Welcome to the World of Lenses!
Have you ever wondered how your smartphone camera takes sharp photos, how spectacles help people read, or how telescopes reveal distant galaxies? The secret behind all of these incredible technologies is the lens. In this chapter of AS 2: Waves, Photons and Astronomy, we will explore how lenses bend light to form images, how to predict image positions using simple mathematics, and how lenses are used to correct human vision.
Don't worry if this seems tricky at first! By breaking down the rules of light rays step by step, you will find that geometric optics is logical, predictable, and very rewarding.
1. Types of Lenses and Essential Terminology
A lens is a shaped piece of transparent material (such as glass or plastic) that refracts light rays to form an image. There are two main types of lenses you need to know:
• Converging (Convex) Lens: Thicker in the middle than at the edges. It brings parallel rays of light together to a single point.
• Diverging (Concave) Lens: Thinner in the middle than at the edges. It spreads parallel rays of light apart so they appear to come from a single point.
Key Terms You Must Master
• Principal Axis: The straight line passing through the centre of curvature of the lens faces, perpendicular to the plane of the lens.
• Optical Centre (\(C\)): The central point of the lens. Any light ray passing through the optical centre travels straight through without deviation.
• Principal Focus (\(F\)): For a converging lens, it is the point on the principal axis where rays initially parallel to the principal axis converge after passing through the lens. For a diverging lens, it is the point from which the refracted rays appear to diverge.
• Focal Length (\(f\)): The distance along the principal axis between the optical centre (\(C\)) and the principal focus (\(F\)). It is measured in metres (\(\text{m}\)).
Did you know? A thicker converging lens bends light more sharply than a thin one, which means it has a shorter focal length \(f\)!
Key Takeaway: Converging lenses bring parallel light rays together at a focal point; diverging lenses spread them apart.
2. Real vs. Virtual Images
When light rays pass through a lens, they intersect or appear to intersect, forming an image. Images are described using three pairs of characteristics:
• Real vs. Virtual:
- A real image is formed when light rays actually meet at a point. It can be projected onto a screen (like a cinema projector or image on a camera sensor).
- A virtual image is formed when light rays only appear to diverge from a point. It cannot be caught on a screen (like looking through a magnifying glass or a bathroom mirror).
• Inverted vs. Upright:
- Inverted: Upside down compared to the object.
- Upright: The same way up as the object.
• Magnified vs. Diminished:
- Magnified: Larger than the original object.
- Diminished: Smaller than the original object.
3. Drawing Ray Diagrams
Ray diagrams allow you to locate the position, size, and nature of an image. To draw an accurate ray diagram, you only need to trace two standard rays originating from the top of the object:
Ray 1: A ray travelling parallel to the principal axis passes through the principal focus (\(F\)) after refraction (or appears to diverge from \(F\) for a concave lens).
Ray 2: A ray passing straight through the optical centre (\(C\)) continues in a straight line without bending.
Image Formation in a Converging Lens
The type of image formed by a converging lens depends entirely on the distance of the object (\(u\)) from the lens relative to the focal length (\(f\)):
• Object beyond \(2f\): Image is between \(f\) and \(2f\), real, inverted, and diminished (e.g., in a photographic camera).
• Object at \(2f\): Image is at \(2f\), real, inverted, and the same size as the object.
• Object between \(f\) and \(2f\): Image is beyond \(2f\), real, inverted, and magnified (e.g., in a film projector).
• Object inside \(f\) (closer than one focal length): Rays diverge after the lens, so their extensions meet behind the object. The image is virtual, upright, and magnified (e.g., a magnifying glass).
Image Formation in a Diverging Lens
A diverging lens always produces an image that is virtual, upright, and diminished, no matter where the object is placed.
Memory Trick: Remember the phrase "Real is upside down, Virtual is right way round!" In single-lens systems, real images are always inverted, and virtual images are always upright.
Key Takeaway: Converging lenses can produce both real and virtual images depending on object distance, whereas diverging lenses only ever produce virtual, upright, diminished images.
4. The Lens Formula and Sign Convention
Rather than drawing scale diagrams every time, we can use the thin lens formula to calculate image distances and focal lengths mathematically.
\(\frac{1}{u} + \frac{1}{v} = \frac{1}{f}\)
Where:
• \(u\) = distance of the object from the optical centre
• \(v\) = distance of the image from the optical centre
• \(f\) = focal length of the lens
The "Real-is-Positive" Sign Convention
In CCEA Physics, we apply the Real-is-Positive convention. Make sure to memorise these simple rules:
• All distances are measured from the optical centre (\(C\)).
• Real distances are positive (\(+\)): Real object \(u > 0\), real image \(v > 0\), converging lens focal length \(f > 0\).
• Virtual distances are negative (\(-\)): Virtual image \(v < 0\), diverging lens focal length \(f < 0\).
Worked Example: Using the Lens Formula
Question: An object is placed \(15\text{ cm}\) in front of a converging lens of focal length \(10\text{ cm}\). Calculate the position and nature of the image.
Step 1: Identify values with signs.
• Object is real: \(u = +15\text{ cm}\)
• Converging lens: \(f = +10\text{ cm}\)
Step 2: Substitute into the lens formula.
\(\frac{1}{u} + \frac{1}{v} = \frac{1}{f}\)
\(\frac{1}{15} + \frac{1}{v} = \frac{1}{10}\)
Step 3: Rearrange for \(\frac{1}{v}\).
\(\frac{1}{v} = \frac{1}{10} - \frac{1}{15}\)
\(\frac{1}{v} = \frac{3}{30} - \frac{2}{30} = \frac{1}{30}\)
\(v = +30\text{ cm}\)
Conclusion: Because \(v\) is positive (\(+30\text{ cm}\)), the image is real and formed \(30\text{ cm}\) on the opposite side of the lens.
Common Mistake to Avoid: Forgetting to take the reciprocal! After finding \(\frac{1}{v} = \frac{1}{30}\), students sometimes write \(v = \frac{1}{30}\). Always remember to flip the fraction at the end to find \(v\).
Key Takeaway: Use the real-is-positive convention: converging lenses have \(+f\), diverging lenses have \(-f\), real images have \(+v\), and virtual images have \(-v\).
5. Linear Magnification
Linear magnification (\(m\)) describes how much larger or smaller an image is compared to the original object. It has no units because it is a ratio.
\(m = \frac{\text{height of image}}{\text{height of object}} = \frac{h_i}{h_o}\)
Using similar triangles from ray diagrams, we also find:
\(m = \frac{v}{u}\)
• 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.
6. Power of a Lens
The power (\(P\)) of a lens measures its ability to bend (refract) light. A more powerful lens bends light through a greater angle and therefore has a shorter focal length.
\(P = \frac{1}{f}\)
Where:
• \(P\) = power of the lens in dioptres (\(\text{D}\)) or \(\text{m}^{-1}\)
• \(f\) = focal length in metres (\(\text{m}\))
Sign Rules for Power:
• Converging lenses have positive focal lengths \(\implies\) positive power (\(+P\)).
• Diverging lenses have negative focal lengths \(\implies\) negative power (\(-P\)).
Lenses in Contact
When two or more thin lenses are placed in close contact, their combined power is simply the algebraic sum of their individual powers:
\(P_{\text{total}} = P_1 + P_2\)
Or in terms of focal lengths:
\(\frac{1}{f_{\text{total}}} = \frac{1}{f_1} + \frac{1}{f_2}\)
Quick Check Example
Question: A converging lens with \(f_1 = +0.20\text{ m}\) is placed in contact with a diverging lens with \(f_2 = -0.50\text{ m}\). What is the total power of the combination?
• \(P_1 = \frac{1}{+0.20} = +5.0\text{ D}\)
• \(P_2 = \frac{1}{-0.50} = -2.0\text{ D}\)
• \(P_{\text{total}} = P_1 + P_2 = +5.0\text{ D} + (-2.0\text{ D}) = +3.0\text{ D}\)
Key Takeaway: Always convert focal length to metres before calculating power in dioptres (\(\text{D}\)). Combine lenses in contact by adding their powers algebraically.
7. Correcting Defects of Vision
The human eye contains a flexible converging lens that focuses light rays onto the retina (the light-sensitive screen at the back of the eye). Sometimes, the eye cannot focus light properly onto the retina, leading to common vision defects.
1. Short-Sight (Myopia)
• The Problem: A person can see nearby objects clearly, but distant objects appear blurred.
• The Cause: The eyeball is too long from front to back, or the eye lens is too powerful (curves too sharply). As a result, parallel rays from distant objects converge and focus in front of the retina.
• The Correction: Corrected using a diverging (concave) lens. The diverging lens spreads the incoming parallel rays slightly before they enter the eye, allowing the eye's lens to focus them precisely onto the retina.
2. Long-Sight (Hypermetropia)
• The Problem: A person can see distant objects clearly, but close objects appear blurred.
• The Cause: The eyeball is too short, or the eye lens is too weak (cannot curve sufficiently). Light rays from nearby objects are focused behind the retina.
• The Correction: Corrected using a converging (convex) lens. The converging lens provides extra converging power, bending the rays inward before they enter the eye so that they focus sharply onto the retina.
Summary of Vision Defects:
• Myopia: Focus in front of retina \(\implies\) Corrected by Diverging lens (Negative power).
• Hypermetropia: Focus behind retina \(\implies\) Corrected by Converging lens (Positive power).
8. Quick Review & Exam Checklist
Before sitting your exam, make sure you can confidently:
• State the definitions of principal axis, optical centre, principal focus, and focal length.
• Accurately sketch ray diagrams for converging and diverging lenses.
• Apply the thin lens formula \(\frac{1}{u} + \frac{1}{v} = \frac{1}{f}\) using the real-is-positive convention without sign errors.
• Calculate magnification using \(m = \frac{v}{u}\).
• Convert focal lengths to metres and calculate lens power using \(P = \frac{1}{f}\).
• Explain the causes of myopia and hypermetropia and identify the lenses needed to correct each defect.