Section 5.1: Magnetic Fields

Magnetic fields arise from moving charges (currents) and magnetic materials. The field is represented by the vector B (measured in tesla, T). Magnetic fields exert forces on moving charges and current-carrying conductors and are described by simple formulae for common geometries.

Key Concepts
  • B is a vector field; direction given by magnetic field lines (north → south outside magnets).
  • Magnetic field lines form closed loops and never begin or end (no magnetic monopoles observed).
  • Moving charges and currents produce magnetic fields; fields exert forces on other moving charges and currents.
Important formulae
Magnetic field from a long straight wire (distance r):\n\n\( B = \dfrac{\mu_0 I}{2\pi r} \)
Magnetic field at center of a circular loop (radius R, N turns):\n\n\( B = \dfrac{\mu_0 N I}{2R} \)
Magnetic permeability of free space:\n\n\( \mu_0 = 4\pi \times 10^{-7}\ \text{T·m/A} \)
Magnetic field lines around a straight wire and loop (placeholder)

Example 1

A long straight wire carries a current of 6.0 A. Find the magnetic field magnitude at a point located 0.08 m from the wire.

Use \( B = \dfrac{\mu_0 I}{2\pi r} \). With \( \mu_0 = 4\pi\times10^{-7} \):

\( B = \dfrac{4\pi\times10^{-7}\times 6.0}{2\pi\times0.08} = \dfrac{24\pi\times10^{-7}}{0.16\pi} = \dfrac{24\times10^{-7}}{0.16} \)

\( B = 150\times10^{-7} = 1.5\times10^{-5}\ \text{T} \)

Answer: \(1.5\times10^{-5}\ \text{T}\).

Example 2

A single circular loop of radius 0.10 m carries a current of 4.0 A. Find the magnetic field at the centre of the loop.

Use \( B = \dfrac{\mu_0 N I}{2R} \) with \(N=1\):

\( B = \dfrac{4\pi\times10^{-7}\times 1\times 4.0}{2\times0.10} = \dfrac{16\pi\times10^{-7}}{0.20\pi} = \dfrac{16\times10^{-7}}{0.20} \)

\( B = 80\times10^{-7} = 8.0\times10^{-6}\ \text{T} \)

Answer: \(8.0\times10^{-6}\ \text{T}.\)

Practice Problems

  1. A long straight wire carries 10 A. Calculate the magnetic field 0.05 m away.
  2. A circular coil (N = 20 turns) radius 0.05 m carries 3 A. Find the magnetic field at the centre.
  3. Compare B at r = 0.1 m and r = 0.2 m from a straight wire carrying 8 A. How does B scale with r?
  4. Sketch the magnetic field pattern for a bar magnet and for a straight current-carrying wire.
  5. A coaxial cable carries equal currents in opposite directions in inner and outer conductors. Explain qualitatively the resulting external magnetic field.
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