V60 – Lens experiment

An object of height G stands in front of a converging lens; its distance g from the lens is called the object distance. Light leaves each of its points in all directions; the lens collects it again in one point each — that is where the image forms, with the height B. Its distance b from the lens is the image distance. The lens has two focal points F₁ and F₂, both at the distance of the focal length f from its centre.

Three distinguished rays are enough to find the image — they can be drawn without any calculation, because for each of them it is known how it leaves the lens:

Figure: ray diagram for g = 140 and f = 84 (arbitrary units). The drawing is to scale — the three rays meet in one point.

1. Parallel ray: travels towards the lens parallel to the axis and leaves it through the focal point F₂.
2. Focal ray: travels through F₁ towards the lens and leaves it parallel to the axis. It is the parallel ray in reverse — light has no preferred direction.
3. Central ray: passes through the centre of the lens and is there not deflected. At a thin lens it meets front and back surface at the same angle.

The tip of the image lies where the rays intersect. Two rays already fix that point; the third one is the check.

What the drawing shows

Real, inverted — and magnified by b/g magnified

The image forms where real light meets — it can be caught on a screen. That is exactly what you do later in the experiment. It stands upside down, and its size ratio can be read straight off the drawing: the central ray crosses the axis at the centre of the lens, so B to G as b to g.

This ratio is called the magnification. In the drawing above it is 210/140 = 1.5 — the image is one and a half times the size of the object.

The drawing contains four right-angled triangles which are pairwise similar : they have the same angles, hence the same length ratios. From these ratios follows the relation between g, b and f — without any measurement.

Imaging equation

What the equation achieves

It links exactly the three quantities this experiment is about: the two distances you can measure on the optical bench, and the focal length you are looking for. Knowing two of them fixes the third.

A single pair of values of g and b therefore already yields a focal length. A Measurement series yields more: it averages out the scatter and shows whether the thin-lens model holds at all. How that works is in the next chapters.

Light source, object, converging lens and screen sit on a common optical axis. Object and screen stay in place, their distance e is set before each measurement. Only the lens is moved, until the image is sharp on the screen.

This works at two different lens positions: one puts a large image on the screen, the other a small one. Each position has its own object distance, which is read off the setup.

Figure: the same setup in both positions. The screen shows the image of the solid position; in the dashed one a smaller image appears in the same place. The drawing is to scale.

• e — solid Object-screen distance (is set before each measurement)
• g₁, b₁ — Object and image distance in the position with the large image
• g₂, b₂ — the same distances in the position with the small image

The same distance lies between object and screen in both positions, it is merely divided differently:

What stands out

Two positions, one distance

Compare the two dimension lines in the drawing: what is the object distance in one position is the image distance in the other. That is no coincidence, because light has no preferred direction, and the same lens images the path sharply in reverse as well.

That these two positions already hold everything needed for the focal length is the subject of the next chapter.

The two lens positions from chapter 3 belong to a single distance e. Their separation is called d, and these two distances alone determine the focal length — without measuring the object and image distances individually.

Figure: the same setup as in chapter 3, now with the separation d of the two lens positions that image sharply.

This is more than a shortcut. Object and image distance are measured from the principal plane of the lens, and that lies somewhere inside the glass, not at the scale of the lens carrier. This unknown offset is contained in every single measurement of g and b.

It is not contained in the separation of the two positions: d is a difference of two carriage positions, and the offset cancels out. And e is a fixed distance between two components that do not move during the measurement.

Your task

Derive the relationship yourself

Everything you need is already there: the lens equation from chapter 2 and the fixed distance from chapter 3.

You are looking for the focal length, expressed by the two quantities you can read off the setup. Pen and paper are enough. One route: replace the image distance, solve the equation for the object distance and look at how the two solutions relate to each other.

The three checks each reveal one intermediate step. Use them only once you are stuck at that point.

Check 1: the quadratic equation

Replacing the image distance by the remainder of the fixed distance leaves only one unknown in the lens equation. Multiplying both sides by the common denominator and sorting gives:

Check 2: the two lens positions

The quadratic equation has two solutions:

These are exactly the two positions from chapter 3. Their separation is the difference of the two solutions, and the halved distances cancel out:

Check 3: when there are two images at all

The square root is real only for a non-negative radicand. This yields a condition on the distance that is set:

If the distance is smaller, no sharp image appears on the screen at all. At the limit the two positions coincide, so their separation becomes zero. This is why the digital setup requires a minimum distance, and why one measures with a clear margin above it: only then can the two sharp settings be told apart cleanly.

Check 4: the last step

The last step is solving the equation found in check 2 for f. Nothing more remains to be done.

Show the finished formula (without derivation)

So only two distances are measured per data point: the distance set between object and screen and the separation of the two lens positions.

Each measurement scatters. One Straight line can be determined robustly and transparently from scattered data points, whereas fitting a curved relationship is more difficult. A nonlinear relationship is therefore linearised by choosing suitable variables for the axes. The quantity being determined is then contained in the Increase or in Axle section the straight line.

The lens equation from Chapter 2 links g and b in a non-linear way: plotting the image distance over the object distance gives a hyperbola. The example series of the test bench shows it:

Figure: image distance b over the object distance g for the lens of the example data set. The closer the object moves to the focal point, the further away the image moves — at g = f (dashed) it disappears to infinity. No straight line can be fitted anywhere here. Only the right rearrangement of the lens equation turns this curve into a straight line; which one that is, you decide yourself in the experiment.

Linear regression determines that straight line y = m·x + cthat best fits your measurement points. Slope m and intercept c only carry a physical meaningif you derive the axes from the basic equation of the experiment. That choice is yours to make.

On the digital setup, set the distance e , find both sharp settings, record a series of measurements and evaluate it with the linear regression.

A button then takes you straight on to station 9, the guiding questions.

Drawing

The ray diagram, once more

You know the drawing from chapter 1. Now the labels are missing: drag them to the right place — with the mouse, with your finger, or by tapping first a label and then a field.

Your answers

Questions on the evaluation

Lens A — linear regression

1. What quantities do you need to plot against each other on a graph to perform a linear regression? (In addition to the conscientious measurement, this is the crucial exercise of this partial experiment.)

2. How can the value of the focal length be determined from the parameters of the linear regression (slope, intercept). f determine?

3. What error estimate Δf results from the uncertainties of slope and/or intercept for the focal length?

Lens B — Bessel process

4. Derive the relationship for determining the focal length of a lens with the Bessel method. Why must e > 4f hold?

5. Use Gauss's error propagation law to determine the relationship for the focal length error. What are the actual measurements?

Your answers are stored locally and are transferred into your Preparation report taken over. There is no automatic evaluation — discuss your thoughts in the lab and in the colloquium.

Your result

Calculating the quantities you are after

Finally, complete the evaluation: the same measurement series yields the focal length in two ways — once via the regression parameters, once with the Bessel method directly from a single row of measurements. State both relationships symbolically, evaluate them with your own values and compare the two numbers.

1. Focal length f from the linear regression

   Relationship, symbolic Numerical value from your measurements
   cm

2. Focal length f from the Bessel method

   Relationship, symbolic Numerical value from your measurements
   cm

There is no automatic assessment here either. Both entries go into your preparation report — so that in the end it contains not only the approach but also the result.