Reflection and Transmission at a Boundary
Raise the angle of incidence to reach total internal reflection
Light travels from a dense medium (below) into a rare one (above). As you raise the angle of incidence, the refracted ray (green) lies down ever closer to the boundary and the reflected ray (gold) grows stronger. Past a certain angle the refracted ray vanishes and the light bounces back whole. Find where that critical angle is.
Light splits at the boundary
When light crosses the boundary between two media — air to glass, or water to air — one beam splits into two. Part becomes the reflected ray that bounces back, part the refracted ray that bends and enters. The energy the two share adds up to the energy of the original beam. A mirror showing an image, and your faint reflection in a windowpane, are both this reflection; a straw in water looking bent is refraction.
Refractive index and slowing light
Light is fastest in vacuum but slows inside a medium. How much it slows is the refractive index n, and the speed inside is v = c/n. Glass has an index of about 1.5, so light moves through it at two-thirds of c. In a non-magnetic medium n is set by the permittivity as √ε_r. The frequency stays the same but the speed drops, so the wavelength shortens. This difference in how light slows from medium to medium is exactly why it bends.
Snell's law
The angle of refraction follows Snell's law n₁ sinθ₁ = n₂ sinθ₂. Going from a dense medium (large n) into a rare one (small n), light bends away from the normal; the other way, it bends toward the normal. Reflection is simpler: the angle of reflection always equals the angle of incidence (θ_r = θ₁). Why these rules? Along the boundary the crests of the incident and transmitted waves must keep in step. Since the wavelength differs in each medium, matching the crest spacing forces the angles to change exactly as Snell's law says.
Total internal reflection
Going from a dense medium to a rare one, raising the angle of incidence makes the angle of refraction grow even faster, reaching 90° first. The angle of incidence at that instant is the critical angle θ_c, with sinθ_c = n₂/n₁. Beyond the critical angle, Snell's law would demand sinθ₂ > 1, and no such angle exists. So the light cannot transmit at all and is entirely reflected. This is total internal reflection. Optical fibres trapping light to carry it far, the sparkle of a diamond, and a desert mirage all rely on it.
Back to the first screen
On the first screen, the higher you pushed the angle of incidence, the more the refracted ray (green) lay down toward the boundary, until past one angle it vanished without trace and the light went back whole as the reflected ray (gold). That angle is the critical angle θ_c, the one satisfying sinθ_c = n₂/n₁. Below it, Snell's law n₁ sinθ₁ = n₂ sinθ₂ set the angle of refraction while the angle of reflection always equalled the angle of incidence. Because light runs at a different speed in each medium, it splits and bends at the boundary.
So far light has been a wave spreading freely. From the next unit (EM-26) we treat transmission lines, which confine an electromagnetic wave between two conductors and send it one way. Inside a cable, voltage and current run as waves, and a new quantity, the characteristic impedance, appears. We move into how high-frequency signals travel through a circuit.