refraction and lenses explained
The straw that looks bent in a glass of water, the way glasses and a camera form an image, and the way an optical fiber traps light to carry a signal all come from one fact. When light changes medium its speed changes, and so it bends at the boundary. Write that refraction as Snell's law alone, and predict the place and the size of the image with the lens equation 1/f = 1/u + 1/v. The bend at the boundary is the place of the image.
When light changes medium its speed changes, and that is why it bends at the boundary. What governs the bend is Snell's law, n₁ sinθ₁ = n₂ sinθ₂. The slower the medium it enters, the closer the light hugs the normal. The refractive index is the speed of light in vacuum divided by its speed in that medium, a measure of how much the light slows down there. Move the angle of incidence and n₂ and the way the angle of refraction follows shows itself. Entering the slower side means hugging closer.
When light leaves a slow medium for a fast one, the angle of refraction is larger than the angle of incidence. Keep raising the angle of incidence and there comes a moment when the angle of refraction reaches 90 degrees. That angle is the critical angle θc = asin(n₂/n₁). Beyond it, no light gets out and all of it reflects. An optical fiber sends light far by trapping it with exactly this total internal reflection. A diamond has a very high refractive index, so its critical angle is small, and light that enters glints through many total reflections. Trapping is how it travels far.
A thin lens forms an image by refraction alone. Between the object distance u, the image distance v, and the focal length f there holds the relation 1/f = 1/u + 1/v. When the object is farther than the focus, an inverted real image forms on the other side. When it comes inside the focus, an upright virtual image forms on the same side. A magnifying glass, with the object held inside the focus, is an example that uses this virtual image. Push and pull the object and the image follows, showing where it forms and how large. Inside or outside the focus splits real from virtual.
The sign of the focal length decides the character of the lens. A positive focal length is a convex lens that gathers light, and a negative one is a concave lens that spreads it. A concave lens always makes only a small upright virtual image. The magnification is m = -v/u. The sign tells whether the image is upright or inverted, and the size tells whether it is enlarged or reduced. Glasses that correct nearsightedness are a concave lens of negative focal length. They spread the incoming light in advance so the image lands on the retina. Move the focal length left and right and the image passes between real and virtual. The sign decides gathering and spreading.
Where the image forms can be constructed with three representative rays. A ray coming parallel to the axis passes through the focus after the lens. A ray through the center of the lens goes straight without bending. A ray that has passed the front focus leaves parallel to the axis behind the lens. The point where the three rays meet again is the image point. Two rays are enough to fix the place of the image, and the third ray is a check that the result is right. Move the object and the three rays always meet in the same place. Two rays fix the place and the third confirms it.