Images by a thin lens: relation and magnification
A thin lens forms an image of an object. With the Cartesian sign convention (distances measured from the lens, along the light direction positive):
1/v − 1/u = 1/f and m = v/u
Here u is the object distance (negative for a real object), v the image distance and f the focal length (positive for a convex lens, negative for a concave one). If m is negative, the image is inverted; if |m| is more than 1, it is bigger than the object. The power of a lens is P = 1/f in dioptres when f is in metres. The full ray diagrams are in the lesson on the thin lens formula.
White light and complementary colours
White light is a mixture of the colours of the spectrum. A prism splits it because glass bends each colour by a different amount (violet most, red least). Two colours that add up to white light are complementary: red and cyan, green and magenta, blue and yellow.
A coloured filter lets its own colour through and absorbs the rest, mostly its complement. A yellow filter passes red and green and absorbs blue. An object looks red in white light because it reflects red and absorbs the rest. More on colour mixing is in light and colour.
Absorption, scattering and transmission
When light meets matter, three things can happen to its photons:
- Transmission: light passes through (clear glass, clean air, water).
- Absorption: the photon's energy is taken in and becomes heat or another form (black cloth, coloured filters).
- Scattering: photons are thrown in many directions (milk, fog, clouds). Tiny air molecules scatter short waves (blue) much more than long waves (red), so the daytime sky looks blue and a sunset, where light travels a long way through air, looks red.
Reflection is the fourth case: a smooth surface sends light back in one direction. A material usually does a mix of these, which is why it has its particular look.
The electromagnetic spectrum
Light is one kind of electromagnetic wave. All of them travel at c = 3 × 10⁸ m/s in vacuum and obey c = fλ. The domains, from long wavelength to short, are roughly:
- Radio (longer than about 10 cm)
- Microwave (1 mm to 10 cm)
- Infrared (about 750 nm to 1 mm)
- Visible (about 380 to 750 nm: violet to red)
- Ultraviolet (10 to 380 nm)
- X-rays (about 0.01 to 10 nm)
- Gamma rays (shorter than about 0.01 nm)
Limits are approximate and neighbours overlap. Going from radio to gamma, frequency and photon energy rise, wavelength falls.
Photon energy
Light is emitted and absorbed in packets called photons. One photon has energy
E = hf = hc/λ
where h = 6.63 × 10⁻³⁴ J·s is Planck's constant. In electron-volts (1 eV = 1.6 × 10⁻¹⁹ J) and nanometres there is a handy shortcut: E (eV) ≈ 1240 ÷ λ (nm). So red light (700 nm) has about 1.8 eV and violet (400 nm) about 3.1 eV. A bright beam has many photons per second; a dim beam has few. The colour fixes the energy of each photon, the brightness fixes how many.
This is why UV can harm skin or cause the photoelectric effect while a powerful radio beam cannot: each UV photon carries a lot more energy.
Atomic energy levels and spectra
The electrons of an atom can have only certain energies, called energy levels, like steps of a ladder, not the heights in between. An electron jumping from a higher level E₂ to a lower one E₁ gives out one photon with
hf = E₂ − E₁
The reverse jump needs a photon of exactly that energy to be absorbed. A hot gas therefore shows a few bright lines (an emission spectrum). White light passing through a cool gas shows dark lines at the same places (an absorption spectrum). Every element has its own lines, so spectra work like fingerprints for finding the elements in a flame, a lamp or a star.
For hydrogen, E = −13.6 eV ÷ n². The jump from n = 3 to n = 2 gives 1.89 eV, a red line at 656 nm. The jump from n = 2 to n = 1 gives 10.2 eV, an ultraviolet line at 122 nm.
Try it
In the 3D: on free play, slide λ to find the wavelength where the domain changes from infrared to visible. Then press "Levels" and jump the electron down in two steps.
At home: look at a CD or a clear glass of water in sunlight and find a rainbow. Cover a phone torch with red, then green cellophane and see which colour reaches the wall. Check the energy of a photon at 500 nm: 1240 ÷ 500.
Key formulas and definitions
- Thin lens: 1/v − 1/u = 1/f; magnification m = v/u; power P = 1/f (f in m, P in dioptre)
- Wave: c = fλ, c = 3 × 10⁸ m/s
- Photon energy: E = hf = hc/λ, h = 6.63 × 10⁻³⁴ J·s
- E (eV) ≈ 1240 ÷ λ (nm); 1 eV = 1.6 × 10⁻¹⁹ J
- Atomic transition: hf = E(upper) − E(lower); hydrogen E(n) = −13.6 eV ÷ n²
Worked examples
1. An object is 30 cm from a convex lens of focal length 10 cm. Find the image distance and magnification.
u = −30 cm, f = +10 cm. 1/v = 1/f + 1/u = 1/10 − 1/30 = 2/30, so v = +15 cm. m = v/u = 15 / (−30) = −0.5. The image is real, inverted and half the size.
2. Find the energy of a photon of green light of wavelength 500 nm, in eV and joules.
E = 1240 / 500 = 2.48 eV. In joules: 2.48 × 1.6 × 10⁻¹⁹ = 3.97 × 10⁻¹⁹ J.
3. A hydrogen atom drops from n = 3 to n = 2. Find the photon energy, wavelength and colour.
E = 13.6 × (1/4 − 1/9) = 13.6 × 5/36 = 1.89 eV. λ = 1240 / 1.89 ≈ 656 nm, which is red.
4. How many photons per second does a 1 W beam of 500 nm light carry?
Energy per photon = 3.97 × 10⁻¹⁹ J. Photons per second = 1 / 3.97 × 10⁻¹⁹ ≈ 2.5 × 10¹⁸.
5. Which has more energy per photon: red light (700 nm) or violet light (400 nm)? Give the numbers.
Red: 1240 / 700 = 1.77 eV. Violet: 1240 / 400 = 3.1 eV. Violet photons carry more energy.
6. A yellow filter is placed in front of white light. What colours come out, and which is absorbed?
Yellow is red + green. Red and green pass; blue (the complement of yellow) is absorbed.
Common mistakes
- Thinking brighter light means higher-energy photons. Brightness is the number of photons; colour (frequency) sets the energy of each photon.
- Mixing up which end of the spectrum has more energy. Short wavelength and high frequency mean high energy (violet, UV, X-ray, gamma).
- Forgetting the sign convention in the lens formula. For a real object u is negative, so 1/v = 1/f + 1/u.
- Saying atoms can have any energy. An electron can sit only on its allowed levels, so only certain photon energies are absorbed or emitted.