The Lens Equation, Magnification and Power of a Lens

1What does "−2.00" on a spectacle card mean?

When an optometrist prescribes spectacles, the card might read "Right −2.00, Left −2.25". Those numbers are not sizes or prices. Each is the power of a lens in dioptres, and the minus sign says the lens is diverging. In lesson 3 you found images by drawing; in this lesson you will calculate them, measure a focal length, and see how lenses correct eyesight.

2The lens equation

For a thin lens of focal length $ff$, with an object at distance $uu$ and its image at distance $vv$, both measured from the optical centre:

It has the same form as the mirror equation you used in S2, with the same sign convention, "real is positive":

• Distances to real objects and real images are positive. Distances to virtual images are negative.
• The focal length of a converging lens is positive; that of a diverging lens is negative.

The linear magnification compares the size of the image with the size of the object:

Use the sizes of $vv$ and $uu$ here; the sign of $vv$ has already told you whether the image is real or virtual. If $mm$ is greater than 1 the image is magnified; less than 1, diminished.

3Experiment: measuring the focal length of a converging lens

Apparatus: a converging lens in a holder, an illuminated object (a lamp behind a card with a cross-wire cut-out), a white screen and a metre rule.

1. Place the object at a measured distance $uu$ from the lens, more than one focal length away.
2. Move the screen until the image on it is sharp. Measure the image distance $vv$.
3. Repeat for at least five values of $uu$.
4. Work out $f=\frac{uv}{u+v}f\; =\; \backslash frac\{uv\}\{u\; +\; v\}$ for each pair (this is the lens equation rearranged), or plot $\frac{1}{v}\backslash frac\{1\}\{v\}$ against $\frac{1}{u}\backslash frac\{1\}\{u\}$.

These results are illustrative. The mean focal length is 15.0 cm. On the graph, the points lie on a straight line, because $\frac{1}{u}+\frac{1}{v}\backslash frac\{1\}\{u\}\; +\; \backslash frac\{1\}\{v\}$ is the same, $\frac{1}{f}\backslash frac\{1\}\{f\}$, for every pair. The line meets each axis at $\frac{1}{f}=0.067\mathrm{c}{\mathrm{m}}^{-1}\backslash frac\{1\}\{f\}\; =\; 0.067\backslash ,\backslash mathrm\{cm^\{-1\}\}$, giving $f=\frac{1}{0.067}=15\mathrm{c}\mathrm{m}f\; =\; \backslash frac\{1\}\{0.067\}\; =\; 15\backslash ,\backslash mathrm\{cm\}$.

The lamp gets hot: let it cool before you move it, and keep its cable away from the screen and the metre rule.

4Power of a lens

A lens with a short focal length bends light strongly. Opticians describe this with the power of the lens:

The unit is the dioptre (D), equal to ${\mathrm{m}}^{-1}\backslash mathrm\{m^\{-1\}\}$. Converging lenses have positive power and diverging lenses negative power, following the sign of $ff$. When thin lenses are placed in contact, their powers add:

5Lenses at work: the eye and its defects

The eye is a converging lens system. The curved cornea and the eye lens form a real, inverted, diminished image on the retina, just like the "object beyond 2F" diagram in lesson 3. The brain turns the picture the right way up. To focus on near or far objects, the ciliary muscles change the shape, and so the focal length, of the eye lens. This is called accommodation.

So the "−2.00" spectacles in the opening are diverging lenses of focal length $\frac{1}{2.00}=0.50\mathrm{m}\backslash frac\{1\}\{2.00\}\; =\; 0.50\backslash ,\backslash mathrm\{m\}$ for a short-sighted eye. From about the age of 40 to 45 the eye lens also stiffens and cannot focus on near objects, and most people then need converging reading glasses. A pupil who cannot read the board, gets headaches or screws up their eyes should have an eye test at a health centre.