Determine the focal length of a given concave mirror by the u-v method and verify the mirror formula 1/v + 1/u = 1/f.
Determine the focal length of a given concave mirror by the u-v method and verify the mirror formula 1/v + 1/u = 1/f.
To determine the focal length of a given concave mirror using an optical bench by the u-v method, and to verify the mirror formula 1/v + 1/u = 1/f.
For a spherical mirror, the object distance (u), the image distance (v), and the focal length (f) — all measured from the pole of the mirror — are related by the mirror formula:
1/v + 1/u = 1/f
Note the plus sign, in contrast to the thin lens formula 1/v − 1/u = 1/f, which carries a minus sign. This difference arises because a mirror forms an image by reflection (the image lies on the same side as the object), whereas a lens forms an image by refraction (light passes through to the far side).
All distances in this experiment are measured from the pole (P) of the mirror using the Cartesian sign convention, with the direction of the incident light taken as positive and distances measured against it taken as negative:
f = R / 2
For a mirror of small aperture, the focal length is exactly half the radius of curvature (R). This relation follows from the geometry of paraxial rays reflecting close to the pole, and provides a quick way to relate the two quantities once either is known.
In this method, the illuminated object is placed beyond the centre of curvature (C) of the concave mirror, so that a real, inverted image forms on the same side. The screen is then moved back and forth until the sharpest, most clearly focused inverted image of the object is obtained on it — care being taken not to let the screen coincide with the object itself. The distance of the object from the pole is noted as u, and the distance of the screen (image) from the pole is noted as v. Since both are measured against the direction of the incident light, they are recorded as negative in the sign convention, though for simplicity the observation table below records their magnitudes and applies the formula f = uv/(u+v) using positive magnitudes throughout.
When the object is placed exactly at the centre of curvature (u = R), the reflected rays converge to form the image also at the centre of curvature (v = R). The image is real, inverted, and exactly the same size as the object. This special position is a quick, independent check — locating the point where object and image coincide in size immediately gives the radius of curvature R, and hence f = R/2, without needing to apply the full mirror formula.
Figure: Optical bench setup showing the illuminated object placed beyond the centre of curvature C, the concave mirror at the pole P, and the screen positioned at the sharp, real, inverted image. Object distance u and image distance v are measured from the pole.
Least count of the bench scale: 0.1 cm
| S.No. | Mirror position (cm) | Object position (cm) | u (cm) | Screen position (cm) | v (cm) | f = uv/(u+v) (cm) |
|---|---|---|---|---|---|---|
| 1 | 80.0 | 35.0 | 45.0 | 57.4 | 22.6 | 15.04 |
| 2 | 80.0 | 40.0 | 40.0 | 55.9 | 24.1 | 15.04 |
| 3 | 80.0 | 45.0 | 35.0 | 53.7 | 26.3 | 15.02 |
| 4 | 80.0 | 50.0 | 30.0 | 49.8 | 30.2 | 15.05 |
| 5 | 80.0 | 55.0 | 25.0 | 42.4 | 37.6 | 15.02 |
Mean focal length: favg = (15.04 + 15.04 + 15.02 + 15.05 + 15.02) / 5 = 15.03 cm ≈ 15.0 cm
Rough check (distant-object / infinity method): Mirror-to-screen distance for a sharp image of a distant object was found to be approximately 15.1 cm, in close agreement with the u-v method.
Given:
Using f = uv / (u + v):
f = (45.0 × 22.6) / (45.0 + 22.6)
f = 1017.0 / 67.6
f = 15.04 cm
Taking the mean of all five readings:
Mean f = (15.04 + 15.04 + 15.02 + 15.05 + 15.02) / 5 = 15.03 ≈ 15.0 cm
Substituting the mean f back into the mirror formula 1/v + 1/u = 1/f for each pair of u and v confirms the relation holds to within the limits of experimental error, verifying the mirror formula.
The focal length of the given concave mirror, determined by the u-v method:
f = 15.0 ± 0.2 cm
The radius of curvature of the mirror (R = 2f):
R = 30.0 cm
The mirror formula 1/v + 1/u = 1/f is verified within experimental error for all five sets of observations.
The mirror formula relates the object distance (u), image distance (v), and focal length (f) of a spherical mirror as 1/v + 1/u = 1/f. This differs from the thin lens formula, 1/v − 1/u = 1/f, which has a minus sign. The difference arises because a mirror forms an image by reflection, with the image lying on the same side as the object, whereas a lens forms an image by refraction, with light passing through to the far side, so the sign convention treats the distances differently.
Using the Cartesian sign convention, all distances are measured from the pole of the mirror, with the direction of the incident light taken as positive. Since a concave mirror's principal focus lies in front of the mirror, on the same side as the incident light, its focal length is measured against the positive direction and is therefore assigned a negative value. A convex mirror, whose focus lies behind the reflecting surface, has a positive focal length by the same convention.
For a spherical mirror of small aperture, the focal length (f) is exactly half the radius of curvature (R), expressed as f = R/2. This relation follows from the geometry of a paraxial ray reflecting close to the pole of the mirror, where the reflected ray crosses the principal axis at a point midway between the pole and the centre of curvature. Knowing either f or R therefore immediately gives the other.
When an object is placed exactly at the centre of curvature (u = R) of a concave mirror, the image also forms at the centre of curvature (v = R). The image is real, inverted, and exactly the same size as the object, because rays striking the mirror along the radius are reflected straight back along the same path. This special case gives a quick, independent check of the radius of curvature and hence the focal length.
Rays coming from a very distant object (effectively at infinity) are almost parallel to the principal axis by the time they reach the mirror. A concave mirror converges parallel rays to a point at its principal focus. By pointing the mirror toward a distant object, such as a window or a tree far away, and adjusting a screen until the sharpest, smallest image is obtained, the distance from the mirror to the screen at that position gives an approximate value of the focal length directly, without needing to solve the mirror formula. This serves as a useful rough cross-check on the u-v method.
