The power of a lens

How strong is a lens?

A lens of a short focal length bends light a lot. A lens of a long focal length bends it a little. The power of a lens is a number for this. It is large when the lens bends light strongly.

Power and focal length

The power $P$ of a lens is the reciprocal of its focal length, with $f$ in metres:

The unit of power is the dioptre, with the symbol D. One dioptre is the power of a lens of focal length $1\ \text{m}$: $1\ \text{D} = 1\ \text{m}^{-1}$.

• A convex lens has a positive $f$, so a positive power.
• A concave lens has a negative $f$, so a negative power.

A lens of a smaller focal length has a larger power. A lens with $f = 0.125\ \text{m}$ has $P = +8\ \text{D}$. A lens with $f = 2\ \text{m}$ has $P = +0.5\ \text{D}$.

Worked example 1

Find the power of a convex lens with $f = 50\ \text{cm}$, and of a concave lens with $f = 20\ \text{cm}$.

1. Change the focal lengths to metres: $0.5\ \text{m}$ and $0.2\ \text{m}$.
2. The convex lens: $P = \dfrac{1}{0.5\ \text{m}} = +2\ \text{D}$.
3. The concave lens has a negative $f$, so $P = \dfrac{1}{-0.2\ \text{m}} = -5\ \text{D}$.

The most common mistake is to put $f$ in centimetres into the formula. The answer would be $100$ times too small.

Lenses in contact

When two thin lenses are placed close together, one after the other, the power of the combination is the sum of the powers:

The focal length of the combination is $f = \dfrac{1}{P}$. An optician uses this, and the makers of cameras use it too, because a camera lens is made of several lenses together.

You should find that a weak concave lens only lengthens the focal length of the pair. A strong one wins, and the pair spreads the light. The focus jumps to the near side when $P$ turns negative.

Worked example 2

A convex lens of $+4\ \text{D}$ is placed in contact with a concave lens of $-1.5\ \text{D}$. Find the power and the focal length of the combination.

1. Add the powers. $P = +4\ \text{D} + (-1.5\ \text{D}) = +2.5\ \text{D}$.
2. The focal length. $f = \dfrac{1}{P} = \dfrac{1}{2.5\ \text{D}} = 0.4\ \text{m} = 40\ \text{cm}$.
3. So the combination acts as a convex lens, because the power is positive.

Spectacles

An optician writes the power of a spectacle lens in dioptres. A prescription of $-1.25\ \text{D}$ means a concave lens of focal length $\dfrac{1}{-1.25\ \text{D}} = -0.8\ \text{m}$, which is $-80\ \text{cm}$. It is used for a person who cannot see distant things clearly. A prescription of $+1.5\ \text{D}$ means a convex lens, for a person who cannot see near things clearly. We study these two eye defects in the next chapter.

Practise with new numbers

Try it in Python

The program finds the power of four lenses, and it adds the powers of two lenses in contact.

Remember these

• Power $P = 1/f$, with $f$ in metres. The unit is the dioptre (D).
• A convex lens has a positive power. A concave lens has a negative power.
• For thin lenses in contact, the powers add: $P = P_1 + P_2$.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. Find the power of a convex lens of focal length $16\ \text{cm}$.

2. A spectacle lens has the power $-2\ \text{D}$. What is its focal length, and what kind of lens is it?

3. Two lenses of $+3\ \text{D}$ and $+2\ \text{D}$ are in contact. What is the power of the pair?

4. Which bends light more strongly: a lens of $+2\ \text{D}$ or a lens of $-6\ \text{D}$?