understand that a magnetic field is an example of a field of force produced either by moving charges or by permanent magnets

Magnetic Fields – Cambridge IGCSE/A‑Level Physics (9702)

20.1 What is a Magnetic Field?

• A magnetic field 𝐁 is a vector field that can exert a force on

  • moving electric charges (e.g. electrons, ions) and
  • magnetic dipoles (e.g. a tiny bar magnet, the Earth’s magnetic moment).

• Unit: tesla (T) (1 T = 10⁴ gauss). Conversion to gauss can be placed in a sidebar for revision.
• Direction of the field is given by the right‑hand rule for currents or by the orientation of the north‑seeking pole of a magnetic dipole.
• Key field‑line properties (Cambridge emphasis):

  • Lines are continuous and form closed loops – they never start or end.
  • Density of lines is proportional to the magnetic field strength.
  • Outside a bar magnet they emerge from the north pole and re‑enter at the south pole.

• Important concept: the magnetic field is solenoidal – its divergence is zero (∇·𝐁 = 0). This underlies electromagnetic induction.

20.2 Magnetic Field Produced by Moving Charges (Currents)

20.2.1 Straight current‑carrying wire

For an (ideal) infinitely long straight conductor carrying a current I, the magnetic field at a perpendicular distance r is

Direction: right‑hand thumb rule – point the thumb in the direction of conventional current; the curled fingers give the direction of 𝐁.

20.2.2 Circular loop (single turn)

For a circular loop of radius R carrying current I, the field on the axis a distance x from the centre is

20.2.3 Solenoid (long tightly wound coil)

Inside a long solenoid with n turns per unit length the field is essentially uniform and given by

Direction: use the right‑hand grip rule** – curl the fingers in the sense of the current around the coil; the thumb points in the direction of the magnetic field inside the solenoid.

20.2.4 Biot–Savart Law (general form) – optional/advanced

For any infinitesimal current element Id𝐥 the contribution to the magnetic field at a point with position vector r is

Integrating over the whole conductor yields the total field. (Not required for the exam, but useful for deeper study.)

20.2.5 Toroid – optional/advanced

A toroidal coil (donut‑shaped) of N turns, mean radius r, carrying current I produces a field confined to the core:

Included only as an extension topic.

20.3 Magnetic Field of Permanent Magnets

• Permanent magnets consist of many atomic‑scale magnetic dipoles m aligned in the same direction.
• Field‑line pattern of a bar magnet: lines emerge from the north pole, loop through space, and re‑enter at the south pole.
• Advanced – dipole formula (optional)
  \$\mathbf{B}(\mathbf{r}) = \frac{\mu_0}{4\pi}\,\frac{3(\mathbf{m}\!\cdot\!\hat{\mathbf{r}})\hat{\mathbf{r}}-\mathbf{m}}{r^{3}}\$

  This expression describes the field of a single magnetic dipole; it is not required for the syllabus.

• Example of a natural magnet – Earth (optional context)

  The Earth behaves like a giant bar magnet with a magnetic moment pointing roughly from geographic south to north; the surface field is about 25–65 µT (0.25–0.65 G).

20.4 Force on a Moving Charge in a Magnetic Field

The magnetic part of the Lorentz force on a charge q moving with velocity v in a magnetic field B is

• Magnitude: F = q v B \sin\theta, where θ is the angle between v and B.
• Direction: given by the right‑hand rule (thumb = v, fingers = B, palm pushes the force on a positive charge).
• The force is always perpendicular to both the velocity and the magnetic field; consequently the speed of the charge does not change, only its direction.

Uniform magnetic field – circular motion

If v is perpendicular to a uniform field, the charge moves in a circle of radius

and angular (cyclotron) frequency

Period

Check‑your‑understanding: A proton (m = 1.67 × 10⁻²⁷ kg, q = +e) enters a uniform magnetic field of 2 T with a speed of 1.0 × 10⁶ m s⁻¹ perpendicular to the field. Calculate the radius of its circular path.

20.5 Magnetic Flux and Electromagnetic Induction (Brief)

• Magnetic flux through a surface of area A whose normal makes an angle θ with the field is
  \$\Phi = B A \cos\theta\$

  Unit: weber (Wb), where 1 Wb = 1 T·m².

• Faraday’s law: a change of magnetic flux through a closed conducting loop induces an emf
  \$\mathcal{E}= -\frac{\mathrm{d}\Phi}{\mathrm{d}t}\$

  (The symbol 𝓔 or ε is commonly used for emf.)

• Lenz’s law (explicit statement required by the syllabus):

  The induced emf produces a current whose magnetic field opposes the change in magnetic flux.

Summary Table – Sources and Key Equations

Practical Examples for Revision

• Cathode‑ray tube: An electron beam is deflected by a known magnetic field; the deflection allows determination of the charge‑to‑mass ratio e/m.
• Mass spectrometer: Ions travel in a uniform magnetic field; the radius of curvature separates ions according to their mass‑to‑charge ratio.
• Electromagnet (solenoid): Increasing the current strengthens the field – the principle behind lifting magnets and MRI machines.
• Compass: The magnetic needle aligns with the horizontal component of the Earth’s magnetic field.
• Generator (rotating coil): Rotating a coil in a uniform magnetic field produces an alternating emf (Faraday’s law).

Key Take‑aways

1. A magnetic field is a vector field that exerts forces on moving charges and magnetic dipoles.
2. It is produced either by electric currents (moving charges) or by the aligned atomic dipoles of permanent magnets.
3. Field strength is measured in tesla (1 T = 10⁴ G); magnetic flux in weber (1 Wb = 1 T·m²).
4. Right‑hand rules give the direction of 𝐁 for currents and the direction of the magnetic force.
5. In a uniform magnetic field a charged particle moves in a circular path with radius \(r = mv/|q|B\).
6. Lenz’s law: the induced emf always opposes the change in magnetic flux that produces it.