Lenses

Q: What is the difference between a converging lens and a diverging lens in terms of shape and effect on parallel light?

A converging (convex) lens is thicker at the centre and focuses parallel rays to a real principal focus; a diverging (concave) lens is thinner at the centre and causes parallel rays to spread out from a virtual principal focus.

Q: What distinguishes a real image from a virtual image?

A real image is formed where light rays physically intersect and can be projected onto a screen, whereas a virtual image is formed where rays only appear to diverge from and cannot be projected onto a screen.

Q: What three standard rays can be drawn to locate an image using a ray diagram?

1. A parallel ray that refracts through or from the principal focus; 2. A central ray passing straight through the optical centre undeviated; 3. A focal ray passing through the near focus that emerges parallel.

Q: What three descriptors must always be included when describing the nature of an image?

You must state whether the image is Real or Virtual , Inverted or Upright , and Magnified, Diminished, or Same size .

Q: What type of image is formed by a converging lens when an object is placed inside the focal length ((u?

A virtual , upright , and magnified image formed on the same side of the lens as the object (magnifying glass setup).

Q: What is the thin lens equation and the "Real is Positive" sign convention?

The equation is \(\frac{1}{u} + \frac{1}{v} = \frac{1}{f}\), where real object/image distances and converging focal lengths are positive, while virtual image distances and diverging focal lengths are negative.

Q: How is linear magnification defined and calculated for a thin lens?

Linear magnification is the ratio of image height to object height, given by \(m = \frac{h_i}{h_o} = \frac{v}{u}\) (with no units).

Q: How is lens power defined and what is its standard SI unit?

Lens power is defined as \(P = \frac{1}{f}\) (where \(f\) is in metres), measured in Dioptres (\(\text{D}\)) or \(\text{m}^{-1}\).

Q: How do you find the total power and effective focal length of thin lenses in contact?

Total power is the algebraic sum \(P_{\text{total}} = P_1 + P_2 + \dots\), and the combined focal length satisfies \(\frac{1}{f_{\text{total}}} = \frac{1}{f_1} + \frac{1}{f_2} + \dots\)

Q: How can the focal length of a converging lens be determined from a graph of \(\frac{1}{v}\) against \(\frac{1}{u}\)?

Plotting \(\frac{1}{v}\) on the y-axis versus \(\frac{1}{u}\) on the x-axis yields a straight line of gradient \(-1\), where both the y-intercept and x-intercept equal \(\frac{1}{f}\).

Physics 1210 · CCEA AS Level · 10 min read · Updated: 31 Aug 2026

Chapter 2.3: Lenses

Welcome to the study notes for Lenses, part of your CCEA AS 2 module (Waves, Photons and Astronomy). Whether you are looking through a pair of glasses, snapping a photo on your phone, or using a magnifying glass, lenses are at work bending light to help us see the world clearly. Don't worry if optical diagrams or sign conventions seem intimidating at first—by breaking them down into simple rules and step-by-step methods, you will master every aspect of this topic for your exam!

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1. Types of Lenses and Essential Terminology

A lens is a shaped piece of transparent material (usually glass or plastic) that refracts light rays to form images. In AS Physics, we focus on two primary types of thin lenses.

A. Converging (Convex) Lens

A converging lens is thicker at the centre than at the edges. When parallel rays of light pass through it, they bend inward (converge) to meet at a single point on the far side called the real principal focus.

B. Diverging (Concave) Lens

A diverging lens is thinner at the centre than at the edges. When parallel rays of light pass through it, they spread outward (diverge). If you trace these spreading rays backward, they appear to come from a single point on the near side called the virtual principal focus.

Key Optical Definitions

Make sure you learn these exact definitions, as examiners frequently ask for them:

• Principal Axis: The straight line passing through the optical centre of the lens, perpendicular to the plane of the lens.
• Principal Focus (Focal Point, \(F\)):
– For a converging lens: The point on the principal axis where rays initially parallel to the principal axis converge after passing through the lens.
– For a diverging lens: The point on the principal axis from which rays initially parallel to the principal axis appear to diverge after passing through the lens.
• Focal Length (\(f\)): The distance along the principal axis from the optical centre of the lens to the principal focus \(F\).
• Real Image: An image formed where light rays physically intersect and converge. Real images can be captured on a screen and are inverted (upside-down) relative to the object when formed by a single converging lens.
• Virtual Image: An image formed where light rays only appear to diverge from. Virtual images cannot be projected onto a screen because the rays do not physically meet. They are always upright relative to the object.

Quick Key Takeaway: Real images can be caught on a screen because rays truly cross; virtual images cannot be projected because rays only appear to come from behind the lens.

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2. Standard Ray Tracing Rules

To find where an image forms and what it looks like, we draw a ray diagram. You only ever need to draw at least two of the following three standard principal rays from the top of the object:

1. The Parallel Ray: A ray travelling parallel to the principal axis refracts through the principal focus \(F\) on the far side (for a converging lens) or appears to diverge from the principal focus \(F\) on the near side (for a diverging lens).
2. The Central Ray: A ray passing directly through the optical centre of the lens continues straight through without any deviation.
3. The Focal Ray: A ray passing through the near principal focus \(F\) refracts through the lens and emerges parallel to the principal axis on the far side.

The Image Description Protocol

Whenever an exam question asks you to "describe the nature of the image", you must state all three of the following characteristics:

1. Nature: Is it Real or Virtual?
2. Orientation: Is it Inverted (upside down) or Upright?
3. Size (Scale): Is it Magnified (larger than object), Diminished (smaller than object), or Same size?

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3. Image Formation by a Converging Lens

The type of image formed by a converging lens depends entirely on where the object is placed relative to the focal length \(f\). Here are the five key scenarios you need to know:

Scenario 1: Object beyond \(2f\) (\(u > 2f\))
• Image position: Formed between \(f\) and \(2f\) on the opposite side.
• Image properties: Real, Inverted, and Diminished.
• Everyday use: Cameras and the human eye (where large distant scenes are shrunk down onto a small sensor or retina).

Scenario 2: Object exactly at \(2f\) (\(u = 2f\))
• Image position: Formed exactly at \(2f\) on the opposite side (\(v = 2f\)).
• Image properties: Real, Inverted, and Same size.

Scenario 3: Object between \(f\) and \(2f\) (\(f < u < 2f\))
• Image position: Formed beyond \(2f\) on the opposite side (\(v > 2f\)).
• Image properties: Real, Inverted, and Magnified.
• Everyday use: Projectors (enlarging a small slide/display onto a big wall screen).

Scenario 4: Object at the principal focus (\(u = f\))
• Image position: Refracted rays emerge parallel to each other.
• Image properties: The image is formed at infinity (no image can be focused onto a nearby screen).

Scenario 5: Object closer than the principal focus (\(u < f\))
• Image position: Refracted rays spread apart on the far side; tracing them backward shows them meeting on the same side as the object (\(v < 0\)).
• Image properties: Virtual, Upright, and Magnified.
• Everyday use: A simple magnifying glass.

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4. Mathematical Formulae & Sign Conventions

A. The Thin Lens Equation

To calculate the exact positions of objects and images, we use the thin lens equation:

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

Where:
• \(u\) = distance from the object to the optical centre of the lens
• \(v\) = distance from the image to the optical centre of the lens
• \(f\) = focal length of the lens

The "Real is Positive" Sign Convention

CCEA uses the standard Real is Positive convention. Memorise these signs carefully:

• Real object distance: \(u > 0\) (always positive for real objects)
• Real image distance: \(v > 0\) (positive)
• Virtual image distance: \(v < 0\) (negative)
• Converging lens focal length: \(f > 0\) (positive)
• Diverging lens focal length: \(f < 0\) (negative)

B. Linear Magnification (\(m\))

Linear magnification describes how much bigger or smaller the image is compared to the object:

\(m = \frac{\text{image size}}{\text{object size}} = \frac{h_i}{h_o} = \frac{v}{u}\)

Where \(h_i\) is the image height and \(h_o\) is the object height.
• Magnification is a pure ratio, so it has no units.
• If \(m > 1\), the image is magnified.
• If \(m < 1\), the image is diminished.
• If \(m = 1\), the image is the same size as the object.

C. Lens Power (\(P\))

The power of a lens measures how strongly it bends light rays. A shorter focal length means a stronger, more powerful lens:

\(P = \frac{1}{f}\)

Crucial Unit Warning: For this formula, the focal length \(f\) must be in metres (\(\text{m}\)).
• Unit of Power: Dioptres (\(\text{D}\)) or \(\text{m}^{-1}\) (\(1\text{ D} = 1\text{ m}^{-1}\)).
• Power is positive for converging lenses (e.g. \(+5.0\text{ D}\)).
• Power is negative for diverging lenses (e.g. \(-2.5\text{ D}\)).

D. Thin Lenses in Contact

When two or more thin lenses are placed touching each other along the same axis, their total power is simply the sum of their individual powers:

\(P_{\text{total}} = P_1 + P_2 + P_3 + \dots\)

In terms of focal lengths:

\(\frac{1}{f_{\text{total}}} = \frac{1}{f_1} + \frac{1}{f_2} + \frac{1}{f_3} + \dots\)

Step-by-Step Worked Example

Question: An object of height \(4.0\text{ cm}\) is placed \(15.0\text{ cm}\) away from a converging lens with a focal length of \(10.0\text{ cm}\). Calculate the image distance \(v\), the magnification \(m\), and the height of the image \(h_i\).

Step 1: Identify the known values and check signs.
• Object distance \(u = +15.0\text{ cm}\)
• Converging lens focal length \(f = +10.0\text{ cm}\)

Step 2: Use the thin lens equation to find \(v\).
\(\frac{1}{u} + \frac{1}{v} = \frac{1}{f}\)
\(\frac{1}{15.0} + \frac{1}{v} = \frac{1}{10.0}\)
\(\frac{1}{v} = \frac{1}{10.0} - \frac{1}{15.0} = \frac{3}{30.0} - \frac{2}{30.0} = \frac{1}{30.0}\)
\(v = +30.0\text{ cm}\) (Since \(v\) is positive, it is a real image formed \(30.0\text{ cm}\) behind the lens).

Step 3: Calculate linear magnification \(m\).
\(m = \frac{v}{u} = \frac{30.0}{15.0} = 2.0\)

Step 4: Calculate the image height \(h_i\).
\(m = \frac{h_i}{h_o} \implies h_i = m \times h_o = 2.0 \times 4.0\text{ cm} = 8.0\text{ cm}\)

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5. Required Practical: Determining Focal Length & Verifying the Lens Formula

In both Unit AS 2 and Unit AS 3, you are expected to know how to determine the focal length of a converging lens experimentally.

Apparatus

• Illuminated light source (such as an illuminated crosswire target)
• Converging lens in a lens holder
• White screen
• Metre rule / optical bench

Experimental Procedure

1. Place the illuminated crosswire object at one end of the optical bench.
2. Position the converging lens at a chosen object distance \(u\) (making sure \(u > f\)).
3. Move the white screen back and forth until a sharply focused image of the crosswire appears on the screen.
4. Measure and record the object distance \(u\) (from object to lens centre) and image distance \(v\) (from lens centre to screen).
5. Repeat for several different values of \(u\) over a wide range.

Graphical Analysis

To verify the thin lens formula graphically, rearrange the equation into the straight-line form \(y = mx + c\):

\(\frac{1}{u} + \frac{1}{v} = \frac{1}{f} \implies \frac{1}{v} = -\frac{1}{u} + \frac{1}{f}\)

• Plot a graph with \(\frac{1}{v}\) on the \(y\)-axis and \(\frac{1}{u}\) on the \(x\)-axis.
• Gradient: The slope of the line equals \(-1\).
• \(y\)-intercept: The point where the line crosses the vertical axis is \(\frac{1}{f}\).
• \(x\)-intercept: The point where the line crosses the horizontal axis is also \(\frac{1}{f}\).
• To find \(f\), calculate the reciprocal of either intercept: \(f = \frac{1}{\text{intercept}}\).

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6. Common Exam Pitfalls to Avoid

Make sure you do not lose easy marks by falling into these common traps highlighted by CCEA examiners:

1. Forgetting to convert centimetres to metres when calculating Power:
Always check that \(f\) is in metres before calculating \(P = \frac{1}{f}\). If \(f = 20\text{ cm} = 0.20\text{ m}\), then \(P = \frac{1}{0.20} = +5.0\text{ D}\). Using \(20\) directly will give an answer that is out by a factor of 100!

2. Sign errors with virtual images and diverging lenses:
Remember that virtual image distances are negative (\(v = -d\)) and diverging lens focal lengths are negative (\(f < 0\)). Always insert the negative sign explicitly when rearranging the lens formula.

3. Incomplete descriptions of images:
If asked to describe an image, always state all three characteristics: (1) real or virtual, (2) inverted or upright, (3) magnified or diminished.

4. Sloppy ray diagrams:
Always use a sharp pencil and a ruler for straight rays. Add arrows to show the direction of light. Draw real rays and real images as solid lines, and virtual rays/extensions as dashed lines.

5. Misunderstanding \(u = f\):
When an object is placed at the focal point (\(u = f\)), light rays emerge parallel and meet at infinity. No image can be formed on a nearby screen.

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7. Quick Review Summary

• Converging Lens: Thicker in the middle; \(f > 0\); \(P > 0\); converges light to a real focus.
• Diverging Lens: Thinner in the middle; \(f < 0\); \(P < 0\); diverges light from a virtual focus.
• Thin Lens Formula: \(\frac{1}{u} + \frac{1}{v} = \frac{1}{f}\) (Real is positive).
• Magnification: \(m = \frac{v}{u} = \frac{h_i}{h_o}\) (No units).
• Power: \(P = \frac{1}{f}\) (Units: \(\text{D}\) or \(\text{m}^{-1}\), with \(f\) in metres).
• Lenses in Contact: \(P_{\text{total}} = P_1 + P_2\) and \(\frac{1}{f_{\text{total}}} = \frac{1}{f_1} + \frac{1}{f_2}\).
• Graph of \(\frac{1}{v}\) vs \(\frac{1}{u}\): Straight line with gradient \(-1\) and intercepts equal to \(\frac{1}{f}\).

Quick check

Can you answer these now?

Open each question to check the key ideas from this chapter.

What is the difference between a converging lens and a diverging lens in terms of shape and effect on parallel light?

A converging (convex) lens is thicker at the centre and focuses parallel rays to a real principal focus; a diverging (concave) lens is thinner at the centre and causes parallel rays to spread out from a virtual principal focus.

What distinguishes a real image from a virtual image?

A real image is formed where light rays physically intersect and can be projected onto a screen, whereas a virtual image is formed where rays only appear to diverge from and cannot be projected onto a screen.

What three standard rays can be drawn to locate an image using a ray diagram?

1. A parallel ray that refracts through or from the principal focus; 2. A central ray passing straight through the optical centre undeviated; 3. A focal ray passing through the near focus that emerges parallel.

What three descriptors must always be included when describing the nature of an image?

You must state whether the image is Real or Virtual, Inverted or Upright, and Magnified, Diminished, or Same size.

What type of image is formed by a converging lens when an object is placed inside the focal length ((u < f))?

A virtual, upright, and magnified image formed on the same side of the lens as the object (magnifying glass setup).

What is the thin lens equation and the "Real is Positive" sign convention?

The equation is \(\frac{1}{u} + \frac{1}{v} = \frac{1}{f}\), where real object/image distances and converging focal lengths are positive, while virtual image distances and diverging focal lengths are negative.

How is linear magnification defined and calculated for a thin lens?

Linear magnification is the ratio of image height to object height, given by \(m = \frac{h_i}{h_o} = \frac{v}{u}\) (with no units).

How is lens power defined and what is its standard SI unit?

Lens power is defined as \(P = \frac{1}{f}\) (where \(f\) is in metres), measured in Dioptres (\(\text{D}\)) or \(\text{m}^{-1}\).

How do you find the total power and effective focal length of thin lenses in contact?

Total power is the algebraic sum \(P_{\text{total}} = P_1 + P_2 + \dots\), and the combined focal length satisfies \(\frac{1}{f_{\text{total}}} = \frac{1}{f_1} + \frac{1}{f_2} + \dots\)

How can the focal length of a converging lens be determined from a graph of \(\frac{1}{v}\) against \(\frac{1}{u}\)?

Plotting \(\frac{1}{v}\) on the y-axis versus \(\frac{1}{u}\) on the x-axis yields a straight line of gradient \(-1\), where both the y-intercept and x-intercept equal \(\frac{1}{f}\).

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