Big picture
A changing magnetic field produces an electric field (Faraday's law). Maxwell asked: does a changing electric field produce a magnetic field? He showed it does, through a displacement current. Together these two effects let electric and magnetic fields keep regenerating each other and travel through space as an electromagnetic wave, at the speed of light.
So light, radio waves, X-rays and microwaves are all the same kind of wave, differing only in wavelength. This chapter is short but heavily tested for definitions, properties and the EM spectrum.
Learning goals
Explain the need for displacement current and write the Ampere–Maxwell law.
Explain how electromagnetic waves are produced.
State the properties of electromagnetic waves, including c = 1/√(μ₀ε₀) and E₀ = cB₀.
Describe the electromagnetic spectrum: order, wavelength ranges, sources and uses.
Core ideas
1. Displacement current
Ampere's circuital law states ∮ B · dl = μ₀ I. Consider a charging capacitor. A loop around the wire encloses the conduction current I; but a surface passing between the plates encloses no current, because no charge flows across the gap. Applying the law to the two surfaces gives different answers: a contradiction.
Maxwell resolved this. Between the plates, the electric field (and electric flux) is changing as the capacitor charges. He proposed that a changing electric flux acts like a current, the displacement current:
I_d = ε₀ dΦE/dt
Inside the capacitor, the displacement current equals the conduction current in the wire, so current is continuous throughout the circuit.
Ampere–Maxwell law:
∮ B · dl = μ₀ (I_c + ε₀ dΦE/dt)
This means a magnetic field is produced both by conduction current and by a changing electric field.
Example. A capacitor is charged by a current of 0.15 A. The displacement current between its plates is also 0.15 A, since total current must be continuous.
2. Maxwell's equations (statement only)
Law | Meaning |
|---|---|
Gauss's law for electricity | Electric flux through a closed surface = enclosed charge ÷ ε₀ |
Gauss's law for magnetism | Magnetic flux through any closed surface is zero (no magnetic monopoles) |
Faraday's law | A changing magnetic flux produces an electric field (emf) |
Ampere–Maxwell law | A current or a changing electric flux produces a magnetic field |
3. How electromagnetic waves are produced
A stationary charge produces only an electric field. A charge moving with constant velocity produces a steady current, a steady magnetic field. An accelerated charge produces changing electric and magnetic fields, which regenerate each other and travel outward as an EM wave.
An oscillating charge (for example, electrons oscillating in an antenna) radiates EM waves at the same frequency as its oscillation. Hertz first produced and detected EM waves in 1887; J.C. Bose produced waves of much shorter wavelength (25 mm to 5 mm).
4. Properties of electromagnetic waves
The electric field E and magnetic field B oscillate perpendicular to each other and perpendicular to the direction of propagation: EM waves are transverse.
For a wave travelling along z: Ex = E₀ sin(kz − ωt) and By = B₀ sin(kz − ωt). E and B are in phase.
They don't need a medium; they travel through vacuum.
Speed in vacuum: c = 1/√(μ₀ε₀) = 3 × 10⁸ m/s, the speed of light. In a medium, v = 1/√(με), which is less than c.
E₀ = cB₀: the ratio of amplitudes of the electric and magnetic fields equals c.
c = νλ: frequency × wavelength = speed.
They carry energy and momentum, so they exert pressure (radiation pressure) on surfaces they hit. Energy is shared equally between the electric and magnetic fields.
They are not deflected by electric or magnetic fields, since they carry no charge.
Example 1. An EM wave has an electric field amplitude of 48 V/m. B₀ = E₀/c = 48 ÷ (3 × 10⁸) = 1.6 × 10⁻⁷ T.
Example 2. A radio wave has a wavelength of 3 m. ν = c/λ = (3 × 10⁸) ÷ 3 = 10⁸ Hz = 100 MHz (in the FM radio band).
5. The electromagnetic spectrum
Listed from longest wavelength (lowest frequency) to shortest wavelength (highest frequency):
Type | Approximate wavelength | Produced by | Main uses |
|---|---|---|---|
Radio waves | > 0.1 m | Accelerated motion of charges in conducting wires (antennas) | Radio and TV broadcasting (AM 530–1710 kHz; FM 88–108 MHz), mobile communication |
Microwaves | 0.1 m to 1 mm | Klystrons, magnetrons, Gunn diodes | Radar (aircraft navigation), microwave ovens (heat water molecules), satellite communication |
Infrared | 1 mm to 700 nm | Hot bodies and molecules | Heat waves; physical therapy; remote controls; night vision; infrared detectors in satellites; maintaining Earth's warmth (greenhouse effect) |
Visible light | 700 nm (red) to 400 nm (violet) | Electrons in atoms changing energy levels | Seeing; photography |
Ultraviolet | 400 nm to 1 nm | Very hot bodies (the Sun), special lamps | Killing germs in water purifiers; LASIK eye surgery; detecting fake notes. Mostly absorbed by the ozone layer; overexposure harms skin |
X-rays | 1 nm to 10⁻³ nm | Bombarding a metal target with high-energy electrons | Medical diagnosis (bones), cancer treatment, crystal structure studies |
Gamma rays | < 10⁻³ nm | Radioactive decay of nuclei | Cancer treatment (radiotherapy), sterilising medical equipment, nuclear reactions |
Memory aid (increasing frequency): Radio, Microwaves, Infrared, Visible, Ultraviolet, X-rays, Gamma.
The ranges overlap at the edges; the classification depends on how the waves are produced and detected.
A reliable problem-solving routine
For displacement current, use I_d = ε₀ dΦE/dt, or use the fact that it equals the conduction current in the connecting wires.
For field amplitudes, use E₀ = cB₀.
For frequency and wavelength, use c = νλ, with c = 3 × 10⁸ m/s.
For identification questions, match the wavelength or use to the correct region of the spectrum.
Connections that matter
Electromagnetic induction supplies Faraday's law, half of the wave mechanism.
Moving charges and magnetism supplies Ampere's law, which Maxwell corrected.
Ray and wave optics treat light, which is an EM wave.
Dual nature of radiation shows that EM radiation also behaves as photons.
Common mistakes
Calling EM waves longitudinal. They are transverse.
Saying EM waves need a medium.
Writing B₀ = cE₀. It is E₀ = cB₀.
Mixing up the order of the spectrum, such as placing microwaves above infrared in frequency.
Thinking displacement current involves moving charges. It comes from a changing electric flux.
Revision checklist
I can explain why displacement current is needed and write I_d = ε₀ dΦE/dt.
I can state the Ampere–Maxwell law.
I can explain how accelerated charges produce EM waves.
I can list the properties of EM waves and use E₀ = cB₀ and c = νλ.
I can list the EM spectrum in order with ranges, sources and uses.