Radius is twice the focal length

The idea

For spherical mirrors of small aperture (the diameter of the reflecting surface), the radius of curvature is twice the focal length, R = 2f, so the principal focus lies midway between the pole and the centre of curvature.

A bright spot from a lamp

Parallel rays of light from a far away lamp fall on a concave mirror. The centre of curvature C of the mirror is 40 cm in front of it. The mirror gathers the rays into one small bright spot.

How far in front of the mirror will the bright spot be?

• About 20 cm, halfway between the mirror and C
• About 40 cm, at C
• About 80 cm, beyond C

Change the radius

Drag the slider to change the radius of curvature R. Watch where F lies compared with C.

Half of R

As R changed, F always stayed halfway between the pole P and C. So the focal length f was always half of R.

For a spherical mirror of small aperture (the aperture is the diameter of the reflecting surface), the radius of curvature is twice the focal length: R = 2f. So the principal focus lies midway between the pole and the centre of curvature.

• R (cm) | f = R/2 (cm)
• 20 | 10
• 30 | 15
• 40 | 20
• 60 | 30

Notes

For a spherical mirror of small aperture, R = 2f: the principal focus lies midway between the pole and the centre of curvature.

Check yourself

A concave mirror has a radius of curvature of 30 cm. What is its focal length, in cm?

Answer: 15 cm

R = 2f, so f = 30 cm divided by 2 = 15 cm.

A mirror has a focal length of 12 cm. What is its radius of curvature, in cm?

Answer: 24 cm

R = 2f = 2 times 12 cm = 24 cm.

Where does the principal focus of a spherical mirror lie?

For which mirrors is R = 2f found to hold?

• At the centre of curvature. C is at a distance R from the pole. The focus is at a distance f = R/2, so it is nearer to the mirror than C.
• Midway between the pole and the centre of curvature — correct. Yes!
• At the pole. The pole P is on the mirror. The focal length is the distance from P to F, so F is not at P.
• Only mirrors with a very small radius of curvature. The condition is about the aperture (the diameter of the reflecting surface), not about R being small.
• Mirrors of any aperture, however wide. The relation is found for spherical mirrors of small aperture, and the chapter deals only with those.
• Mirrors of small aperture, whose diameter is much smaller than R — correct. Yes!