recall that wavelengths in the range 400–700 nm in free space are visible to the human eye

Electromagnetic Spectrum – Cambridge International AS & A Level Physics (9702)

1. Key Physical Quantities

• Wavelength, λ: metre (m). Common prefixes – nanometre (nm = 10⁻⁹ m), micrometre (µm = 10⁻⁶ m).
• Frequency, f: hertz (Hz). Common prefixes – terahertz (THz = 10¹² Hz).
• Speed of light in vacuum, c: \(c = 3.00\times10^{8}\ \text{m s}^{-1}\). In any transparent medium the speed is reduced to \[ v = \frac{c}{n}, \] where \(n\) is the refractive index.
• Fundamental relation: \(c = \lambda f\) (or \(v = \lambda f\) in a medium).
• Photon energy, E: \(E = hf\) with \(h = 6.626\times10^{-34}\ \text{J s}\). Using \(c = \lambda f\), \(E = \dfrac{hc}{\lambda}\).
• Photon momentum, p: \(p = \dfrac{h}{\lambda}\).
• Intensity, I: power per unit area. For a plane sinusoidal wave \[ I = \frac{1}{2}\,c\varepsilon_{0}E_{0}^{2}\;\; \text{or simply}\;\; I \propto A^{2}, \] where \(E_{0}\) (or \(A\)) is the electric‑field amplitude.

Typical experimental uncertainties: A modern spectrometer resolves wavelengths to ±0.1 nm (≈0.03 % at 500 nm); frequency is then obtained from \(f=c/λ\) with a comparable relative error.

2. Electromagnetic Spectrum Overview (All Regions)

Region	Wavelength (nm)	Wavelength (m)	Frequency (Hz)	Typical Sources
Radio	≥ 10⁶ nm	≥ 10⁻³ m	≤ 3×10⁸ Hz	Broadcast transmitters, radar
Microwave	10³ – 10⁶ nm	10⁻⁶ – 10⁻³ m	3×10⁸ – 3×10¹¹ Hz	Satellite communication, microwave ovens
Infrared (IR)	700 – 10⁶ nm	7×10⁻⁷ – 10⁻³ m	3×10¹¹ – 4.3×10¹⁴ Hz	Thermal emitters, remote controls
Visible	400 – 700 nm	4×10⁻⁷ – 7×10⁻⁷ m	4.3×10¹⁴ – 7.5×10¹⁴ Hz	Sunlight, LEDs, lasers
Ultraviolet (UV)	10 – 400 nm	1×10⁻⁸ – 4×10⁻⁷ m	7.5×10¹⁴ – 3×10¹⁶ Hz	Sun, UV lamps, black lights
X‑ray	0.01 – 10 nm	1×10⁻¹¹ – 1×10⁻⁸ m	3×10¹⁶ – 3×10¹⁹ Hz	Medical imaging, astronomical sources
Gamma ray	< 0.01 nm	< 1×10⁻¹¹ m	> 3×10¹⁹ Hz	Radioactive decay, nuclear reactions

3. Polarisation (Topic 7.5)

• Definition: Because electromagnetic waves are transverse, the direction of the electric‑field vector defines the plane of polarisation. Unpolarised light has electric‑field vectors in all directions perpendicular to the propagation direction; polarised light has a single preferred direction.
• Linear polarisation – the electric field oscillates in one fixed plane.
• Malus’s law (quantitative description of a polariser‑analyser system): \[ I = I_{0}\cos^{2}\theta, \] where \(I_{0}\) is the intensity after the first polariser and \(\theta\) is the angle between the transmission axes of the two polarisers.
• Applications (relevant to the syllabus and everyday life):
  • Polarising sunglasses – reduce glare by transmitting only one polarisation.
  • LCD (liquid‑crystal display) pixels – use controlled rotation of polarisation to modulate light.
  • Stress analysis in transparent plastics – photoelasticity produces fringe patterns that are visible only with a polariser.
  • Optical isolators and lasers – prevent back‑reflected light from destabilising a laser cavity.

4. Visible Light (400 nm – 700 nm)

The human eye is sensitive only to photons whose wavelengths lie between roughly 400 nm (violet) and 700 nm (red). Below 400 nm the cornea and lens absorb strongly (ultraviolet), and above 700 nm the retina is essentially blind (infra‑red).

4.1 Photoreceptor Sensitivity

Cone type	Peak wavelength (nm)	Colour sensation
S (short‑wavelength)	≈ 420	Blue
M (medium‑wavelength)	≈ 534	Green
L (long‑wavelength)	≈ 564	Red

Colour perception is the brain’s interpretation of the relative stimulation of these three cone types.

4.2 Energy, Momentum and Radiation Pressure

• Photon energy: \(E = hf = \dfrac{hc}{\lambda}\). Example: a 500 nm photon has \(E = 3.97\times10^{-19}\ \text{J}\;(2.48\ \text{eV})\).
• Photon momentum: \(p = \dfrac{h}{\lambda}\). Example: a 600 nm photon carries \(p = 1.10\times10^{-27}\ \text{kg m s}^{-1}\).
• Radiation pressure on a surface: \[ P = \frac{I}{c}\quad(\text{perfect absorber}),\qquad P = \frac{2I}{c}\quad(\text{perfect reflector}). \] This principle underlies solar‑sail propulsion (Topic 4).

4.3 Intensity and Amplitude

For a plane EM wave the intensity is proportional to the square of the electric‑field amplitude:

\[ I = \frac{1}{2}\,c\varepsilon_{0}E_{0}^{2}\;\; \text{or}\;\; I \propto A^{2}. \]

Safety note: a laser that appears faint (low amplitude) can still be hazardous because intensity scales with the square of the amplitude.

4.4 Doppler Shift for Light

Relative motion between source and observer changes the observed wavelength:

\[ \frac{\Delta\lambda}{\lambda} = \frac{v}{c}\qquad (v\ll c), \] where \(v\) is the radial speed (positive = recession → red‑shift; negative = approach → blue‑shift). This effect is essential in astrophysics and in laser‑based speed measurements.

5. Links to Other Syllabus Topics

5.1 Work, Energy & Power (Topic 5)

• LEDs: electrical power \(P_{\text{elec}} = VI\); optical output power \(P_{\text{opt}} = \eta P_{\text{elec}}\) (η ≈ 30 % for modern devices). Photon energy \(E = hf\) demonstrates energy quantisation.

5.2 Forces, Density & Pressure (Topic 4)

• Radiation pressure \(P = I/c\) provides a clear example of a force exerted without contact.

5.3 Superposition – Interference & Diffraction (Topic 8)

• Young’s double‑slit experiment: fringe spacing \(y = \dfrac{\lambda D}{d}\) (D = screen distance, d = slit separation).
• Diffraction gratings: separate white light into its constituent wavelengths; the basis of spectrometers.

5.4 Photoelectric Effect (Topic 9 & 22)

• Electrons are emitted if \(hf > \phi\) (work function). This directly links photon energy to electrical concepts of current and potential.

5.5 Ionising Radiation (Topic 11)

• UV, X‑rays and gamma rays have photon energies > 10 eV and can ionise atoms. Although invisible, they are covered under “radiation” and are important for health‑physics and detector technology.

6. Practical Implications

1. Lighting design: LEDs are engineered to emit within 400–700 nm for visual efficiency. Colour‑temperature specifications (e.g., 2700 K, 6500 K) correspond to different spectral power distributions.
2. Colour measurement: Spectrophotometers scan the 400–700 nm band; typical resolution is 1 nm, setting the uncertainty in colourimetric data.
3. Eye safety: Protective eyewear must attenuate UV (< 400 nm) and IR (> 700 nm) because these regions can cause photochemical or thermal damage despite being invisible.
4. Laser safety: Hazard classification uses intensity‑amplitude relationship and the wavelength‑dependent eye‑absorption curve (maximum hazard near 400 nm).
5. Solar sails: Radiation pressure from sunlight (≈ 9 µN m⁻² at 1 AU) provides continuous thrust for spacecraft, illustrating the mechanical effect of EM waves.
6. Polarisation applications: Polarising filters in photography, LCD screens, and stress‑analysis equipment rely on Malus’s law and the transverse nature of light.

7. Summary

• The EM spectrum spans from radio waves (> 10⁶ nm) to gamma rays (< 0.01 nm). All propagate at \(c = 3.00\times10^{8}\ \text{m s}^{-1}\) in vacuum; in a medium the speed is reduced to \(v = c/n\).
• Human vision is limited to the visible band, 400–700 nm (430–750 THz). This range is set by the absorption spectra of retinal photopigments.
• Key relations:
  • \(c = \lambda f\) (or \(v = \lambda f\) in a medium)
  • \(E = hf = \dfrac{hc}{\lambda}\)
  • \(p = \dfrac{h}{\lambda}\)
  • \(I \propto A^{2}\)
  • \(\dfrac{\Delta\lambda}{\lambda} = \dfrac{v}{c}\) (Doppler shift)
  • \(I = I_{0}\cos^{2}\theta\) (Malus’s law for polarisation)
• Understanding visible light links to many syllabus topics: energy conversion (LEDs), forces (radiation pressure), wave phenomena (interference, diffraction), quantum effects (photoelectric effect), polarisation, and safety considerations.