Magnetic Field

SYLLABUS

Magnetic field lines and magnetic flux; Oersted’s experiment, Force on moving charge; Force on a conductor, Force and Torque on rectangular coil, Moving coil galvanometer, Hall effect, Magnetic field of a moving charge, Biot and Savert law and its application to (i) a circular coil (ii) a long straight conductor (iii) a long solenoid, Ampere’s law and its applications to (i) a long straight conductor (ii) a straight solenoid (ii) a long straight conductor (iii) a long solenoid, Ampere’s law and its applications to (i) a long straight conductor (ii) a straight solenoid (iii) a toroidal solenoid, Force between two parallel conductors carrying current-definition of ampere

What is This Chapter About?

Hold two bar magnets near each other — like poles repel, opposite poles attract. Pass an electric current through a wire near a compass and the compass needle deflects. Place a current-carrying wire between the poles of a strong magnet and the wire experiences a force. These observations reveal one of the most beautiful and fundamental connections in physics: electricity and magnetism are intimately linked. Moving electric charges create magnetic fields, and magnetic fields exert forces on moving charges.

Chapter 16 of NEB Grade 12 Physics is one of the most extensive and mathematically rich chapters in the entire course, carrying 9 teaching hours — the highest of any Grade 12 chapter. It covers the nature of magnetic fields, forces on charges and currents in magnetic fields, the laws governing the magnetic fields produced by current-carrying conductors, and the Hall effect. Together these topics form the foundation of electromagnetism — the unified understanding of electricity and magnetism that underlies motors, generators, transformers, and virtually all modern electrical technology.

Key Concepts You Will Learn

Magnetic Field Lines and Magnetic Flux

A magnetic field is a region of space in which a magnetic force can be experienced. It is represented by magnetic field lines — continuous curves whose direction at any point gives the direction of the magnetic field, and whose density indicates the field strength. Magnetic field lines always form closed loops (unlike electric field lines, which start and end on charges) — this reflects the fact that isolated magnetic poles (monopoles) do not exist.

Magnetic flux ([latex]\Phi[/latex]) through a surface is the total number of magnetic field lines passing through it: [latex]\Phi = BA\cos\theta[/latex], where B is the magnetic flux density, A is the area, and [latex]\theta[/latex] is the angle between B and the normal to the surface. The SI unit of magnetic flux is the weber (Wb), and the unit of flux density B is the tesla (T) = Wb/m².

Oersted’s Experiment

In 1820, Hans Christian Oersted made the pivotal discovery that an electric current creates a magnetic field — placing a compass near a current-carrying wire showed the compass needle deflecting. This was the first demonstration of a connection between electricity and magnetism. The magnetic field lines around a long straight current-carrying conductor form concentric circles centred on the wire, with direction given by the right-hand rule: if the thumb points in the direction of current, the curled fingers show the direction of the field.

Force on a Moving Charge

A charge q moving with velocity v in a magnetic field B experiences a force called the Lorentz force:

[latex]\vec{F} = q\vec{v} \times \vec{B}[/latex]

The magnitude is [latex]F = qvB\sin\theta[/latex], where [latex]\theta[/latex] is the angle between v and B. The direction of the force is perpendicular to both v and B, given by the right-hand rule (or left-hand rule for negative charges). Key consequences: the magnetic force does no work (always perpendicular to motion), and a charge moving parallel to B experiences no force.

A charged particle moving perpendicular to a uniform magnetic field follows a circular path, with the magnetic force providing centripetal acceleration: [latex]r = \frac{mv}{qB}[/latex]. This principle underlies particle accelerators, mass spectrometers, and the aurora borealis.

Force on a Current-Carrying Conductor

Since a current is a flow of charges, a current-carrying conductor in a magnetic field experiences a force. For a conductor of length L carrying current I in a field B:

[latex]F = BIL\sin\theta[/latex]

The direction is given by Fleming’s Left-Hand Rule: if the index finger points along B, the middle finger along I, the thumb gives the direction of force. This is the principle behind electric motors — a current-carrying coil in a magnetic field experiences a torque that causes it to rotate.

Torque on a Rectangular Coil

A rectangular coil of N turns, area A, carrying current I in a uniform magnetic field B experiences a torque:

[latex]\tau = NIAB\sin\alpha[/latex]

where [latex]\alpha[/latex] is the angle between the plane of the coil and the magnetic field. Maximum torque occurs when the coil plane is parallel to B ([latex]\alpha = 90°[/latex] to the normal). This torque is the operating principle of moving coil galvanometers, electric motors, and loudspeakers.

Moving Coil Galvanometer

A moving coil galvanometer consists of a rectangular coil suspended in a radial magnetic field (produced by curved pole pieces and a cylindrical soft iron core). Current through the coil produces a torque that is balanced by a restoring spring torque. At equilibrium: [latex]NIAB = k\phi[/latex], where k is the spring constant and [latex]\phi[/latex] is the angular deflection. The deflection is proportional to current — making the galvanometer a linear current-measuring instrument.

Hall Effect

When a current-carrying conductor is placed in a transverse magnetic field, a potential difference develops across the conductor perpendicular to both the current and the field. This is the Hall effect, discovered by Edwin Hall in 1879.

The physical explanation: the magnetic Lorentz force deflects charge carriers sideways, causing charge to accumulate on one face of the conductor until the resulting electric field exactly balances the magnetic force. At equilibrium, the Hall voltage is:

[latex]V_H = \frac{BI}{ntq}[/latex]

where n is the charge carrier density, t is the thickness of the conductor, and q is the charge of the carriers. The Hall effect allows determination of both the density and sign of charge carriers — confirming that electrons (negative charges) are the carriers in metals. Hall probes use this effect to measure magnetic flux density accurately.

Biot-Savart Law

The Biot-Savart Law gives the magnetic field produced by a small element of current-carrying conductor:

[latex]d\vec{B} = \frac{\mu_0}{4\pi}\frac{I d\vec{l} \times \hat{r}}{r^2}[/latex]

where [latex]\mu_0 = 4\pi \times 10^{-7}[/latex] T m A⁻¹ is the permeability of free space. Applications of this law give the magnetic field at the centre of a circular coil: [latex]B = \frac{\mu_0 NI}{2R}[/latex], along the axis of a long straight conductor: [latex]B = \frac{\mu_0 I}{2\pi r}[/latex], and inside a long solenoid: [latex]B = \mu_0 nI[/latex], where n is the number of turns per unit length.

Ampere’s Law

Ampere’s Law states that the line integral of the magnetic field around any closed loop equals [latex]\mu_0[/latex] times the total current enclosed by the loop:

[latex]\oint \vec{B} \cdot d\vec{l} = \mu_0 I_{enc}[/latex]

This is the magnetic analogue of Gauss’s Law for electric fields. It provides an elegant and efficient method for calculating magnetic fields in highly symmetric situations. Applications give the same results as Biot-Savart for a long straight conductor and a solenoid, and also give the field inside a toroidal solenoid: [latex]B = \frac{\mu_0 NI}{2\pi r}[/latex].

Force Between Parallel Current-Carrying Conductors

Two parallel conductors carrying currents in the same direction attract each other; in opposite directions they repel. The force per unit length between them is:

[latex]\frac{F}{L} = \frac{\mu_0 I_1 I_2}{2\pi d}[/latex]

where d is the separation between the conductors. This result is the basis for the definition of the ampere: one ampere is the current that, flowing in each of two infinitely long parallel conductors separated by one metre in vacuum, produces a force per unit length of [latex]2 \times 10^{-7}[/latex] N/m between them.

Why This Chapter Matters for NEB Exams

Magnetic Field is the most extensively examined chapter in Grade 12 Physics. Force on a moving charge and current-carrying conductor, torque on a coil, Hall effect derivation, Biot-Savart Law applications (field at centre of coil, along a straight wire, inside solenoid), Ampere’s Law applications, and force between parallel conductors are all standard examination topics. Numerical problems on magnetic force, field calculations, Hall voltage, and galvanometer sensitivity appear in virtually every board paper. The derivation of the definition of the ampere from force between parallel conductors is a classic long-answer question.

Important Topics to Focus On

• Magnetic field lines and magnetic flux: [latex]\Phi = BA\cos\theta[/latex]
• Oersted’s experiment — discovery and implications
• Lorentz force: [latex]F = qvB\sin\theta[/latex] — direction and applications
• Circular motion of charged particle in magnetic field: [latex]r = \frac{mv}{qB}[/latex]
• Force on current-carrying conductor: [latex]F = BIL\sin\theta[/latex] — Fleming’s Left-Hand Rule
• Torque on rectangular coil: [latex]\tau = NIAB\sin\alpha[/latex]
• Moving coil galvanometer — principle and working
• Hall effect — derivation of [latex]V_H = \frac{BI}{ntq}[/latex] and applications
• Biot-Savart Law — statement and applications to circular coil, straight conductor, solenoid
• Ampere’s Law — statement and applications to straight conductor, solenoid, toroid
• Force between parallel conductors: [latex]\frac{F}{L} = \frac{\mu_0 I_1 I_2}{2\pi d}[/latex]
• Definition of the ampere
• Numerical problems on all of the above

Connection to Other Chapters

Magnetic Field builds on Grade 11 Chapter 19 (Electric Charges) and Chapter 20 (Electric Field) — the Lorentz force combines electric and magnetic forces on a charge. It connects to Grade 11 Chapter 23 (DC Circuits) — Ohm’s Law and current concepts are prerequisites. The moving coil galvanometer connects back to Chapter 14 (Electrical Circuits) of Grade 12 — where it is converted to ammeter and voltmeter. Chapter 17 (Magnetic Properties of Materials) extends the study of magnetic fields to materials. Most importantly, Chapter 18 (Electromagnetic Induction) builds directly on this chapter — a changing magnetic field induces EMF, which is the basis of all electrical generators and transformers. The circular motion of charges in magnetic fields connects to Chapter 23 (Quantization of Energy) — where the Bohr model uses circular electron orbits — and to Chapter 20 (Electrons) where Thomson’s experiment uses magnetic deflection of electron beams.