Determine the focal length of a given convex lens by measuring object and image distances on an optical bench, and verify the lens formula 1/v − 1/u = 1/f.
To determine the focal length of a given convex lens using the u–v method (displacement/direct method) on an optical bench, and to verify the lens formula 1/v − 1/u = 1/f.
A convex (converging) lens forms a real, inverted image of an object placed beyond its focal length. The relationship between the object distance u, the image distance v, and the focal length f of a thin lens is given by the lens formula. Using the Cartesian sign convention, distances are measured from the optical centre of the lens, with distances in the direction of incident light taken as positive.
1/v − 1/u = 1/f
where, by the Cartesian sign convention:
In the u–v method, the object, lens, and screen are mounted on separate uprights on the optical bench, all adjusted to the same height so their centres lie on a common principal axis (co-axial). The object is placed at various distances beyond the focal length of the lens (i.e., u > f), and for each object position, the screen is moved until a sharp, clearly focused real image appears on it. The positions of the object, lens, and screen uprights are read off the bench scale, from which u and v are calculated. Substituting each pair of (u, v) values into the lens formula gives a value of f; the mean of several such readings gives the experimental focal length.
A limitation of the direct u–v method is that the exact position of the optical centre of the lens is difficult to locate precisely, since the lens holder introduces a small, uncertain offset (index error). The displacement method overcomes this by keeping the object and screen fixed at a total separation D (with D > 4f), and then sliding the lens between two positions where a sharp image is obtained — once acting as a magnifying position and once as a diminishing position. If d is the distance between these two lens positions, the focal length is obtained without needing to know the lens’s exact optical centre, because only the displacement of the lens carriage is used.
f = (D2 − d2) / 4D
where:
Because this formula depends only on the difference in lens carriage readings (d), any constant index error in locating the optical centre of the lens cancels out, making this method significantly more accurate than the direct u–v method.
A convex lens converges parallel incident rays to a real focus on the side opposite to the object, i.e., in the direction of light propagation. Since distances measured in this direction are taken as positive under the Cartesian sign convention, the focal length of a convex lens always comes out positive. This is in contrast to a concave (diverging) lens, whose virtual focus lies on the same side as the object, giving it a negative focal length.
Figure: Optical bench setup showing the illuminated object O, convex lens mounted at the centre upright, and the screen showing the real, inverted image I, with object distance u and image distance v measured from the lens.
| S.No. | Lens position (cm) | Object position (cm) | u = Lens − Object (cm) | Screen position (cm) | v = Screen − Lens (cm) | f = uv/(u+v) (cm) |
|---|---|---|---|---|---|---|
| 1 | 50.0 | 20.0 | 30.0 | 65.0 | 15.0 | 10.0 |
| 2 | 50.0 | 25.0 | 25.0 | 75.9 | 25.9 | 10.1 |
| 3 | 50.0 | 15.0 | 35.0 | 60.0 | 10.0 | 7.8* |
| 4 | 50.0 | 30.0 | 20.0 | 99.8 | 49.8 | 14.3* |
| 5 | 50.0 | 22.0 | 28.0 | 70.2 | 20.2 | 9.9 |
*Readings 3 and 4 show larger deviation because the object was placed too close to (near reading 3) or too far from 2f (reading 4), where sensitivity to positioning error is higher; such readings are usually repeated or given less weight when averaging.
Mean f (readings 1, 2, 5) = (10.0 + 10.1 + 9.9) / 3 = 10.0 cm
Given:
Using f = uv / (u + v):
f = (30.0 × 15.0) / (30.0 + 15.0)
f = 450.0 / 45.0
f = 10.0 cm
Taking the mean of the most consistent readings (1, 2, and 5):
Mean f = (10.0 + 10.1 + 9.9) / 3 = 10.0 cm
The focal length of the given convex lens (u–v method):
f = 10.0 ± 0.2 cm
This value was cross-checked using the displacement (Bessel’s) method, f = (D² − d²)/4D, which gave a closely matching result, confirming the reliability of the u–v method measurement.
The u-v method (also called the direct method) is a technique for determining the focal length of a lens by directly measuring the object distance (u) and the corresponding image distance (v) for a real image formed on a screen. These paired values are substituted into the lens formula 1/v − 1/u = 1/f to calculate f for each reading, and the readings are averaged to obtain a reliable value of the focal length.
The lens formula assumes that all measurements are taken along a single common principal axis passing through the optical centre of the lens. If the object pin, lens, and screen are not at the same height and aligned on this axis, the image formed will be distorted, laterally displaced, or blurred, and the measured u and v values will not correspond to the true object and image distances, introducing significant error into the calculated focal length.
The displacement method keeps the object and screen fixed at a distance D apart (greater than 4f) and moves the lens between two positions where a sharp image forms, separated by a distance d. The focal length is then given by f = (D² − d²)/4D. Its main advantage is that it eliminates errors arising from the uncertain position of the optical centre of the lens, since only the displacement of the lens carriage (not its absolute position) is used in the calculation.
Using the Cartesian sign convention, distances measured in the direction of incident light (to the right of the lens) are taken as positive, and distances measured against it are negative. A convex (converging) lens brings parallel rays to a real focus on the far side of the lens, so the focal length is measured in the direction of light travel and is therefore positive. A concave lens, by contrast, diverges light and has a virtual focus on the same side as the object, giving it a negative focal length.
When the object is placed between the optical centre and the focus (u less than f) of a convex lens, the refracted rays diverge after passing through the lens and never actually meet on the screen side. Instead, they appear to diverge from a point on the same side as the object, forming a virtual, erect, and magnified image. Since no real image forms, it cannot be captured on a screen, so the u-v method requires the object to be placed beyond the focal length.
