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# Measurement of the Focal Length of a Concave Mirror

This may be obtained optically by means of the formula(1)

f being the focal length, and r the radius of the surface, u and v respectively the distances from the surface of an object and its image; u and v can be measured, and then r or f calculated.

In practice the following modification of the method will be found most convenient.

It depends on the fact that when the image of an object formed by concave mirror coincides with the object itself, then the object is at the geometrical centre of the spherical surface.

Place a needle in a clip and set it in front of the mirror; place the eye some distance further away from the mirror -than the needle. An inverted image of the needle will be seen, unless the needle has been placed too close to the mirror. Adjust the position of the needle relatively to the mirror, so that the point of the image coincides with the point of the needle. When this is the case the image will be of the same size as the object.

The adjustment can be made as finely as necessary, either by moving the eye about and noting whether the relative positions of image and needle vary, or by using a strong magnifying lens, and noticing whether both needle and image are in focus at the same time.

If the aperture of the mirror be very large, and its surface not perfectly spherical, it may be impossible to see the image when using the lens, in consequence of the aberration of the rays from the outer portions of the surface. These defects may, in some cases, be corrected by covering the mirror with black paper, leaving at the centre only a small hole, which may be either oblong or circular.

When the position of the needle has been carefully adjusted, measure its distance from the reflecting surface by means of a pair of compasses and a scale, if the radius be small; or by the method already described if the mirror be fitted to the optical bench.

The result gives the length of the radius of the mirror surface. Half of it is the focal length.

Experiment. - Determine the radius of curvature of the given mirror, and check your result by the use of the sphero-meter.

Enter results thus:

```Radius of curvature by optical observations: 19.52 cm
Radius of curvature by spherometer: 19.8 cm
```

 (1) For the formulae required in this and the next chapter we may refer to Glazebrook, Physical Optics, chap. iv.

Last Update: 2011-03-27