Honors Physics Unit 15 Review: Light

Visible light is one small slice of the electromagnetic (EM) spectrum. In a plane EM wave, the electric and magnetic fields oscillate perpendicular to each other and to the direction of travel. EM waves need no medium, so they can cross the vacuum of space.

All EM waves move at the same speed in a vacuum, $c = 3.00 \times 10^8$ m/s. Radio waves and gamma rays differ in frequency and wavelength, which are linked by $c = \lambda f$. Since $c$ is fixed, a higher frequency means a shorter wavelength.

Light also comes in packets called photons. Each photon carries energy $E = hf$, where $h = 6.626 \times 10^{-34}$ J·s. Higher frequency means more energy per photon. Light behaves like a wave in some experiments (interference, diffraction) and like a stream of particles in others (the photoelectric effect), which is called wave-particle duality.

From lowest to highest frequency, the usual order is radio, microwave, infrared, visible, ultraviolet, X-ray, gamma. The regions have approximate boundaries, and X-rays and gamma rays can overlap in wavelength. X-rays are typically produced by electron interactions, while gamma rays are often produced by nuclear processes. Visible light runs from about 700 nm (red) down to 380 nm (violet).

Worked example. Find the frequency and photon energy of green light with $\lambda = 500$ nm. Convert first, so $500 \text{ nm} = 5.00 \times 10^{-7}$ m. Then $f = c/\lambda = (3.00 \times 10^8)/(5.00 \times 10^{-7}) = 6.00 \times 10^{14}$ Hz. The energy is $E = hf = (6.626 \times 10^{-34})(6.00 \times 10^{14}) = 3.98 \times 10^{-19}$ J, about 2.5 eV.

Visible light travels more slowly in ordinary transparent materials than in a vacuum. The refractive index $n$ relates the wave's phase speed $v$ to the vacuum speed through $v = c/n$. Water has $n \approx 1.33$ and typical glass has $n \approx 1.5$ for visible light.

That speed change bends light that hits a boundary at an angle. Snell's law gives the new direction, $n_1 \sin\theta_1 = n_2 \sin\theta_2$, with both angles measured from the normal (the line perpendicular to the surface). Going into higher $n$, light bends toward the normal. Going into lower $n$, it bends away.

Frequency stays the same across a stationary boundary, so wavelength changes by the same factor as speed. It shrinks when speed decreases and increases when speed increases. Starting in air, $\lambda_2 \approx \lambda_1 / n$, where $n$ is the second material's refractive index.

When light goes from high $n$ to low $n$ beyond the critical angle $\theta_c = \sin^{-1}(n_2/n_1)$, none of it escapes. That's total internal reflection, which keeps signals inside fiber optic cables. For glass into air, $\theta_c = \sin^{-1}(1/1.5) \approx 41.8°$.

An ideal polarizing filter passes only the electric field component along its axis. It transmits half the intensity of unpolarized light. For already polarized light, Malus's law gives the transmitted intensity as $I = I_0 \cos^2\theta$, where $I_0$ is the incident intensity and $\theta$ is the angle between the light's polarization direction and the filter's axis.

Worked example. Light in air hits water at $30°$ from the normal. Snell's law gives $\sin\theta_2 = (1.00)(\sin 30°)/1.33 = 0.500/1.33 = 0.376$, so $\theta_2 \approx 22.1°$. Its speed in water is $(3.00 \times 10^8)/1.33 = 2.26 \times 10^8$ m/s. A 600 nm beam in air becomes about 451 nm in water.