). A coil suspended in a radial field turns until the magnetic torque balances the restoring torque of the spring, $NIAB=k\varphi$, so the deflection is proportional to the current.

$\text{Current sensitivity}=\frac{\varphi }{I}=\frac{NAB}{k},\text{Voltage sensitivity}=\frac{\varphi }{V}=\frac{NAB}{kR}$

Raising current sensitivity need not raise voltage sensitivity. If more turns are added, $N$ rises but so does the resistance $R$ of the longer wire, and the two effects can cancel exactly. Exercise 4.10 is built on precisely this: the current sensitivity ratio is 1.4 while the voltage sensitivity ratio is 1.0.

Summary

• $\vec{F}=q(\vec{v}\times \vec{B})$, and with the electric term the Lorentz force is $\vec{F}=q(\vec{E}+\vec{v}\times \vec{B})$.
• The magnetic force is always perpendicular to $\vec{v}$, so it does no work and never changes the speed — only the direction.
• No magnetic force acts on a stationary charge, or on one moving parallel to the field.
• On a conductor, $\vec{F}=I(\vec{l}\times \vec{B})$, with magnitude $BIl\sin \theta$.
• Perpendicular entry gives circular motion with $r=mv/qB$; the frequency ${\nu }_c=qB/2\pi m$ is independent of speed.
• A parallel velocity component makes the path a helix of pitch $p=2\pi m{v}_{\parallel }/qB$.
• Biot-Savart: $d\vec{B}=\frac{{\mu }_0}{4\pi }\frac{Id\vec{l}\times \hat{r}}{r^2}$, with ${\mu }_0=4\pi \times 10^{-7}$ T m A${}^{-1}$.
• Straight wire: $B={\mu }_{0}I/2\pi a$. Circular loop centre: $B={\mu }_{0}NI/2R$. On the axis: $B={\mu }_{0}I{R}^2/2(R^2+x^2{)}^{3/2}$.
• Ampere's law $\oint \vec{B}\cdot d\vec{l}={\mu }_0{I}_{\text{enc}}$ counts only the enclosed current, and is quick only when the loop is chosen to exploit symmetry.
• Solenoid: $B={\mu }_{0}nI$ with $n$ the turns per unit length; the radius does not enter.
• Parallel currents attract and antiparallel currents repel, with $F/l={\mu }_0{I}_1{I}_2/2\pi d$ — the opposite convention to like charges.
• A current loop is a magnetic dipole of moment $\vec{m}=NI\vec{A}$, feeling zero net force but a torque $\vec{\tau }=\vec{m}\times \vec{B}$ in a uniform field.
• Torque depends on the enclosed area alone, not the shape of the loop.
• Galvanometer: current sensitivity $NAB/k$, voltage sensitivity $NAB/kR$ — raising one need not raise the other.
• The cyclotron and velocity selector have been removed from this chapter, though the Introduction and Summary still refer to the cyclotron; only the frequency formula remains.
• The toroid has been removed; section 4.7 now covers the solenoid alone.