This AP Physics 2 lesson on thin lens equation notes gives the key ideas first, then worked examples and diagrams. After that, practise with the worksheet on thin lens equation and check yourself with the quiz.
Quick answer: Geometric optics treats light as rays. Reflection obeys θᵣ = θᵢ, refraction obeys Snell’s law n₁ sin θ₁ = n₂ sin θ₂, and mirrors and thin lenses obey 1/dₒ + 1/dᵢ = 1/f with magnification m = −dᵢ/dₒ. Also, the signs of f and dᵢ tell you whether an image is real or virtual.
Level: AP Physics 2 (algebra-based) · Curriculum: AP P2 Unit 13
Thin lens equation notes: the key ideas
How does reflection work?
When light hits a smooth surface, it bounces off in a predictable direction. Angles are always measured from the normal, a line at right angles to the surface at the point of contact.
Law of reflection: the angle of reflection equals the angle of incidence, θᵣ = θᵢ, and the incident ray, reflected ray and normal lie in the same plane.
In addition, a plane mirror forms a virtual image as far behind the mirror as the object is in front. Curved mirrors follow the mirror equation 1/dₒ + 1/dᵢ = 1/f, where the focal length is half the radius of curvature, f = R/2. A concave mirror has f > 0 and a convex mirror has f < 0.
What does Snell’s law say about refraction?
Refraction is the change of direction of light when it passes between media in which it travels at different speeds. The refractive index of a medium is n = c/v, where c = 3.00 × 10⁸ m/s is the speed of light in vacuum.
n₁ sin θ₁ = n₂ sin θ₂
n = refractive index (no unit), θ = angle from the normal (degrees). Light bends towards the normal when it enters a medium with a larger n.
| Medium | Refractive index n | Speed of light (10⁸ m/s) |
|---|---|---|
| Air | 1.00 | 3.00 |
| Water | 1.33 | 2.26 |
| Glass (typical) | 1.50 | 2.00 |
| Diamond | 2.42 | 1.24 |
Light in air hits water at 40° to the normal. Find the angle of refraction.
1.00 × sin 40° = 1.33 × sin θ₂, so sin θ₂ = 0.6428 / 1.33 = 0.4833 and θ₂ = 29°.
When light moves from a higher index to a lower index, it bends away from the normal. Above a certain angle, no light leaves at all and the ray is totally reflected. The smallest angle of incidence that gives total internal reflection is the critical angle, with sin θc = n₂/n₁.
Also, find the critical angle for glass (n = 1.50) in air.
sin θc = 1.00 / 1.50 = 0.667, so θc = 42°. Rays hitting the glass-air boundary at more than 42° are totally reflected, which is how fibre optic cables trap light.
How does a thin lens form an image?
A converging lens bends parallel rays to a focal point at distance f. To locate an image, draw two or three principal rays from the top of the object: one parallel to the axis then through the far focus, one through the centre of the lens undeviated, and one through the near focus then parallel to the axis. Where the rays meet is the image.
1/dₒ + 1/dᵢ = 1/f and m = hᵢ/hₒ = −dᵢ/dₒ
dₒ = object distance, dᵢ = image distance (both in the same unit), f = focal length, m = magnification. In addition, use f > 0 for a converging lens and f < 0 for a diverging lens. A positive dᵢ means a real image; a negative dᵢ means a virtual image. Moreover, a negative m means the image is inverted.
Mei in Singapore uses a converging lens of f = 12 cm. An object is 30 cm from it. Find the image.
1/dᵢ = 1/12 − 1/30 = 3/60, so dᵢ = 20 cm. m = −20/30 = −0.67: the image is real, inverted and reduced.
The same lens, f = 12 cm, with the object at 8.0 cm.
1/dᵢ = 1/12 − 1/8 = −1/24, so dᵢ = −24 cm (virtual, on the object side). m = +24/8.0 = +3.0: upright and three times larger.
| Object distance (converging lens) | Image | Size |
|---|---|---|
| dₒ > 2f | real, inverted | smaller |
| f < dₒ < 2f | real, inverted | larger |
| dₒ < f | virtual, upright | larger |
Measuring angles from the surface instead of from the normal. Another is using the wrong sign for dᵢ: a virtual image has a negative image distance, so a calculator answer of −24 cm is not an error.
Before any calculation, sketch the three principal rays. If your calculated image does not match your sketch, check the signs of f and dᵢ.
- Law of reflection: θᵣ = θᵢ, measured from the normal
- Snell’s law: n₁ sin θ₁ = n₂ sin θ₂, with n = c/v
- Critical angle: sin θ_c = n₂/n₁ when n₁ is larger than n₂
- Thin lens and mirror equation: 1/dₒ + 1/dᵢ = 1/f
- Magnification: m = hᵢ/hₒ = −dᵢ/dₒ; negative m means inverted
Reflection, refraction and images from mirrors and lenses: frequently asked questions
What is the difference between a real and a virtual image?
In addition, a real image is formed where light rays actually meet and can be projected onto a screen. A virtual image is where rays only appear to come from, so no screen can show it.
Why does light bend when it enters glass?
Light travels more slowly in glass than in air. When a ray meets the boundary at an angle, one side of the wavefront slows first, which turns the ray towards the normal.
When does total internal reflection happen?
It happens when light travels from a medium of higher index to one of lower index and the angle of incidence is larger than the critical angle. No light is refracted out.
What does a negative magnification mean?
It means the image is inverted compared with the object. The size of the magnification, taken without the sign, tells you whether the image is larger or smaller than the object.
Why do converging and diverging lenses have opposite signs of focal length?
A converging lens brings parallel rays to a real focus, so f is positive. A diverging lens spreads rays so they appear to come from a virtual focus, so f is negative.
