Chapter 10 / 10.3 Refraction and Reflection of Plane Waves Using Huygens Principle

🌊 Refraction & Reflection of Plane Waves Using Huygens’ Principle

1. Huygens’ Picture in a Nutshell

Every point on a wavefront sends out tiny “secondary wavelets.” After a short time t, the new wavefront is the surface that just kisses (is tangent to) all those little spheres of radius v t 🌟. Huygens also noted that forward-going wavelets have the biggest amplitude, so the “backwave” practically vanishes :contentReference[oaicite:10]{index=10}.

2. Deriving the Law of Refraction (Snell’s Law)

Set-up: A plane wavefront AB hits the boundary between two media at angle i. Speeds are v₁ (medium 1) and v₂ (medium 2) 💡 :contentReference[oaicite:11]{index=11}.
In time t, the crest at B moves to C, so $BC = v_{1}\,t$.
The crest at A births a sphere of radius $v_{2}\,t$. The tangent CE is the refracted wavefront.
Using the two right triangles gives $ \sin i = \dfrac{v_{1}t}{AC}$ and $ \sin r = \dfrac{v_{2}t}{AC}$ :contentReference[oaicite:12]{index=12}.
Divide to obtain the wave-speed form of Snell’s law: $$\frac{\sin i}{\sin r} = \frac{v_{1}}{v_{2}}\tag{10.3}$$ 🌈 :contentReference[oaicite:13]{index=13}
Because $n = \dfrac{c}{v}$, one gets the familiar $$n_{1}\,\sin i = n_{2}\,\sin r\tag{10.6}$$ where $n_{1}$ and $n_{2}$ are refractive indices :contentReference[oaicite:14]{index=14}.
Wavelength shift: $v_{1}\,\lambda_{1} = v_{2}\,\lambda_{2}$ — speed and wavelength change together, but frequency stays put 🎶 :contentReference[oaicite:15]{index=15}.

3. Refraction into a Rarer Medium 🔄

If v₂ > v₁, rays bend away from the normal. Define the critical angle with $$\sin i_{c} = \frac{n_{2}}{n_{1}}\tag{10.8}$$ When $i \gt i_{c}$ the wave cannot cross the boundary and reflects totally — that’s total internal reflection ✨ .

4. Reflection from a Plane Surface 🪞

A plane wavefront AB meets mirror MN. After time t, point B reaches C (distance $BC = v\,t$).
Draw a sphere of radius $v\,t$ from A; its tangent through C is the reflected front CE.
Congruent triangles give the mirror rule $i = r$ ⚖️ :contentReference[oaicite:16]{index=16}.

5. Quick Wavefront View of Optical Elements

🔺 Thin prism	Lower part of the wave lags more (glass is slower), tilting the emerging front.
🔍 Convex lens	Central rays slow most → emerging front becomes a concave “bowl” that converges to the focus F.
🪞 Concave mirror	Reflection turns a plane front into a spherical one that meets at F.

All rays—central or edge—take the same time from object to image, a neat “equal-time” result ⏱️ :contentReference[oaicite:17]{index=17}.

6. ⭐ High-Yield NEET Nuggets

Snell’s law in index form: $n_{1}\sin i = n_{2}\sin r$ (master this!).
Critical angle formula $ \sin i_{c} = n_{2}/n_{1}$ and the condition for total internal reflection.
Relation $v\,\lambda = \text{constant}$ across media — explains color dispersion.
Equal-angle law of reflection $i = r$ derived from wavefronts, not just “ray” talk.
Wavefront construction: every point (secondary source) + common tangent idea — the heart of Huygens’ principle.

Keep these ideas handy, and wave optics questions will feel like a breeze 😊.