Refraction of light: Snell's law and apparent depth

Laws of refraction

Refraction is the bending of light when it passes at an angle from one medium into another. It happens because light travels at different speeds in different media. (1) The incident ray, the refracted ray and the normal at the point of incidence lie in one plane. (2) For a given pair of media, sin i / sin r is a constant, n₂₁. This is Snell's law. If n₂₁ > 1, medium 2 is optically denser: light slows down and bends towards the normal (r < i). If n₂₁ < 1, light speeds up and bends away from the normal. Optical density is not mass density: turpentine is optically denser than water, yet it floats on water. A ray along the normal (i = 0) goes straight through without bending.

Results you will use

• Refractive index from speeds: n₂₁ = v₁/v₂. The absolute refractive index is n = c/v: about 1.5 for glass and 1.33 for water.
• Reversibility: n₁₂ = 1/n₂₁. For light going from glass into air, n(air with respect to glass) = 1/1.5 = 2/3.
• Chain rule: n₃₂ = n₃₁ × n₁₂, for three media 1, 2 and 3.
• Glass slab: the emergent ray is parallel to the incident ray but shifted sideways. This lateral shift grows with the thickness of the slab and with the angle of incidence.
• What changes: the speed and the wavelength change on refraction. The frequency does not.

Derivation: real depth and apparent depth

An object O lies at depth h below the surface of water of refractive index n. You look at it from directly above, so the rays stay close to the normal.

1. 1
   A ray from O meets the surface at A, a small distance x from the normal through O. In the water it makes an angle r with the normal.
2. 2
   In air it bends away from the normal, at angle i. Traced backwards, it meets the normal at I, at depth h′. I is the image you see.
3. 3
   From the figure: tan r = x/h and tan i = x/h′.
4. 4
   Snell's law, from water into air: n sin r = 1 × sin i.
5. 5
   For near-normal viewing the angles are small, so sin ≈ tan: n x/h = x/h′.
6. 6★
   So, h′ = h/n, that is, real depth / apparent depth = n. The object seems raised by h − h′ = h(1 − 1/n).

★ key step

How to draw the apparent depth diagram

• Draw a horizontal line for the water surface. Shade the water below it, and write 'water, n' below and 'air' above.
• Draw a vertical dotted normal. Mark the object O on it, a depth h below the surface.
• Draw a ray from O to a point A on the surface, close to the normal. Draw the normal at A and mark the angle r.
• In air, draw the refracted ray bending away from the normal at A. Mark the larger angle i, and draw an eye where the ray goes.
• Extend the refracted ray backwards as a dotted line until it meets the vertical normal at I, above O.
• Label the real depth h (surface to O) and the apparent depth h′ (surface to I). Put arrows on the real rays.

Worked example: Snell's law and apparent depth

Activity: refractive index of a glass slab and of a liquid

Two practical experiments that use the idea of real and apparent depth.

1. 1
   Glass slab (travelling microscope): Make a small ink cross on white paper. Focus the microscope on it and note the reading R₁.
2. 2
   Place the glass slab over the cross. The cross now looks raised. Raise the microscope to focus on it again, and note R₂.
3. 3
   Sprinkle a little chalk dust (or lycopodium powder) on top of the slab. Focus on the dust and note R₃.
4. 4
   Real thickness = R₃ − R₁ and apparent thickness = R₃ − R₂. So n = (R₃ − R₁)/(R₃ − R₂). Result: about 1.5 for ordinary glass.
5. 5
   Liquid (concave mirror): Lay the concave mirror flat, facing up. Move a pin up or down above it until the pin and its inverted image coincide with no parallax. The pin is then at C, a height R above the pole.
6. 6
   Pour a thin layer of the liquid into the mirror and repeat. The pin now coincides with its image at a smaller height R′.
7. 7
   For a thin layer of liquid, n = R/R′. Result: for water, n comes out close to 1.33.