Theory — Thin Lenses and Image Formation
Converging (Convex) Lenses
A convex lens bends parallel rays of light so they meet at a single point — the focal point — a distance f (the focal length) from the lens. When light from an object passes through the lens, the rays converge to form an image. Where that image forms, and how big it is, depends on how far the object sits from the lens.
Ray Tracing
Three principal rays locate the image of the tip of the object: a ray parallel to the axis bends to pass through the far focal point; a ray through the centre of the lens continues straight; and a ray through the near focal point emerges parallel to the axis. Where these rays cross is the image of the tip.
The Thin-Lens Equation
The object distance dₒ, the image distance dᵢ, and the focal length f are linked by the thin-lens equation.
f = focal length (cm), dₒ = object distance (cm), dᵢ = image distance (cm)
Magnification and Image Type
The magnification compares image size to object size. A real image (positive dᵢ) forms on the far side of the lens where the rays actually cross, and is inverted; a virtual image (the object inside the focal length) appears upright and on the same side.
M > 1 → image larger · M < 1 → image smaller
Object at the Focal Point
If the object sits exactly at the focal point (dₒ = f), the rays leaving the lens are parallel — they never cross, so no image forms (the image is "at infinity"). This is the boundary between a real image (object beyond f) and a virtual image (object inside f).
Linearising — 1/dᵢ vs 1/dₒ
Rearranging the thin-lens equation gives 1/dᵢ = 1/f − 1/dₒ. So a graph of 1/dᵢ against 1/dₒ is a straight line whose intercept is 1/f — a neat way to recover the focal length from several measurements.
Object far away
Image forms close to the focal point, smaller and inverted.
Object near f
Image forms far away, much larger and inverted.
Larger aperture
More light collected (brighter image) but the image position is unchanged.
| Quantity | Relationship | Notes |
|---|---|---|
| Thin-lens equation | 1/f = 1/dₒ + 1/dᵢ | convex lens, f > 0 |
| Image distance | dᵢ = 1/(1/f − 1/dₒ) | positive ⇒ real image |
| Magnification | M = dᵢ/dₒ | >1 larger, <1 smaller |
| Object at f | dᵢ → ∞ | rays parallel, no image |
| Linear plot | 1/dᵢ vs 1/dₒ | intercept = 1/f |
Instructions — Running the Virtual Experiment
The Ray Tracing tab lets you place the object and watch the rays converge to the image; the Thin-Lens tab records each trial and plots 1/dᵢ against 1/dₒ. Record every reading in your lab report with screenshots of the ray diagrams.
Simulation — Convex Lens
Lens & object
Record object distances (f = 40 cm)
| dₒ (cm) | dᵢ (cm) | 1/dₒ (cm⁻¹) | 1/dᵢ (cm⁻¹) | M = dᵢ/dₒ |
|---|---|---|---|---|
| Click an object distance to record it. | ||||
Team Questions
Example Lab Report
Sample report demonstrating the expected format and level of detail. Use as a guide for your own submission, and include labelled screenshots of your ray diagrams.
Geometric Optics and Image Formation
Physics | Section: [Your Section] | Date: [Date]
Lab Members: [Names of all members present]
Purpose
To use ray tracing through a convex lens to locate images and measure image distances, to verify the thin-lens equation 1/f = 1/dₒ + 1/dᵢ, to determine the magnification and orientation of the image at several object distances, to recover the focal length from a graph of 1/dᵢ against 1/dₒ, and to examine how the lens aperture affects the image.
Theory
For a thin convex lens, the object distance, image distance, and focal length obey 1/f = 1/dₒ + 1/dᵢ, and the magnification is M = dᵢ/dₒ. A real image (positive dᵢ) is inverted; an object placed at the focal point produces parallel emerging rays and no image. Rearranged, 1/dᵢ = 1/f − 1/dₒ, so a plot of 1/dᵢ versus 1/dₒ has intercept 1/f.
1/dᵢ = 1/f − 1/dₒ → intercept = 1/f (f = 40 cm)
Calculations — Sample: dₒ = 60 cm, f = 40 cm
Image distance: 1/dᵢ = 1/f − 1/dₒ = 1/40 − 1/60 = 0.025 − 0.0167 = 0.00833 cm⁻¹ → dᵢ = 120 cm
Magnification: M = dᵢ/dₒ = 120/60 = 2.0 (real, inverted, larger)
Results Table (f = 40 cm)
| Trial | dₒ (cm) | dᵢ (cm) | M = dᵢ/dₒ | Orientation / type |
|---|---|---|---|---|
| 1 | 60 | 120 | 2.0 | real, inverted, larger |
| 2 | 80 | 80 | 1.0 | real, inverted, same size |
| 3 | 50 | 200 | 4.0 | real, inverted, larger |
A plot of 1/dᵢ against 1/dₒ was a straight line with intercept ≈ 0.025 cm⁻¹, giving f ≈ 40 cm.
Discussion
The measured image distances matched the thin-lens equation at every object position: dₒ = 60, 80, and 50 cm gave dᵢ = 120, 80, and 200 cm, with magnifications of 2.0, 1.0, and 4.0. All three images were real and inverted, as expected for an object beyond the focal length, and the image grew larger as the object approached the focal point. A graph of 1/dᵢ versus 1/dₒ was a straight line whose intercept, about 0.025 cm⁻¹, gave a focal length of 40 cm — matching the lens setting.
Reducing the lens diameter from 120 cm to 60 cm did not move the crossing point of the rays: the image stayed in the same place but fewer rays reached it, so it became dimmer. This is why a telescope uses a large aperture — to gather more light from faint stars, not to change where their images form. When the object was moved to exactly the focal point (dₒ = 40 cm), the rays emerged parallel and never crossed, so no image formed (the image is "at infinity"), marking the boundary between real and virtual images.
Conclusion
The thin-lens equation was verified for a convex lens of focal length 40 cm: measured image distances agreed with calculation, magnification followed M = dᵢ/dₒ, and the 1/dᵢ-vs-1/dₒ graph recovered f from its intercept. The aperture controlled image brightness but not position, and an object at the focal point produced no image.
Practice Questions
Show all work and include units. Use the thin-lens equation 1/f = 1/dₒ + 1/dᵢ and M = dᵢ/dₒ.