Convex lens focal length theory
Understand convex lens focal length: the equations, assumptions, supporting concepts and measurement method.
01What focal length describes
A converging lens directs paraxial rays parallel to its axis towards a focus. Focal length is the separation between the lens principal plane and that focus. For the ideal thin lens, the principal planes coincide with the drawn lens plane. Moving the object changes image location, while this lens property stays fixed.
Does refocusing a screen change the lens focal length?
No. Refocusing locates a new image position for a different object distance.
02Distances and the convention used here
For this real-image measurement, u and v are positive distance magnitudes: u runs from the object to the lens plane, and v from the lens plane to the focused screen. Read them from the lens plane rather than from its holder edge. This convention differs from a Cartesian coordinate convention that assigns a negative coordinate to the object side.
If the lens coordinate is 40 cm and the object coordinate is 10 cm, what is u?
u = 30 cm. It is the positive separation, not the object's coordinate.
03Connecting object and image positions
The thin-lens equation relates the two measured separations to f. Rearranging it gives a direct focal-length estimate for each focused pair. Use one distance unit consistently. For u = 30.0 cm and v = 30.0 cm, the calculated focal length is 15.0 cm, rather than 30.0 cm.
Why must an unfocused screen distance not be substituted as v?
v identifies the image plane. An arbitrary screen position does not satisfy the image-formation condition.
04Real images, virtual images and the focal boundary
With u greater than f, rays meet beyond the lens and the image can be caught on a screen. At u = f, outgoing rays are parallel and there is no finite screen position for sharp focus. With 0 < u < f, a converging lens gives an upright virtual image on the object side; the screen method used here cannot measure it. Measurement controls therefore keep u between 22 and 70 cm.
Can moving the screen farther away produce a sharp image when u = f?
No finite screen distance works in the ideal model.
05Image size and ray construction
A ray parallel to the axis refracts towards the far focus; a ray through the ideal thin-lens centre continues without deviation. Their intersection locates the image tip. Magnification is negative for this inverted real image. At u = 2f, v = 2f and image and object have equal size. Between f and 2f the real image is larger; beyond 2f it is smaller.
What does m = −0.5 tell you?
The image is inverted and its height magnitude is half the object's height.
06A graph that tests the whole series
Set x = 1/u and y = 1/v. The lens equation becomes y = −x + 1/f. Plot measured reciprocal distances, fit a line with a free slope, and compare it with the predicted slope of −1. The vertical intercept b has units cm⁻¹ when distances are in cm; its reciprocal gives f in cm. Do not force the slope to −1 and then describe the fitted slope as experimental confirmation.
What focal length corresponds to b = 0.0667 cm⁻¹?
f = 1/b ≈ 15.0 cm.
07Resolution, repeatability and systematic error
Record focused distances to 0.1 cm. Rounding is not a complete uncertainty budget: real focus judgement, scale calibration and lens-plane offsets also matter. A shared wrong lens reference can bias every pair, even if their calculated focal lengths agree closely. Repeating a measurement tests repeatability, while checking the distance origin investigates a different possible error.
Does a small spread of f values prove there is no systematic error?
No. A consistent reference-plane error can shift an entire set of otherwise repeatable measurements.
The lens has a model focal length of 15.0 cm and is treated as thin and paraxial. The scene is a schematic ray diagram. A screen setting within 0.05 cm of the ideal image plane is accepted; this tolerance represents half a 0.1 cm scale step, not optical depth of focus. Spherical/chromatic aberrations, lens thickness and alignment errors are omitted.