Electric and magnetic fields
Treat conductors in electrostatics, apply Gauss’s and Ampère’s laws, and describe electric and magnetic dipoles.
Offline pages
Saved on this device only. Connect at least once every 30 days to confirm Premium. Sign-out or a change of account removes saved copies.
Before you begin
Gauss’s and Ampère’s laws relate a field over a closed surface or around a closed loop to the charge or current inside. Where a charge or current distribution has enough symmetry, they give the field in a few lines. The lessons start with the field in and around conductors, then use Gauss’s law for electric fields and Ampère’s law for magnetic fields, and finish with electric and magnetic dipoles in uniform fields.
Be comfortable with: electric fields and electromagnetic forces.
Learning goals
- Explain that an ideal conductor is an equipotential volume with zero electric field inside
- Explain that charge collects on a conductor's surface and the electric field there is normal to the surface
- Recall and apply Gauss's law for electric and magnetic fields
- Use Gauss's law to relate electric flux through a closed surface to the enclosed charge, and explain why zero magnetic flux through it suggests there are no magnetic monopoles
- Recall and apply Ampère's law to find the magnetic field of symmetric current configurations
- Define electric dipole moment as charge × separation
- Use the torque on and potential energy of an electric dipole to solve problems
- Define the magnetic dipole moment of a current loop as current × loop area
- Use the torque on and potential energy of a magnetic dipole to solve problems
- Explain that electric and magnetic dipoles behave analogously, while the theory at this level admits no magnetic monopoles.
Syllabus statements covered
- show an understanding that ideal conductors form an equipotential volume, and that the electric field within an ideal conductor is zero
- show an understanding that electric charge accumulates on the surfaces of a conductor, and that the electric field at the surface of a conductor is normal to the surface
- recall and apply Gauss’s law 6 for electric and magnetic fields (knowledge of the differential form of Gauss’s law is not required), and (i) solve problems involving symmetric charge distributions by relating the electric flux (in a vacuum) through a closed surface with the charge enclosed by that surface; (ii) show an understanding that the magnetic flux through a closed surface is always zero, suggesting the non-existence of magnetic monopoles
- recall and apply Ampère’s law 7 relating the line integral of the magnetic field (in a vacuum) around a closed loop with the electric current enclosed by the loop to solve problems involving symmetric field configurations (knowledge of the differential form of Ampère’s law is not required) [Note further that candidates are not required to know Maxwell’s generalisation of Ampère’s law including the term related to the rate of change of electric flux, nor the Biot-Savart law.]
- define the magnitude of the electric dipole moment as the product of the charge and the separation
- show an understanding of and use the torque on an electric dipole and the potential energy of an electric dipole to solve related problems
- define the magnitude of the magnetic dipole moment for a current loop as the product of the current and the area of the loop
- show an understanding of and use the torque on a magnetic dipole and the potential energy of a magnetic dipole to solve related problems
- appreciate that while electric and magnetic dipoles behave analogously, the theoretical framework at this level of study does not admit the possibility of magnetic monopoles.
Lessons
Work through them in order.
- Electric fields in conductorsExplain zero internal field, equipotential volume, surface charge, and the normal surface field.
- Gauss’s lawRelate closed-surface flux to enclosed charge and select surfaces using symmetry.
- Ampère’s lawRelate magnetic-field circulation to enclosed steady current in symmetric configurations.
- Electric dipolesUse dipole moment, torque, and potential energy in a uniform electric field.
- Magnetic dipolesTransfer the dipole model to a current loop and identify the analogy’s limit.
Practise and check
Or choose
Topic reference
Quick reference
| Idea | Relation | Notes |
|---|---|---|
| Gauss’s law | ∮E · dA = Q_encl/ε₀ | closed surface, outward normal |
| Gauss’s law for magnetism | ∮B · dA = 0 | no magnetic monopoles |
| Ampère’s law | ∮B · dl = μ₀I_encl | steady currents, vacuum |
| electric dipole | p = qd, τ = p × E, U = -p · E | p points from -q to + q |
| magnetic dipole | μ = NIA, τ = μ × B, U = -μ · B | A by the right-hand rule |
Core knowledge to remember
- In electrostatic equilibrium, the field inside a conductor is zero, any excess charge lies on its surface, and the field just outside is perpendicular to the surface.
- To use Gauss’s law, choose a closed surface matching the symmetry, so that |E| is constant and at a known angle to dA wherever the flux is not zero.
- To use Ampère’s law, choose a loop on which |B| is constant along the parts that contribute, and find the direction with the right-hand rule.
- Only the enclosed charge or current appears in each law.
- A dipole in a uniform field feels a torque but no resultant force. It is in stable equilibrium when aligned with the field and in unstable equilibrium when opposite to it.
Common mistakes
- Choosing a Gaussian surface or Amperian loop that does not match the symmetry.
- Using the total charge or current instead of the enclosed charge or current.
- Confusing the outward normal dA with the path element dl.
- Concluding that the field is zero everywhere because the net flux or circulation is zero.
- Drawing the electric dipole moment from + q to -q.