Draw and use ray diagrams for the formation of a real image by a converging lens

Cambridge IGCSE Physics (0625) – Complete Syllabus Overview & Detailed Notes on Real Image Formation by a Converging Lens

1. Syllabus Mapping – What Must Be Covered

2. Core Physics Refresher Boxes (Useful for All Sections)

2.1 Motion, Forces & Energy

• Key equations:
  • \(v = u + at\)
  • \(s = ut + \tfrac12 a t^{2}\)
  • \(F = ma\)
  • \(W = F s\)
  • \(E_{\text{k}} = \tfrac12 mv^{2}\)
  • \(P = \dfrac{W}{t}\)
  • \(p = \dfrac{F}{A}\)
• Worked example (link to optics): A 0.05 kg candle‑stand falls from a height of 0.30 m onto a screen placed behind a converging lens. Calculate the impact energy and discuss whether the screen will be damaged.

2.2 Thermal Physics

• Particle model, specific heat capacity \(c\), \(Q = mc\Delta T\).
• Conduction, convection, radiation – relevance to the lamp used in the lens experiment (heat loss can affect focal length slightly).

2.3 Waves – General

• Wave speed: \(v = f\lambda\).
• Reflection (angle of incidence = angle of reflection) and refraction (Snell’s law).
• Diffraction (single‑slit, double‑slit) – not examined for IGCSE but useful background.

3. Light – Reflection, Refraction & Thin Lenses

3.1 Plane Mirrors & Refraction Basics

• Plane mirror: Image distance = object distance, image upright, virtual, same size.
• Law of reflection: \(\theta_i = \theta_r\).
• Snell’s law: \(n_1\sin\theta_1 = n_2\sin\theta_2\).
• Critical angle: \(\theta_c = \sin^{-1}\!\left(\dfrac{n_2}{n_1}\right)\) (total internal reflection when \(\theta_1 > \theta_c\)).

3.2 Thin Lenses – Real Image Formation (Converging Lens)

3.2.1 Key Concepts

• Converging (convex) lens: Thicker at the centre; parallel rays converge to a focal point on the opposite side.
• Principal axis: Straight line through the optical centre (O) and both focal points (F and F′).
• Optical centre (O): Point where a ray passes undeviated (central ray).
• Focal length (f): Distance O → F (or O → F′). For a converging lens, **f > 0**.
• Real image: Formed when refracted rays actually meet on the image side; it is always inverted and can be captured on a screen.

3.2.2 Sign Conventions (IGCSE)

3.2.3 Why the Sign Convention Works

The thin‑lens equation

is derived using the paraxial (small‑angle) approximation, where all rays make only a few degrees with the principal axis. By assigning positive values to distances measured in the direction of incident light (object side) and to the image side for real images, the algebra remains consistent for both real and virtual cases. The convention also ensures that the magnification formula

produces a negative \(m\) for an inverted real image.

3.2.4 Constructing a Ray Diagram – Step‑by‑Step

1. Draw a horizontal line – the principal axis.
2. Mark the optical centre O (draw a thin vertical line for the lens).
3. From O, measure a distance **f** on each side and label the focal points F′ (object side) and F (image side).
4. Place the object (upright arrow) on the object side at a distance **u** from O. u > f is required for a real image.
5. From the top of the object draw the three principal rays:
   • Parallel ray: Travels parallel to the principal axis; after refraction it passes through F.
   • Focal ray: Passes through F′ before reaching the lens; after refraction it emerges parallel to the principal axis.
   • Central ray: Passes straight through O without deviation.
6. The point where the refracted rays intersect on the image side gives the image position. Mark the image distance **v** and the image height **h′**. The image will be inverted (h′ < 0).

3.2.5 Mathematical Relationships

All distances are measured from O along the principal axis.

• Thin‑lens equation: \(\displaystyle \frac{1}{f}= \frac{1}{u}+ \frac{1}{v}\)
• Magnification: \(\displaystyle m = \frac{v}{u}= \frac{h'}{h}\)
• For a real image formed by a converging lens:
  • \(v>0\) (image on opposite side)
  • \(h'<0\) (inverted)
  • \(m<0\) (negative sign indicates inversion)

3.2.6 Worked Example (with Units & Rounding)

Problem: An object 3.0 cm tall is placed 30.0 cm in front of a converging lens of focal length \(f = 10.0\) cm. Determine the image distance, size, nature, and verify with a ray diagram.

1. Lens formula (keep cm): \[ \frac{1}{10.0} = \frac{1}{30.0} + \frac{1}{v} \;\Rightarrow\; \frac{1}{v}= \frac{1}{10.0}-\frac{1}{30.0}=0.0667\;\text{cm}^{-1} \] \[ v = 15.0\;\text{cm} \]
2. Magnification (2 s.f. as required): \[ m = \frac{v}{u}= \frac{15.0}{30.0}=0.50\;\; \Rightarrow\; m = -0.50 \]
3. Image height: \[ h' = m\,h = (-0.50)(3.0\;\text{cm}) = -1.5\;\text{cm} \] Negative sign ⇒ inverted.
4. Interpretation:
   • Image distance \(v = 15.0\) cm (real, on the far side of the lens).
   • Image height \(1.5\) cm, inverted, half the object size.
   • Because \(u = 3f\), the image lies between \(f\) and \(2f\) on the image side (characteristic of a reduced real image).
5. Ray‑diagram check: Draw the three principal rays; they intersect 15 cm behind the lens, confirming the calculation.

3.2.7 Summary Table – Typical Real‑Image Scenario

3.2.8 Common Mistakes & How to Avoid Them

• Focal ray direction: It must pass through **F′** (the focal point on the object side) *before* the lens, then emerge parallel to the axis.
• Sign of v: Real image ⇒ \(v>0\). Using a negative sign yields an impossible (virtual) result.
• Central ray: Always passes straight through the optical centre O without bending.
• Object inside focal length: If \(u
• Units & rounding: Keep units throughout the calculation; round only at the final answer (usually 2 s.f. for IGCSE).
• Paraxial approximation limit: The thin‑lens formula is accurate only for rays close to the principal axis (small angles). In the classroom experiment the approximation is excellent; for very wide lenses or large angles a small error appears.

3.2.9 Practical Tip – Verifying Real Image Formation

3.2.10 Practice Questions (with brief answer keys)

1. Question: A converging lens has \(f = 8\) cm. An object is placed 12 cm from the lens. Find \(v\), \(m\), and describe the image.
   Answer: \(1/8 = 1/12 + 1/v \Rightarrow 1/v = 1/8 - 1/12 = 1/24\) → \(v = 24\) cm (real). \(m = v/u = 24/12 = 2\) → \(m = -2\). Image is real, inverted, and twice as tall as the object.
2. Question: Draw a ray diagram for a converging lens with \(f = 5\) cm when the object is 20 cm away. State the image characteristics.
   Answer: Using the lens formula, \(v ≈ 6.7\) cm. Magnification \(m ≈ 0.33\) → \(m = -0.33\). Image is real, inverted, reduced to one‑third the object size, located 6.7 cm behind the lens.
3. Question: If a converging lens produces a real, inverted image that is the same size as the object, what is the object distance in terms of the focal length?
   Answer: Same size ⇒ \(|m| = 1\) ⇒ \(|v| = |u|\). Substituting into \(1/f = 1/u + 1/v\) gives \(1/f = 2/u\) → \(u = 2f\). The object is placed at twice the focal length.

3.3 Electromagnetic Spectrum – Quick Reference

Visible light is used for the lens experiment because it is easily produced, safely handled, and its wavelength is much larger than atomic dimensions, ensuring the paraxial approximation holds.

3.4 Sound – Quick Summary

• Production: Vibrating source creates longitudinal pressure waves in a medium.
• Speed: In air at 20 °C, \(v ≈ 343\) m s\(^{-1}\); depends on medium density and elasticity.
• Pitch ↔ Frequency: Higher frequency → higher pitch.
• Loudness ↔ Amplitude: Measured in decibels (dB).
• Applications: Musical instruments, sonar, medical ultrasound.
• Cross‑topic activity: Record the sound of a falling object (from the optics experiment) with a smartphone; analyse the waveform to discuss energy conversion.

4. Electricity & Magnetism – Overview

• Charge & Current: \(I = \dfrac{Q}{t}\).
• Resistance: \(R = \rho\frac{L}{A}\); Ohm’s law \(V = IR\).
• Power: \(P = VI = I^{2}R = \dfrac{V^{2}}{R}\).
• Series & Parallel Circuits: Rules for total resistance and voltage division.
• Magnetic fields: Right‑hand rule for current‑carrying conductors; force \(F = BIL\sin\theta\).
• Induction: Faraday’s law \( \mathcal{E} = -\dfrac{d\Phi}{dt}\).
• Motors & Transformers: Basic principles and safety precautions (e.g., never touch live wires, use insulated handles).
• Link to optics: The lamp used in the lens experiment is powered from a low‑voltage DC supply; ensure the supply is switched off before adjusting the set‑up.

5. Nuclear Physics – Essentials

• Atomic structure: Protons, neutrons, electrons; atomic number \(Z\), mass number \(A\).
• Radioactivity: Alpha (\(\alpha\)), beta (\(\beta\)), gamma (\(\gamma\)) emissions.
• Half‑life: \(N = N_{0}\left(\tfrac12\right)^{t/t_{1/2}}\).
• Applications: Medical imaging (X‑rays), cancer treatment (radiotherapy), carbon dating.
• Safety: Time, distance, shielding; use of lead aprons and dosimeters when working with sources.

6. Space Physics – Brief Recap

• Earth’s rotation → day/night; tilt → seasons.
• Moon’s phases caused by its position relative to Sun and Earth.
• Solar‑eclipse geometry – alignment of Sun, Moon, Earth.
• Relevance to optics: Astronomical telescopes use converging lenses (or mirrors) to form real images of distant objects.

7. Practical Skills Checklist (for the Lens Experiment)

• Set‑up a stable optical bench; ensure the lens holder is centred.
• Use a white screen with a matte surface to avoid glare.
• Measure distances with a ruler or vernier caliper; record to the nearest 0.1 cm.
• Check that the light source is securely fixed and switched off before repositioning the object.
• Record at least three sets of \(u\) and \(v\) values; calculate the average and compare with theoretical predictions.
• Estimate experimental uncertainties (±0.1 cm for distances, ±0.5 cm for focal length) and propagate them through the lens formula.
• Dispose of candles or hot objects safely; allow them to cool before handling.

8. Final Summary

Real image formation by a converging lens is a cornerstone of the IGCSE optics syllabus. Mastery requires:

1. Understanding the sign conventions and the thin‑lens equation.
2. Being able to construct accurate ray diagrams using the three principal rays.
3. Applying magnification to predict image size and orientation.
4. Linking the theory to a hands‑on experiment and recognising sources of error.
5. Connecting optics to the broader physics curriculum – waves, energy, electricity, and safety.

With the refreshed notes, practice questions, and practical tips, students should be well prepared for both the written examination and laboratory assessments.