Difficulty: Intermediate | Prerequisites: Basic trigonometry, understanding of angles of incidence and reflection, comfort with similar triangles.
Before you work with lenses and mirrors, you need the rules that govern how light travels in straight lines, bounces off surfaces, and bends at boundaries between materials. This section covers the law of reflection, Snell's law of refraction, pinhole (aperture) imaging, and parallax. These are the building blocks: every lens and mirror problem later in the course relies on the same angle relationships and geometry you learn here. If you can draw a ray, measure its angles from the normal, and apply similar triangles, you are in good shape.
The angle of incidence equals the angle of reflection, and rotating a mirror by some angle rotates the reflected beam by twice that angle. Snell's law (n₁ sin θ₁ = n₂ sin θ₂) governs how light bends at a boundary. Pinhole cameras and apertures form images by similar triangles, and parallax tells you which of two aligned objects is closer by shifting your viewpoint sideways.
Law of reflection
The angle of incidence (measured from the normal to the surface) equals the angle of reflection. Think of it as a ball bouncing off a wall at the same angle it arrived.
Snell's law (law of refraction)
n₁ sin θ₁ = n₂ sin θ₂. Describes how light bends when crossing a boundary between two materials with different refractive indices. In simple terms, light slows down in denser materials and bends toward the normal as it enters.
Index of refraction (n)
A dimensionless number describing how much a material slows light relative to vacuum. n = c/v, where c is the speed of light in vacuum and v is the speed in the material. Air is approximately 1.00, water is 1.33, glass is typically 1.5.
Parallax
The apparent shift in position of an object when viewed from two different vantage points. Closer objects shift more than farther ones. In simple terms, hold up your thumb and close one eye, then the other: your thumb jumps against the background.
Pinhole camera / aperture imaging
A small opening in a barrier casts an image of a light source on a screen behind it. The image size and shape are determined by similar triangles (for a point source) or by the shape of the aperture (for an extended source). Think of it as the simplest possible camera: no lens, just a hole.
The angle of incidence equals the angle of reflection, both measured from the normal to the mirror surface.
When a mirror is rotated by an angle α about the point where the laser hits it, the reflected beam rotates by 2α. This is because the normal rotates by α, which changes both the incidence and reflection angles by α, giving a total change of 2α.
Worked example: a laser hits a mirror at 30° from the horizontal, reflecting upward at 30°. The spot lands on a wall 2.50 m away at height y = 2.50 tan(30°) = 1.44 m. If the mirror rotates 10°, the new reflected angle is 50° from horizontal, so y_new = 2.50 tan(50°) = 2.98 m. The spot moves 2.98 - 1.44 = 1.54 m.
To find an unknown refractive index: identify the angle of incidence and the angle of refraction, then solve n₁ sin θ₁ = n₂ sin θ₂ for the unknown.
For a sphere, the geometry matters: a ray entering the sphere refracts at the surface. The angle of incidence is measured from the normal to the sphere at the entry point (which passes through the centre).
Worked example: three laser beams converge at the back of a sphere. The incidence angle is 45° and the refraction angle (inside the sphere) is 22.5°. So n = sin(45°) / sin(22.5°) = 1.85, rounded to 1.8.
A point source of light in front of a mask with an aperture produces an image on a screen behind the mask. The image size is found by similar triangles.
For a circular aperture: d_mask / L_mask = h_image / (L_mask + L_screen). Solve for the unknown.
For a small (point) source and a shaped aperture (T-shape, triangle, circle): the image on the screen is the shape of the aperture.
For an extended (long filament) source and a small aperture: the image on the screen is the shape of the source.
For an extended source and a shaped aperture: the screen shows multiple overlapping images of the aperture, one per point on the source, arranged in the shape of the source.
When two objects at different distances appear aligned from one viewpoint, moving your head sideways breaks the alignment.
The closer object appears to shift more. If the closer object shifts to the left of the farther object when you move right, the closer object is the one that moved.
This technique is used in the course to determine whether an image formed by a lens is closer or farther than a reference object.
\theta_i = \theta_rLaw of reflection: angle of incidence equals angle of reflection.
n_1 \sin\theta_1 = n_2 \sin\theta_2Snell's law: relates incident and refracted angles across a boundary.
\Delta\theta_{\text{reflected}} = 2\alphaWhen a mirror rotates by angle α, the reflected beam rotates by 2α.
\frac{d_{\text{mask}}}{L_{\text{mask}}} = \frac{h_{\text{image}}}{L_{\text{mask}} + L_{\text{screen}}}Similar-triangles relation for pinhole/aperture imaging.
Snell's law is the principle behind fibre optics (total internal reflection keeps light trapped inside the fibre) and explains why a swimming pool looks shallower than it is. Mirror rotation at twice the input angle is how galvanometer-based laser scanners work in barcode readers and laser light shows.
Students often measure the angle of incidence from the surface rather than from the normal. Always measure from the normal.
When a mirror rotates by 10°, students frequently assume the reflected beam also rotates by 10°. It rotates by 20°.
In pinhole problems, students sometimes confuse which triangle to draw. The aperture is where the triangles share a vertex, not the source or the screen.
Students sometimes think a shaped aperture always produces an image of that shape. It depends on whether the source is a point or extended.
⚠️ Mirror rotation problems appear frequently. Remember the 2α rule.
⚠️ Snell's law problems often embed the geometry inside a sphere or prism. Draw the normal carefully before applying the formula.
⚠️ Aperture/pinhole imaging problems test your ability to set up similar triangles. Draw the diagram first.
⚠️ Parallax questions test conceptual understanding. Know the rule: closer objects shift more when you move your viewpoint.
True or false: If a mirror is rotated by 15°, the reflected beam rotates by 15°. (False, it rotates by 30°)
Fill in the blank: Snell's law states n₁ sin θ₁ = ______. (n₂ sin θ₂)
True or false: A point source behind a T-shaped aperture produces a T-shaped image on the screen. (True)
Fill in the blank: If you move your head to the right and a closer object appears to shift to the ______, then it is closer than the object behind it. (left)
True or false: The angle of incidence is measured from the surface of the mirror. (False, from the normal)
Q: A laser beam hits a mirror at 30° from the horizontal and reflects onto a wall 2.50 m away. The mirror is then rotated 10°. By how much does the laser spot move on the wall?
A: Original height: 2.50 tan(30°) = 1.44 m. New angle is 50° (the beam rotates by 2 × 10° = 20°). New height: 2.50 tan(50°) = 2.98 m. Change = 2.98 - 1.44 = 1.54 m.
Q: Three laser beams enter a glass sphere and converge at a point on the back surface. The angle of incidence is 45° and the refraction angle inside the sphere is 22.5°. What is the sphere's refractive index?
A: n = sin(45°) / sin(22.5°) = 0.707 / 0.383 = 1.8.
Q: A point-source light bulb is 50 cm in front of a mask with a circular hole. A screen is 40 cm behind the mask. The image on the screen is 3.60 cm tall. What is the diameter of the hole?
A: By similar triangles: d / 50 = 3.60 / (50 + 40). So d = 50 × 3.60 / 90 = 2.0 cm.
Q: A pen and the image of a box (formed by a convex lens) appear aligned when viewed along the lens axis. When you move your head to the right, the box's image appears to the left of the pen. Which is closer to you?
A: The image of the box is closer. Closer objects shift in the opposite direction to your movement (they appear to move left when you move right).
Snell's law is the foundation for understanding how lenses work (the next topic): a lens is simply two curved refracting surfaces. The law of reflection underpins every mirror problem you will encounter. Parallax connects to how astronomers measure distances to nearby stars, and the pinhole camera is the conceptual ancestor of every lens-based imaging system.
law of reflection, Snell's law, refraction, refractive index, index of refraction, mirror rotation, reflected beam angle, similar triangles, pinhole camera, aperture imaging, parallax, image distance, point source, extended source, normal to surface, angle of incidence, angle of refraction, PHYS 116 midterm 2, geometric optics