01Reflection, Refraction, and Lenses
Apply the law of reflection to plane and curved mirrors, Snell's law to refraction, and the thin-lens equation to analyze converging and diverging lenses.
Optics explains how we see, how cameras work, and how telescopes and microscopes are built. Reflection and refraction are the foundation of all optical instruments.
How do the laws of reflection and refraction explain the behavior of light at surfaces, and how do mirrors and lenses use these laws to form images?
Light travels in straight lines until it strikes a surface or passes into a new medium. At a reflective surface the angle of incidence equals the angle of reflection (θᵢ = θᵣ). When light crosses into a new medium its speed changes, bending the ray according to Snell's law (n₁ sin θ₁ = n₂ sin θ₂). Curved mirrors and lenses exploit both phenomena to converge or diverge rays, forming real or virtual images described by the mirror/thin-lens equation 1/f = 1/d_o + 1/d_i and the magnification ratio m = −d_i/d_o. Understanding these relationships is the foundation of every optical instrument from eyeglasses to the Hubble Space Telescope.
Mirror Types
Plane Mirror
| Image type | Virtual, upright |
| Size | Same as object (m = +1) |
| Focal length | f → ∞ |
| Uses | Bathroom mirrors, periscopes |
Concave Mirror
| Image type | Real & inverted (d_o > f); virtual & upright (d_o < f) |
| Size | Varies with object distance |
| Focal length | f = R/2 > 0 |
| Uses | Telescopes, headlights, makeup mirrors |
Convex Mirror
| Image type | Always virtual, upright |
| Size | Always smaller (|m| < 1) |
| Focal length | f = −R/2 < 0 |
| Uses | Security mirrors, car side mirrors |
Sign Convention
| Quantity | Positive (+) | Negative (−) |
|---|---|---|
| d_o (object distance) | Object in front of mirror/lens (real) | Object behind mirror (virtual — rare) |
| d_i (image distance) | Image in front of mirror / behind lens (real) | Image behind mirror / in front of lens (virtual) |
| f (focal length) | Concave mirror; converging lens | Convex mirror; diverging lens |
| m (magnification) | Image upright | Image inverted |
| h_i (image height) | Image upright | Image inverted |
Key Equations
Law of Reflection
θᵢ = θᵣ
Mirror / Lens Equation
1/f = 1/d_o + 1/d_i
Magnification
m = −d_i/d_o = h_i/h_o
Focal Length
f = R/2
Snell's Law
n₁ sin θ₁ = n₂ sin θ₂
Critical Angle
sin θ_c = n₂/n₁
Worked Examples
A concave mirror has a focal length of 15 cm. An object is placed 45 cm in front of the mirror. Find the image distance and magnification.
Given: f = +15 cm, d_o = +45 cm
Mirror equation: 1/d_i = 1/f − 1/d_o = 1/15 − 1/45
1/d_i = 3/45 − 1/45 = 2/45
d_i = 45/2 = 22.5 cm (positive → real image, in front of mirror)
m = −d_i/d_o = −22.5/45 = −0.5
Interpretation: real, inverted, half the size of the object
A convex mirror has a focal length of −20 cm. An object is placed 30 cm in front of the mirror. Find the image distance and magnification.
Given: f = −20 cm, d_o = +30 cm
1/d_i = 1/f − 1/d_o = 1/(−20) − 1/30
1/d_i = −3/60 − 2/60 = −5/60
d_i = −12 cm (negative → virtual image, behind mirror)
m = −d_i/d_o = −(−12)/30 = +0.4
Interpretation: virtual, upright, 40% the size of the object
A ray of light travels from air (n = 1.00) into water (n = 1.33) at an angle of incidence of 45°. Find the angle of refraction.
Given: n₁ = 1.00, n₂ = 1.33, θ₁ = 45°
Snell's law: n₁ sin θ₁ = n₂ sin θ₂
sin θ₂ = (n₁/n₂) sin θ₁ = (1.00/1.33) × sin 45°
sin θ₂ = (1/1.33) × 0.7071 = 0.5317
θ₂ = arcsin(0.5317) ≈ 32.1°
Light bends toward the normal when entering a denser medium.
Find the critical angle for total internal reflection at a diamond–air interface. (n_diamond = 2.42, n_air = 1.00)
Given: n₁ = 2.42 (diamond), n₂ = 1.00 (air)
Critical angle formula: sin θ_c = n₂/n₁
sin θ_c = 1.00/2.42 = 0.4132
θ_c = arcsin(0.4132) ≈ 24.4°
Any ray inside diamond hitting the surface at θ > 24.4° undergoes total internal reflection.
A converging lens has a focal length of 10 cm. An object is placed 15 cm from the lens. Find the image distance and magnification.
Given: f = +10 cm, d_o = +15 cm
Thin-lens equation: 1/d_i = 1/f − 1/d_o = 1/10 − 1/15
1/d_i = 3/30 − 2/30 = 1/30
d_i = 30 cm (positive → real image, on opposite side of lens)
m = −d_i/d_o = −30/15 = −2
Interpretation: real, inverted, magnified 2×
Guided Problems
A concave mirror has a radius of curvature of 40 cm. An object is placed 60 cm in front of it. Find the image distance.
Hint: First find f = R/2 = 20 cm, then apply 1/d_i = 1/f − 1/d_o.
A convex mirror produces an image 8 cm behind the mirror when the object is 24 cm in front. What is the focal length?
Hint: d_i is negative for a virtual image behind the mirror. Use 1/f = 1/d_o + 1/d_i.
Light travels from glass (n = 1.50) into air (n = 1.00) at θ₁ = 25°. Find the angle of refraction.
Hint: Apply n₁ sin θ₁ = n₂ sin θ₂ and solve for sin θ₂. Is the ray bending toward or away from the normal?
A diverging lens has f = −12 cm. An object is 20 cm from the lens. Find d_i and state whether the image is real or virtual.
Hint: For a diverging lens f is negative. After solving, a negative d_i means the image is virtual (same side as the object).
Two converging lenses are placed 30 cm apart. Lens 1 has f₁ = 10 cm and the object is 15 cm to its left. The image from Lens 1 becomes the object for Lens 2 (f₂ = 8 cm). Find the final image distance from Lens 2.
Hint: Solve Lens 1 first to get d_i1. Then d_o2 = 30 − d_i1. Apply the thin-lens equation again for Lens 2.
Vocabulary
Law of Reflection
The angle of incidence equals the angle of reflection (θᵢ = θᵣ), both measured from the normal to the surface.
Example: A laser beam hitting a flat mirror at 30° bounces off at 30°.
Plane Mirror
A flat, reflective surface that produces a virtual, upright image the same size as the object, located as far behind the mirror as the object is in front.
Example: A bathroom mirror shows your reflection at the same apparent distance.
Concave Mirror
A mirror that curves inward (like the inside of a bowl). It converges parallel rays to a real focal point and can produce real or virtual images depending on object distance.
Example: Makeup mirrors and satellite dish reflectors are concave.
Convex Mirror
A mirror that curves outward. It diverges reflected rays, always producing a virtual, upright, diminished image regardless of object distance.
Example: Parking-lot security mirrors and car side mirrors are convex.
Focal Length (f)
The distance from a mirror or lens to its focal point, where parallel rays converge (or appear to diverge from). For curved mirrors f = R/2.
Example: A concave mirror with R = 30 cm has f = 15 cm.
Index of Refraction (n)
The ratio of the speed of light in a vacuum to its speed in a medium: n = c/v. A higher n means light travels more slowly and bends more.
Example: n_water ≈ 1.33 means light travels 1/1.33 ≈ 75% as fast in water as in vacuum.
Total Internal Reflection
When light in a denser medium strikes a less-dense medium at an angle greater than the critical angle θ_c, all light is reflected back — none is transmitted.
Example: Optical fibers use total internal reflection to carry light signals around bends.
Converging Lens
A lens that is thicker at the center than the edges. It refracts parallel rays to a real focal point on the far side and forms a real image when d_o > f.
Example: A magnifying glass and a camera objective lens are converging lenses.
Workbook Check
Interactive Practice — 5 Questions
A concave mirror has f = 20 cm and d_o = 60 cm. What is d_i?
Light travels from water (n = 1.33) into air (n = 1.00). At what condition does total internal reflection occur?
A diverging lens always produces an image that is:
The magnification produced by a mirror is m = +0.5. This means the image is:
Snell's law states that when light enters a medium with a higher index of refraction, the refracted ray:
Independent Practice
A ray of light strikes a plane mirror at an angle of incidence of 35°. Find the angle of reflection and the angle between the incident and reflected rays.
Light travels from water (n = 1.33) into air (n = 1.00) at an angle of incidence of 30°. Find the angle of refraction.
A converging lens has a focal length of 12 cm. An object is placed 36 cm from the lens. Find (a) the image distance and (b) the magnification.
A diverging lens has a focal length of −20 cm. An object is placed 30 cm from the lens. Find the image distance and describe the image (real/virtual, upright/inverted).
★ A compound microscope has an objective lens (f = 0.80 cm) and an eyepiece (f = 2.5 cm). The object is placed 0.90 cm from the objective. (a) Find the image distance from the objective. (b) Find the magnification of the objective. (c) If the eyepiece is used as a simple magnifier with the image at 25 cm (near point), find the total magnification of the microscope.
ChallengeCommon Mistakes
Using the mirror/lens equation without the correct sign convention for d_i
Real images have positive d_i (same side as reflected/transmitted light); virtual images have negative d_i. Always apply the sign convention before solving
Forgetting that a negative magnification means the image is inverted, not smaller
m = −d_i/d_o: negative m → inverted image; |m| < 1 → smaller; |m| > 1 → larger. Sign and magnitude give different information
Applying Snell's law with angles measured from the surface instead of the normal
All angles in Snell's law (n₁sinθ₁ = n₂sinθ₂) are measured from the NORMAL to the surface, not from the surface itself
Thinking total internal reflection can occur when light travels from a less dense to a more dense medium
TIR only occurs when light travels from a MORE optically dense medium to a LESS dense one (n₁ > n₂), and the angle exceeds the critical angle
Math Tips
Mirror/lens equation: 1/f = 1/d_o + 1/d_i. Rearrange to find the unknown: 1/d_i = 1/f − 1/d_o. For a concave mirror/converging lens, f > 0; convex/diverging, f < 0
Critical angle shortcut: sinθ_c = n₂/n₁ (when n₂ = 1 for air: sinθ_c = 1/n₁). Diamond (n=2.42): θ_c ≈ 24.4°. Water (n=1.33): θ_c ≈ 48.8°
Two-lens system: find the image from lens 1 first (using 1/f₁ = 1/d_o1 + 1/d_i1), then use that image as the object for lens 2. Total magnification = m₁ × m₂
Ray tracing rules for converging lens: (1) parallel ray → through far focal point; (2) ray through center → straight through; (3) ray through near focal point → parallel. Where all three meet = image location