Know the traditional seven colours of the visible spectrum in order of frequency and in order of wavelength

3.2.3 Thin Lenses

Learning Objective

By the end of this unit students should be able to:

• Describe how converging (convex) and diverging (concave) thin lenses bend light.
• Define principal axis, principal focus, focal length and the sign conventions for object distance u, image distance v and focal length f.
• Draw accurate ray diagrams for:

  • real and virtual images formed by a converging lens,
  • virtual images formed by a diverging lens,
  • identify the image characteristics (position, nature, size, orientation).

• Apply the thin‑lens equation \(\displaystyle \frac{1}{f}= \frac{1}{v}+ \frac{1}{u}\) and the magnification relation \(\displaystyle m=\frac{v}{u}= \frac{h'}{h}\).
• Explain why a lens produces a spectrum (dispersion) and recall the traditional seven colours of the visible spectrum in both wavelength and frequency order.
• Relate thin lenses to everyday devices such as spectacles, cameras, microscopes, projectors and spectroscopes.

Key Definitions & Sign Conventions

• Principal axis: straight line through the centre of the lens, perpendicular to its faces.
• Principal focus (F): point on the principal axis where a ray parallel to the axis converges (convex) or appears to diverge from (concave) after passing through the lens.
• Focal length (f): distance between the principal focus and the optical centre (C).
  Positive for converging lenses, negative for diverging lenses.
• Object distance (u): distance from the object to C. Taken as negative when the object is on the same side as the incoming light (standard sign convention).
• Image distance (v): distance from the image to C. Positive for real images (formed on the opposite side to the object), negative for virtual images.
• Magnification (m): \(\displaystyle m=\frac{h'}{h}= \frac{v}{u}\). Positive → upright, Negative → inverted.

Action of Thin Lenses

Converging (Convex) Lens

• Parallel rays are brought to a real focus on the principal axis.
• Can form both real and virtual images depending on the object position relative to the focal length.

Diverging (Concave) Lens

• Parallel rays appear to diverge from a virtual focus on the same side as the incoming light.
• Always produces a virtual, upright, reduced image.

Ray‑Diagram Construction (Thin‑Lens Approximation)

General steps (common to all lenses)

1. Draw the principal axis and mark the optical centre (C) of the lens.
2. Locate the principal focus (F) on each side (positive f for convex, negative f for concave).
3. From the top of the object draw the three principal rays:

   • Parallel ray: travels parallel to the axis, then refracts through (or appears to diverge from) the focus.
   • Focal ray: passes through the near focus before reaching the lens, then emerges parallel to the axis.
   • Central ray: passes straight through the optical centre without deviation.

4. Where the refracted rays (or their extensions) intersect gives the image location.
5. Read off the image characteristics (real/virtual, upright/inverted, magnified/reduced).

Real image with a converging lens (object beyond f)

Example: object at 1.5 f (between f and 2f). The three rays intersect on the opposite side of the lens, producing a real, inverted, enlarged image.

Virtual image with a converging lens (object inside f)

Example: object at 0.5 f. The parallel and focal rays diverge after the lens; extending them backwards meets the central ray on the same side as the object, giving a virtual, upright, enlarged image.

Virtual image with a diverging lens (any object position)

Example: object at 2 f. The parallel ray appears to diverge from the virtual focus on the same side as the object; the focal ray emerges parallel to the axis. Extending the diverging rays backwards meets the central ray, producing a virtual, upright, reduced image on the object side.

Image Characteristics Summary

Lens Formulae (for reference)

Thin‑lens equation (sign convention shown above):

\[

\frac{1}{f}= \frac{1}{v}+ \frac{1}{u}

\]

Magnification:

\[

m = \frac{v}{u}= \frac{h'}{h}

\]

Dispersion – Why a Lens Can Produce a Spectrum

All transparent media have a refractive index that varies slightly with wavelength (dispersion). Because the focal length of a thin lens is given by

\[

f = \frac{R}{(n-1)}

\]

where \(R\) is the radius of curvature and \(n\) the refractive index, light of shorter wavelength (higher frequency) experiences a larger \(n\) and therefore a slightly longer focal length than light of longer wavelength. The separation of the focal points for the different colours creates a spectrum, the same principle that produces a rainbow in a prism.

Traditional Seven Colours of the Visible Spectrum

Mnemonic: ROY GBV

Order of Wavelength (Longest → Shortest)

1. Red
2. Orange
3. Yellow
4. Green
5. Blue
6. Indigo
7. Violet

Order of Frequency (Lowest → Highest)

1. Violet
2. Indigo
3. Blue
4. Green
5. Yellow
6. Orange
7. Red

Typical Wavelength and Frequency Ranges

Everyday Applications of Thin Lenses

• Vision correction: convex lenses for hyperopia (far‑sightedness), concave lenses for myopia (near‑sightedness).
• Camera: a convex lens forms a real image on the sensor; the aperture controls depth of field.
• Microscope (compound): objective (convex) creates a real, enlarged intermediate image; eyepiece (convex) acts as a magnifier.
• Projector: convex lens projects a real, inverted image onto a screen.
• Spectroscope / spectrometer: a thin convex lens together with a slit and a diffraction grating separates white light into its component colours, demonstrating dispersion.

Suggested Diagram

Quick Revision Checklist

• Can you state the definitions of focal length, principal axis and principal focus?
• Do you know the sign conventions for u, v and f in the thin‑lens equation?
• Can you draw the three principal rays for:

  • a real image formed by a converging lens,
  • a virtual image formed by a converging lens,
  • a virtual image formed by a diverging lens?

• Can you read the image characteristics (real/virtual, upright/inverted, size) from your diagram?
• Remember the mnemonic ROY GBV and the opposite order for frequency.
• Explain why the focal length varies with colour (dispersion) and how this produces a spectrum.
• Identify at least two real‑world devices that rely on converging or diverging lenses.