Question bank — Geometric optics — rectilinear propagation, reflection, refraction

Light always travels in a straight line. ::: False — only in a homogeneous medium. Where speed varies smoothly (hot air over a road, atmosphere), rays curve, because the least-time path is no longer the straight one. See Fermat's principle. The angle of incidence equals the angle of reflection for a rough (matte) wall too. ::: True at each tiny facet — every point still obeys ${\theta }_i={\theta }_r$ against its local normal. The facets point every which way, so the reflections scatter (diffuse reflection); the law itself never breaks. When light enters a denser medium its frequency decreases. ::: False — frequency is fixed by the source and is unchanged. The wavelength shrinks to ${\lambda }_{med}={\lambda }_0/n$ (with ${\lambda }_0$ the vacuum wavelength) and the speed drops to $v=c/n$; frequency stays put so colour stays put. A higher refractive index means light moves faster in that medium. ::: False — $n=c/v$, so larger $n$ means smaller $v$. High index = optically denser = slower light. Total internal reflection can happen when light goes from air into glass. ::: False — it only happens going from denser to rarer, i.e. when the starting index $n_1$ exceeds the destination index $n_2$. Air→glass has $n_1<n_2$: it bends toward the normal and always refracts, so there is no critical angle in that direction. See Total internal reflection. Refraction bends the ray, but reflection and refraction come from different physical principles. ::: False — both drop out of the same principle, Fermat's least time. Reflection is the constant-speed special case; refraction is the changing-speed case. At exactly the critical angle the light is already totally internally reflected. ::: False — at the critical angle the refracted ray grazes along the surface (${\theta }_2=90°$). Total internal reflection begins beyond it, where $\sin {\theta }_2$ would exceed 1. If you reverse the direction of a light ray it retraces its exact path back. ::: True — reversibility. Snell's law $n_1\sin {\theta }_1=n_2\sin {\theta }_2$ is symmetric in the two sides, so swapping source and receiver gives the identical bent path. In a pinhole camera a bigger hole gives a sharper image. ::: False — a bigger hole lets many straight rays from one candle point land at different screen points, blurring it. Rectilinear propagation gives sharpness only when the hole is small.

"I measured the angle of incidence as 60° from the mirror surface, so I use 60° in the law." ::: Error — all optics angles are measured from the normal, not the surface. The true angle of incidence is $90°-60°=30°$. Draw the normal first, always. "Light slows in glass, so its energy is destroyed and it dims." ::: Error — slowing changes speed and wavelength, not energy conservation. Dimming (if any) comes from reflection/absorption losses, not from the speed change itself. "Snell's law is $n_1\cos {\theta }_1=n_2\cos {\theta }_2$." ::: Error — it uses sine, not cosine. In the figure at the top, the least-time derivative turns the horizontal offset $x/\sqrt{x^2+a^2}$ (opposite over hypotenuse at $P$) into $\sin {\theta }_1$, never cosine. "The straw in water looks bent because water magnifies it." ::: Error — it is refraction, not magnification. Rays from the submerged part bend at the surface, so they appear to come from a shallower position; the break is a shift in apparent direction. "For the critical angle I used $\sin {\theta }_c=n_1/n_2$ with $n_1=1.5$, $n_2=1$." ::: Error — the formula is $\sin {\theta }_c=n_2/n_1$ (destination index over starting index, rarer over denser). Your version gives $1.5>1$, an impossible sine — a clear sign the ratio is flipped. "The image in the pinhole is upright because light travels straight." ::: Error — straight travel is exactly why it is inverted: top rays cross through the single hole to the bottom of the screen. Straightness causes inversion, it does not prevent it. "At the boundary the light picks the shortest distance, so it goes straight through." ::: Error — least time, not least distance, once the speeds differ. Spending less distance in the slow medium beats a straight line, which is why the ray bends.

Why does light bend toward the normal entering a denser medium? ::: The side of the wavefront entering the slow medium first lags, so the front pivots toward the boundary's normal — like a marching row swinging as it hits mud. Slower medium ⇒ smaller angle. See Wavefronts and Huygens' principle. Why does the least-time principle give reflection with equal angles (show the math)? ::: One constant speed means least time = least path length $L(x)=\sqrt{x^2+a^2}+\sqrt{(d-x{)}^2+b^2}$. Setting $dL/dx=0$ gives $\frac{x}{\sqrt{x^2+a^2}}=\frac{d-x}{\sqrt{(d-x{)}^2+b^2}}$; by the figure the left side is $\sin {\theta }_i$ and the right is $\sin {\theta }_r$, so $\sin {\theta }_i=\sin {\theta }_r$, hence ${\theta }_i={\theta }_r$. Why does white light split into colours in a prism but not colour-shift in refraction generally? ::: Colour is tied to frequency, which never changes. Splitting happens because $n$ depends slightly on frequency (dispersion), so each colour bends by a different angle. See Dispersion. Why can an optical fibre guide light around bends without leaking? ::: Each time light meets the wall above the critical angle it is totally internally reflected — no refracted ray escapes — so it stays trapped. See Optical fibres. Why do we even bother with rays instead of full waves? ::: When objects are far larger than the wavelength, diffraction is negligible and energy flows along straight arrows. Rays are the simple, accurate limit of waves at large scale. Why is $n$ always at least 1? ::: Because $n=c/v$ and nothing carrying a signal outruns $c$ in a medium, so $v\le c$ and the ratio is $\ge 1$.

What happens to a ray that hits the surface exactly along the normal (0° incidence)? ::: It passes straight through with no bend: $\sin 0°=0$ on both sides of Snell's law, so ${\theta }_2=0°$ too. The speed still changes, but the direction does not. What happens as ${\theta }_1\to 90°$ (grazing incidence) going into a denser medium ($n_2>n_1$)? ::: The refracted angle approaches the largest value it can — $\sin {\theta }_2\to n_1/n_2<1$ — so even grazing light enters at a finite angle. This defines the cone of light seen by a fish looking up. What if the two media have equal refractive index ($n_1=n_2$)? ::: No bending at all: Snell's law gives ${\theta }_1={\theta }_2$. Optically the boundary is invisible — this is how index-matching liquids hide objects. What if light goes from denser to rarer ($n_1>n_2$) at less than the critical angle? ::: It refracts normally, bending away from the normal (${\theta }_2>{\theta }_1$), and only partially reflects. Total internal reflection needs the angle to exceed ${\theta }_c$. What does a ray do when the medium's index changes continuously (a gradient), not in a sudden step? ::: It curves smoothly, bending toward the higher-index (slower) region at every point. This is the mirage: hot low-index air near the road bends rays upward. Does reflection change the wavelength or frequency of light? ::: Neither — the medium is the same before and after bouncing, so $v$, $\lambda$ and $f$ are all unchanged. Only the direction flips.