Wavefronts, Interference and Young's Experiment: Quick Revision

A brief note on Wavefronts, Interference and Young's Experiment. Read it before a test. Try each question at the end before you open its answer.

Wavefronts and Huygens' principle

• Wavefront: a surface of equal phase. A ray is at right angles to it.
• Huygens' principle: every point on a wavefront sends out secondary wavelets. The new wavefront is the surface that touches them.
• Reflection: $i = r$ (two equal right-angled triangles).
• Refraction: $n_1\sin i = n_2\sin r$, since $\sin i/\sin r = v_1/v_2$.
• In a medium: $f$ is unchanged, $v = c/n$ and $\lambda = \lambda_0/n$.

Interference

• Phase and path: $\varphi = \dfrac{2\pi}{\lambda}\Delta$.
• Bright: $\Delta = n\lambda$. Dark: $\Delta = (n + \tfrac12)\lambda$.
• Two waves: $I = I_1 + I_2 + 2\sqrt{I_1I_2}\cos\varphi$. For equal waves, $I = 4I_0\cos^2(\varphi/2)$.
• Extremes: $I_{\max}/I_{\min} = \left(\dfrac{a_1 + a_2}{a_1 - a_2}\right)^2$.
• Coherent sources have a fixed phase difference. Divide one wavefront or one amplitude to get them.

Young's double slit

• Path difference: $\Delta \approx yd/D$ for $D \gg d$.
• Bright fringes: $y_n = n\beta$. Dark fringes: $y = (n + \tfrac12)\beta$.
• Fringe width: $\beta = \dfrac{\lambda D}{d}$. All fringes are equally wide.
• In a liquid: $\beta \to \beta/n$. White light: a white centre, then coloured fringes.

A worked example

Light of wavelength $650$ nm lights two slits $0.40$ mm apart. The screen is $1.00$ m away. Find the fringe width, the 3rd bright fringe and the 3rd dark fringe.

1. Fringe width. $\beta = \dfrac{650 \times 10^{-9}\ \text{m} \times 1.00\ \text{m}}{0.40 \times 10^{-3}\ \text{m}} = 1.63\ \text{mm}$.
2. 3rd bright fringe. $3\beta = 4.88\ \text{mm}$.
3. 3rd dark fringe. $2.5\beta = 4.06\ \text{mm}$.

Practise with new numbers

Check yourself

1. Two coherent beams have amplitudes in the ratio $3 : 1$. What is $I_{\max}/I_{\min}$?

2. A double-slit experiment is moved from air into water of index $1.33$. What happens to the fringe width?

3. Why does a pair of torches not give fringes?

Sources

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