MathSci Problems
Intro to Curved Mirrors
Mirrors don’t have to be flat. Curve them, and they start doing even more interesting things with light. Spherical mirrors, shaped like a slice of a sphere, come in two flavours: concave (curving inward, like a cave) and convex (bulging outward). Let’s unpack how they work, why they behave the way they do, and what that means for the images we see.
Concave mirrors are the “focusing” type. Think of a makeup mirror or a car headlight. They bend light inward to a point, because the inward curve, like a bowl, gathers the light rays and forces them to meet. Convex mirrors, in contrast, spread light out, like those security mirrors in stores that let you see around corners. This is because the outward bulge pushes rays apart so they never meet.
To understand these mirrors, we need to know their parts. Imagine a giant sphere, and our mirror is a piece of its surface. The center of curvature (C) is the center of that imaginary sphere. The radius of curvature (R) is the distance from the mirror’s surface to C. Now, picture a line cutting straight through the mirror’s middle. That’s the principal axis, which acts like the mirror’s backbone.
A neat fact: f = R/2. Why? Picture a ray of sunlight hitting the curved mirror. Draw a helper line from the point of contact to the sphere’s center; this line is perpendicular to the surface there, so the ray bounces off at the same angle, creating a big right-angled triangle that reaches the center (length R) and a little one that ends at the focal point (length f). The two triangles have the same shape, so their sides scale by two, which means the focal distance is half the radius, giving f = R/2.
Ray Diagrams and Image Properties
Images can be real (rays physically meet and can be projected) or virtual (rays only appear to meet). Real images happen with concave mirrors when the object is outside the focal point. Virtual images happen with convex mirrors, or with concave mirrors when the object is inside the focal point. Real images are inverted, while virtual images are upright. The size of the image depends on magnification: it appears larger if the image is farther than the object and smaller if it’s closer.
Here’s why. Imagine two rays from the top of the object: one travels straight toward the mirror’s center and reflects back on itself, while another travels in parallel to the axis and then passes through (or seems to come from) the focal point. Where those two reflected rays meet tells you where the top of the image lands. If the object sits far from the mirror, those rays meet close to the mirror, so a long object distance pairs with a short image distance and the image looks small. Slide the object closer, and the rays cross farther away, stretching the image and making it look large. Inside the focal point the rays never truly cross, so our eyes trace them backward, creating an upright virtual image that always looks smaller in a convex mirror and larger in a concave mirror inside its focal length.
The Mirror Equation
The mirror equation is: 1/f = 1/dₒ + 1/dᵢ
Here, f is the focal length, dₒ is the object distance (from mirror to object), and dᵢ is the image distance (from mirror to image). For concave mirrors, f is positive. For convex mirrors, it is negative. Real images have positive dᵢ, while virtual images have negative dᵢ.
Where does this equation come from?
Choose one handy ray: it leaves the top of the object, reaches the mirror parallel to the axis, then reflects through the focal point. The incoming and outgoing parts of that ray form two overlapping triangles that share all their angles: one triangle stretches from the object to the mirror, the other from the image to the mirror.
Because the triangles are similar, the ratios of their side lengths must match, so the three distances along the axis (object, image, and focal) fit into a single proportional statement. Rewrite that proportion in reciprocal form and it simplifies to 1/f = 1/dₒ + 1/dᵢ, which is the mirror equation.
Reflect & Explore
Here are some open-ended questions to help you think more deeply about this material and connect it to related ideas.
Hold a shiny metal spoon at arm’s length and slowly move it toward your eye, first looking at the shiny inner bowl (concave) then the outer back (convex). Describe how the image flips, grows, or shrinks on each side and connect each change to object distance, focal point, and ray behaviour.
A driver replaces the factory convex side mirror with a flat one to “see bigger images.” Explain why the flat mirror actually gives less useful coverage even though objects look larger.
Car headlights use concave reflectors so that light from the bulb (close to the focal point) emerges in nearly parallel beams. If a manufacturer accidentally places the bulb exactly at the center of curvature instead of at the focal point, sketch or describe the outgoing rays and what the road illumination would look like.
Magicians sometimes create the “floating coin” trick with a pair of facing concave mirrors. Describe how two identical mirrors, each with its focal point at the other’s surface, can make a coin appear to hover in mid-air, and identify whether the hovering coin is a real or virtual image.