Describe the use of a single lens as a magnifying glass

3.2.3 Thin Lenses – Using a Single Lens as a Magnifying Glass

1. Key Terminology

• Principal axis – the straight line that passes through the centre of the lens and is perpendicular to both lens surfaces.
• Principal focus (F) – the point on the principal axis where rays that were originally parallel to the axis converge (converging lens) or appear to diverge from (diverging lens).
• Focal length (f) – distance between the lens centre (O) and its principal focus. Positive for a converging (convex) lens, negative for a diverging (concave) lens.
• Near point (N) – the closest distance at which a normal, relaxed eye can focus comfortably, ≈ 25 cm.

2. Sign‑Convention (Cambridge IGCSE)

3. Ray‑Diagram Construction

3.1 Real image by a converging lens (object > f)

1. Draw the principal axis and mark the lens centre \(O\), the two principal focusses \(F\) (left) and \(F'\) (right), and the optical centre.
2. Place the object upright on the axis at a distance \(u\) from \(O\) (with \(u>f\)).
3. Draw the three standard rays from the top of the object:

   • Parallel ray – travels parallel to the axis, then refracts through the far focus \(F'\).
   • Focal ray – passes through the near focus \(F\) before the lens, then emerges parallel to the axis.
   • Central ray – passes straight through the optical centre (undeviated).

4. The point where any two refracted rays intersect gives the image position. The image is real, inverted and can be projected onto a screen.

3.2 Virtual, upright image by a converging lens (object < f)

1. Set the object inside the focal length (\(u

2. Use the same three rays:

   • Parallel ray – after the lens it diverges as if it came from \(F'\) on the same side as the object.
   • Focal ray – emerges parallel to the axis after passing through \(F\).
   • Central ray – passes straight through the centre.

3. The extensions of the refracted rays behind the lens intersect at a point on the same side as the object. This point is the virtual, upright, magnified image.

3.3 Diverging lens (any object position)

1. Draw a converging lens diagram first, then reverse the direction of the refracted rays, or simply:

   • Parallel ray – after the lens it diverges as if it originated from the near focus \(F\).
   • Focal ray – directed toward the far focus \(F'\) before the lens, emerges parallel to the axis.
   • Central ray – passes straight through the centre.

2. All three rays appear to diverge from a common point on the same side as the object. The image is always virtual, upright and reduced.

4. Image Characteristics

5. Lens Types

• Converging (convex) lens – positive focal length; can produce real or virtual images depending on object distance.
• Diverging (concave) lens – negative focal length; always produces a virtual, upright, reduced image, regardless of object position. (Not used as a magnifier but included for completeness.)

6. Thin‑Lens Equation

The relationship between object distance \(u\), image distance \(v\) and focal length \(f\) is

\[

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

\]

All distances are measured from the lens centre and must obey the sign‑convention table above.

7. Magnification

7.1 Linear (transverse) magnification

\[

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

\]

• \(m<0\) → image inverted (real image).
• \(m>0\) → image upright (virtual image).

7.2 Angular magnification (magnifying glass)

When the virtual image is placed at the near point (≈ 25 cm), the angular magnification is

\[

M{\text{ang}} = \frac{\theta{\text{with lens}}}{\theta_{\text{without lens}}}

\approx 1 + \frac{25\text{ cm}}{f}

\]

This is the formula required for the IGCSE exam question “describe the use of a single lens as a magnifying glass”.

8. Using a Single Convex Lens as a Magnifying Glass

1. Select a lens with a short focal length (typical 5 – 10 cm). Shorter focal lengths give higher magnification but increase spherical aberration.
2. Place the object (e.g., a printed letter) at a distance \(u\) inside the focal length (\(u

3. Adjust the lens‑to‑eye distance so that the virtual image appears at the near point (≈ 25 cm). In practice, move the lens until the image is clear and comfortably focused.
4. Read the enlarged image. The angular magnification you experience is roughly \(M_{\text{ang}} = 1 + 25/f\).

9. Conditions for an Effective Magnifier

• Object must be within the focal length of the lens (\(u

• Virtual image should be at or beyond the near point (≥ 25 cm) to avoid eye strain.
• Lens should be free from major spherical or chromatic aberrations; a small aperture (stop) reduces peripheral‑ray errors.
• The eye should be close to the lens (≈ 2–3 cm) to minimise accommodation effort.

10. Example Calculations

Negative \(v\) indicates a virtual image on the same side as the object; the resulting \(m\) is positive, giving an upright, enlarged image.

11. Practical Tips & Common Aberrations

• Spherical aberration – rays farther from the axis focus nearer the lens. Use a small‑diameter lens or a stop to limit peripheral rays.
• Chromatic aberration – different colours focus at slightly different points. For low‑power magnifiers the effect is usually negligible, but achromatic lenses are used in higher‑quality instruments.
• Hold the lens so that its optical centre aligns with the eye; this avoids parallax and maximises comfort.
• For a higher magnification, increase the eye‑to‑lens distance slightly, but ensure the virtual image remains at the near point.

12. Other Common Uses of Thin Lenses (Illustrative)

• Simple microscope – a magnifying glass combined with an illuminated specimen.
• Refracting telescope – objective converging lens + eyepiece (magnifying glass).
• Camera – converging lens forms a real image on film or a sensor.
• Projector – converging lens projects a real image onto a screen.

13. Optional Extended Material (Higher‑Level Study)

• Lens‑maker’s formula (useful when the glass index \(n\) is known):

  \[

  \frac{1}{f} = (n-1)\!\left(\frac{1}{R1}-\frac{1}{R2}\right)

  \]

  where \(R1\) and \(R2\) are the radii of curvature of the two surfaces.

• Combined lenses in contact – effective focal length:

  \[

  \frac{1}{f{\text{eq}}}= \frac{1}{f1}+ \frac{1}{f_2}

  \]

  (Relevant for “compound magnifier” questions.)

• Accommodation problem – calculate the change in dioptre power when the virtual image is placed at 20 cm instead of the near point.