Waves and optics tools

Measure angles from the normal

Every angle in refraction is measured from the normal, the line perpendicular to the surface, not from the surface itself. A ray described as striking glass at 20 degrees to the surface has an angle of incidence of 70 degrees. This single convention accounts for most wrong answers in optics, and it is worth checking whenever a result bends the wrong way.

It is also the ratio of the sines that stays fixed, not the ratio of the angles. Doubling the angle of incidence does not double the refraction angle, and the gap between the two widens as you approach grazing incidence.

Total internal reflection follows directly, and only in one direction. A critical angle exists only where light travels from a denser medium into a less dense one, given by the arcsine of the ratio of the indices: 48.6 degrees for water into air, 41.1 degrees for typical crown glass (index about 1.52) into air, and none at all for air into water. Past it nothing refracts and everything reflects, which is the basis of optical fibre.

Wavelength, frequency and what stays constant

When light crosses into a denser medium its speed drops and its wavelength shortens, but its frequency does not change. Frequency is set by the source; the medium can only affect how fast the wave travels and therefore how far apart the crests are. Colour follows frequency, which is why an object does not change colour underwater.

Photon energy follows frequency for the same reason: E = hf. A photon energy quoted from a wavelength is only meaningful if you say which medium it was measured in, and by convention that is vacuum unless stated. Usefully, hc is 1239.84 eV·nm, so energy in electronvolts is 1240 divided by the wavelength in nanometres, making a 620 nm red photon almost exactly 2 eV.

Energy per mole compares directly against chemistry. A mole of 550 nm green photons carries about 217 kJ, below a typical carbon to carbon single bond at around 350 kJ/mol, so visible light does not break one, while a mole of 300 nm ultraviolet photons carries close to 400 kJ and does.

Lens sign conventions differ between textbooks

The thin lens equation is written here as 1/f = 1/dₒ + 1/dᵢ, the convention in which real objects and real images both take positive distances. Many physics textbooks use the Cartesian convention instead and write 1/f = 1/dᵢ minus 1/dₒ, with distances signed by direction along the axis. Both are correct and they disagree about the sign of almost everything, so mixing a formula from one with a sign rule from the other gives a confidently wrong answer.

Under the convention used here, a negative image distance means the image is virtual and sits on the same side of the lens as the object. Your eye sees it through the lens but you cannot catch it on a screen, which is what a magnifying glass does when held closer to the object than its focal length. Magnification is negative dᵢ over dₒ, where the sign carries orientation rather than size. Two checks catch a sign slip: an object at exactly the focal point produces no image at all, because the rays leave parallel, and an object at twice the focal length produces an image at twice the focal length, same size and inverted.

Decibels do not add

The decibel is a logarithm of a ratio, so ordinary addition does not apply. Two 60 dB sources together give 63 dB, not 120, because doubling the intensity adds about 3 dB. Unequal sources barely combine at all: 60 dB alongside 70 dB gives about 70.4 dB, since the quieter one contributes a tenth of the intensity. This is why silencing the second loudest machine in a room changes nothing.

Perception adds another layer. Ten decibels is a tenfold rise in intensity but only around a doubling in perceived loudness, and 3 dB, a genuine doubling of intensity, is close to imperceptible. Nor is 0 dB silence: it is the reference intensity of 1 picowatt per square metre, the quietest sound a healthy young ear detects.

Distance and exposure follow the same logarithm. Intensity from a point source falls with the square of distance, so each doubling costs about 6 dB. Exposure runs the other way: around 85 dB is the usual limit for an eight-hour day, and every 3 dB above that halves the safe time, putting 100 dB at roughly 15 minutes. Damage above about 120 dB can be immediate, and it does not heal.