Why a Glass Lens Loses Most of Its Power Underwater

Look into the inside of a steel spoon and your face appears small and upside down; turn it over and you look upright. A few formulas, built from the geometry of rays and one law of refraction, predict every such image.

This lesson covers the mirror formula, Snell's law with total internal reflection, and the lens-maker's and thin-lens formulas.

**For a spherical mirror, object distance u, image distance v and focal length f are related by $\frac{1}{v}+\frac{1}{u}=\frac{1}{f}$, with $f=R/2$, and the magnification is $m=-\frac{v}{u}$.

The derivation for a concave mirror.** An object AB forms a real image A'B'. For a small aperture, similar triangles give

$$\frac{A^{\prime }{B}^{\prime }}{AB}=\frac{B^{\prime }F}{FP}\frac{A^{\prime }{B}^{\prime }}{AB}=\frac{B^{\prime }P}{BP}$$

So $\frac{B^{\prime }F}{FP}=\frac{B^{\prime }P}{BP}$. With the Cartesian sign convention, $B^{\prime }P=-v$, $FP=-f$, $BP=-u$ and $B^{\prime }F=-v+f$, which gives $\frac{v-f}{f}=\frac{v}{u}$. Dividing $uv-uf=vf$ by $uvf$ gives the mirror formula.

Worked example 1 — concave. $f=-15$ cm and $u=-25$ cm:

$$\frac{1}{v}=-\frac{1}{15}+\frac{1}{25}=-\frac{2}{75}\Rightarrow v=-37.5\text{ cm}m=-\frac{-37.5}{-25}=-1.5$$

The image is real, inverted and magnified.

Worked example 2 — convex. $f=+20$ cm and $u=-20$ cm give $\frac{1}{v}=\frac{1}{20}+\frac{1}{20}$, so $v=+10$ cm and $m=+0.5$ — virtual, upright and smaller.

An everyday example. The rear-view mirror on a scooter is convex, giving a small upright image of a wide stretch of road.

The substance. A negative m means inverted, and a real image of a real object always has negative m — the sign carries as much information as the size.

**Snell's law states $n_1\sin i=n_2\sin r$, and when light travels from a denser to a rarer medium at an angle greater than the critical angle C, where $\sin C=\frac{n_2}{n_1}$, all of it reflects back — total internal reflection.

Two conditions for total internal reflection:

- Light must travel from the optically denser medium towards the rarer one
- The angle of incidence must exceed the critical angle

Worked example 1 — refraction.** Light strikes water, with $n=1.33$, at $60^{\circ }$:

$$\sin r=\frac{\sin 60^{\circ }}{1.33}=0.651\Rightarrow r=40.6^{\circ }$$

Worked example 2 — an optical fibre. A core of $n_1=1.50$ inside cladding of $n_2=1.45$:

$$\sin C=\frac{1.45}{1.50}=0.967\Rightarrow C=75.2^{\circ }$$

Any ray meeting the core wall at more than $75.2^{\circ }$ stays trapped, bouncing along the fibre.

Totally reflecting prisms. For glass, $\sin C=1/1.5$, so $C=41.8^{\circ }$. A right-angled prism receives light at $45^{\circ }$ on its long face, which exceeds $41.8^{\circ }$, so it turns the beam through $90^{\circ }$ or $180^{\circ }$ without the dimming of a silvered mirror.

An everyday example. Fibre broadband connections to homes carry data as pulses of light guided by total internal reflection.

The substance. Total internal reflection can never happen when light enters a denser medium — the refracted ray then always bends towards the normal and exists.

**The lens-maker's formula is $\frac{1}{f}=(n-1)\left(\frac{1}{R_1}-\frac{1}{R_2}\right)$, the thin-lens formula is $\frac{1}{v}-\frac{1}{u}=\frac{1}{f}$, and thin lenses in contact combine as $\frac{1}{F}=\frac{1}{f_1}+\frac{1}{f_2}$, so powers add.

The derivation.** Refraction at a single spherical surface obeys $\frac{n_2}{v}-\frac{n_1}{u}=\frac{n_2-n_1}{R}$. Applying this at the first surface, then using that image as the object for the second surface, and adding the two equations for a thin lens in air gives the lens-maker's formula.

Worked example 1. A biconvex lens with $n=1.5$, $R_1=+20$ cm and $R_2=-20$ cm:

$$\frac{1}{f}=0.5\left(\frac{1}{20}+\frac{1}{20}\right)=\frac{1}{20}\Rightarrow f=20\text{ cm}$$

Worked example 2 — lenses in contact. Lenses of $+20$ cm and $-40$ cm:

$$\frac{1}{F}=\frac{1}{20}-\frac{1}{40}=\frac{1}{40}\Rightarrow F=40\text{ cm},P=\frac{100}{40}=+2.5\text{ D}$$

An object at $u=-60$ cm gives $\frac{1}{v}=\frac{1}{40}-\frac{1}{60}=\frac{1}{120}$, so $v=120$ cm and $m=v/u=-2$.

An everyday example. An optician adds lens powers directly when combining a trial lens with another in the testing frame, because powers of thin lenses in contact add.

The substance. A lens's focal length depends on its surroundings — a glass lens in water has $(n-1)$ replaced by $(1.5/1.33-1)$, making f about four times longer.