Polarisation, Scattering and Thin Films: Quick Revision

A brief note on Polarisation, Scattering and Thin Films. Read it before a test. Try each question at the end before you open its answer.

Polarisation

• Transverse waves can be polarised. Sound in air cannot.
• Unpolarised light through a Polaroid: $I = I_0/2$.
• Malus' law: $I = I_p\cos^2\theta$. Crossed Polaroids ($90^\circ$) give no light.
• Partly polarised: $I_{\max} = I_p + I_u/2$ and $I_{\min} = I_u/2$.
• Three Polaroids with the first and last crossed and the middle one at $\theta$: $I = \dfrac{I_0}{2}\cos^2\theta\sin^2\theta$, at most $I_0/8$.

Brewster's angle and scattering

• Brewster's law: $\tan i_B = n$ (air to material), and $\tan i_B = n_2/n_1$ in general.
• At $i_B$: the reflected light is fully plane polarised, and the reflected and refracted rays are at $90^\circ$.
• Scattering by small particles: strength $\propto 1/\lambda^4$. Blue sky, red sunset. Clouds are white.

Thin films and the changes of Young's fringes

• Film in air, straight on: bright when $2nt = (m + \tfrac12)\lambda$, dark when $2nt = m\lambda$. A very thin film is dark.
• Quarter-wave coating: $t = \lambda/4n$.
• A sheet over one slit: shift $= (\mu - 1)tD/d$ towards the covered slit. The fringe width is the same.
• In a liquid: $\beta \to \beta/n$.

A worked example

Unpolarised light of intensity $120\ \text{W/m}^2$ goes through two Polaroids. The second is turned by $45^\circ$ from the first. Find the intensity that comes out, and Brewster's angle for glass of index $1.62$.

1. After the first. $\dfrac{120}{2} = 60\ \text{W/m}^2$.
2. After the second. $60\ \text{W/m}^2 \times \cos^2 45^\circ = 30\ \text{W/m}^2$.
3. Brewster's angle. $i_B = \tan^{-1}1.62 = 58.3^\circ$.

Practise with new numbers

Check yourself

1. A soap film that is $360$ nm thick has index $1.33$. What is $2nt$, in nm?

2. At Brewster's angle, what is the angle between the reflected and the refracted ray?

3. A sheet is put over one slit of a double slit. What happens to the pattern?

Sources

• OpenStax, University Physics Volume 3, Chapters 1 and 3.
• NCERT Physics Class 12, Chapter 10 (Wave Optics).