Ray Model of Light

In the ray model, light travels in straight lines within a uniform medium – which is why objects cast sharp shadows and pinhole cameras form images. Rays only change direction when they reflect or refract.

A plane mirror forms a virtual image (the light only appears to come from behind the mirror), upright, the same size as the object, and as far behind the mirror as the object is in front. It is also laterally inverted (left–right swapped).

Refractive index is $n = \dfrac{c}{v}$. A larger $n$ means a smaller speed $v$, so light is slower in a high-index material. The bigger the slow-down, the more the light bends on entering.

Total internal reflection needs two things: light going from an optically denser (higher-$n$) medium into a less dense one, and an angle of incidence greater than the critical angle. Then no light refracts out – it is all reflected back.

Dispersion is the separation of white light into its colours, because each colour refracts by a slightly different amount in the glass. It produces the spectrum you see from a prism (and, via raindrops, a rainbow).

A converging lens bends parallel rays to meet at its focal point, a distance of one focal length beyond the lens. A diverging (concave) lens would instead spread the rays apart.

$v = \dfrac{c}{n} = \dfrac{3.0\times10^{8}}{1.5} = 2.0\times10^{8}\text{ m/s}$. (Multiplying instead of dividing gives $4.5\times10^{8}$, which is faster than light and impossible.)

$\sin\theta_2 = \dfrac{n_1\sin\theta_1}{n_2} = \dfrac{(1.00)\sin 30^\circ}{1.33} = 0.376$, so $\theta_2 = 22^\circ$. The ray bends toward the normal on entering the denser water.

$\sin\theta_c = \dfrac{1}{n} = \dfrac{1}{1.50} = 0.667$, so $\theta_c = 42^\circ$. At exactly this angle the refracted ray runs along the surface; beyond it, the light is totally internally reflected.

Your image is always as far behind the mirror as you are in front. As you move $2\text{ m/s}$ toward the mirror, the image moves $2\text{ m/s}$ toward it as well, so the gap between you closes at $2 + 2 = 4\text{ m/s}$. (Counting only your own speed gives the $2\text{ m/s}$ trap.)

The frequency is fixed and the light slows by the factor $n$, so the wavelength shrinks by the same factor: $\lambda_\text{glass} = \dfrac{\lambda_\text{air}}{n} = \dfrac{600}{1.5} = 400\text{ nm}$. (The colour you see is set by the unchanged frequency, not this shorter wavelength.)

Violet has the shortest wavelength and the highest refractive index in glass, so it slows the most and bends the most – it is deviated furthest. Red, with the longest wavelength, bends least. That is why violet sits at the most-deviated edge of the spectrum.

The angle of incidence $50^\circ$ is greater than the critical angle $42^\circ$, so no light can escape into the air: it is all totally internally reflected back into the glass.

When the object is beyond the focal point, a converging lens forms a real, inverted image on the far side of the lens (this is how a camera or the eye works). Only when the object is inside the focal length does the image become virtual and upright.

$\text{apparent depth} = \dfrac{2.0}{1.33} = 1.5\text{ m}$. Refraction at the surface bends the light so the coin looks nearer the surface than it really is. (Multiplying by $n$ instead, giving $2.7\text{ m}$, would make it look deeper – the wrong way.)

Rearranging, $n = \dfrac{1}{\sin\theta_c} = \dfrac{1}{\sin 40^\circ} = \dfrac{1}{0.643} = 1.56$. (A smaller critical angle means a higher refractive index – dense materials like diamond have very small critical angles.)

First the index: $n = \dfrac{c}{v} = \dfrac{3.0\times10^{8}}{2.0\times10^{8}} = 1.5$. Then Snell's law, $(1.5)\sin 20^\circ = (1.0)\sin\theta_2$, gives $\sin\theta_2 = 0.513$, so $\theta_2 = 31^\circ$ – bending away from the normal into the air. (Since $20^\circ$ is below the $42^\circ$ critical angle, the light does get out.)

A glass's refractive index varies slightly with the colour (wavelength) of light – it is a little higher for violet than for red. Because $n$ differs, each colour bends by a different amount at the surfaces, so they separate. Their frequency (which sets the colour) does not change.

$\dfrac{1}{v} = \dfrac{1}{f} - \dfrac{1}{u} = \dfrac{1}{10} - \dfrac{1}{30} = \dfrac{3-1}{30} = \dfrac{2}{30}$, so $v = 15\text{ cm}$ on the far side of the lens.

The magnification is $m = \dfrac{v}{u} = \dfrac{15}{30} = 0.5$, so the image is half the size. Because it is a real image from a single converging lens, it is inverted.

Refract each colour separately. Red: $\sin\theta = \dfrac{\sin 60^\circ}{1.50} = 0.577$, so $35.3^\circ$. Violet: $\dfrac{\sin 60^\circ}{1.54} = 0.562$, so $34.2^\circ$. They differ by about $1^\circ$ – violet bends slightly more (higher $n$). That small spread, widened over distance, is dispersion. (Answering $0^\circ$ misses that the two indices differ at all.)

$\sin\theta_c = \dfrac{n_2}{n_1} = \dfrac{1.33}{1.50} = 0.887$, so $\theta_c = 62^\circ$. Because the two indices are close, the critical angle is large. (The $42^\circ$ trap uses $1/n$, which is the critical angle for glass-to-air, not glass-to-water.)

Rearranging $\text{apparent} = \dfrac{\text{real}}{n}$ gives $n = \dfrac{\text{real}}{\text{apparent}} = \dfrac{1.5}{1.2} = 1.25$. (Dividing the other way gives $0.80$, which is less than 1 and impossible for a refractive index.)

$\dfrac{1}{v} = \dfrac{1}{10} - \dfrac{1}{5} = -\dfrac{1}{10}$, so $v = -10\text{ cm}$. The negative sign means a virtual image on the same side as the object. Its magnification is $\left|\dfrac{v}{u}\right| = \dfrac{10}{5} = 2$, so it is upright and twice as large – exactly what a magnifying glass does.

The $45^\circ$ angle at the sloping face exceeds the critical angle of $42^\circ$, so the ray is totally internally reflected – turning through $90^\circ$ and leaving through the bottom face. This is exactly how prisms redirect light in binoculars and periscopes, without the light loss of a mirror.