Electromagnetic Waves — Light Is Magnetism in Motion
Electromagnetic Waves
James Clerk Maxwell made one of the greatest intellectual leaps in the history of science when he showed in the 1860s that changing electric fields produce magnetic fields, and changing magnetic fields produce electric fields. This mutual creation means that oscillating electric and magnetic fields can sustain each other and propagate through space as waves — electromagnetic waves. Maxwell calculated the speed of these waves and found it exactly equaled the known speed of light. Light is an electromagnetic wave.
The electromagnetic spectrum spans an extraordinary range: radio waves with wavelengths of kilometers, microwaves at centimeters, infrared, the narrow band of visible light from about 400 to 700 nanometers, ultraviolet, X-rays, and gamma rays with wavelengths smaller than an atomic nucleus. All are the same phenomenon — oscillating electric and magnetic fields — differing only in frequency and wavelength. They all travel at the speed of light in vacuum: c equals 1 over the square root of mu-zero times epsilon-zero, which equals approximately 3 times 10 to the 8th meters per second.
The energy carried by electromagnetic waves connects directly to the electric and magnetic field amplitudes. The intensity — power per unit area — is given by the Poynting vector, whose time-averaged magnitude is I equals E-zero squared over 2 mu-zero c, where E-zero is the peak electric field amplitude. The relationship between the electric and magnetic field amplitudes in a wave is E equals c times B, linking the two fields in a precise mathematical relationship.
The mathematics of electromagnetic waves draws on everything you have learned: algebra, trigonometry for the sinusoidal oscillation, calculus for Maxwell's equations, and dimensional analysis for the physical constants. Understanding electromagnetic waves means understanding how your phone communicates, how the Sun warms the Earth, how MRI machines image your body, and how the universe itself is structured at its most fundamental level.
$$c = \frac{1}{\sqrt{\mu_0 \epsilon_0}} \approx 3 \times 10^8 \text{ m/s}$$
where $\mu_0 = 4\pi \times 10^{-7}$ T·m/A and $\epsilon_0 = 8.854 \times 10^{-12}$ F/m.
For all electromagnetic waves: $c = f\lambda$, where $f$ is frequency and $\lambda$ is wavelength.
Electric and magnetic field amplitudes are related by: $E_0 = cB_0$.
The average intensity (power per unit area) of an EM wave:
$$I = \frac{E_0^2}{2\mu_0 c} = \frac{c B_0^2}{2\mu_0} = \frac{E_0 B_0}{2\mu_0}$$
An FM radio station broadcasts at 100 MHz. What is the wavelength?
Step 1: Use $\lambda = c/f$.
$$\lambda = \frac{3 \times 10^8}{100 \times 10^6} = 3 \text{ m}$$
FM radio waves are about 3 meters long — roughly the height of a room.
Sunlight has an average intensity of about $I = 1{,}370$ W/m² at Earth's distance. Find the peak electric field.
Step 1: Solve for $E_0$ from $I = E_0^2/(2\mu_0 c)$:
$$E_0 = \sqrt{2\mu_0 c I} = \sqrt{2(4\pi \times 10^{-7})(3 \times 10^8)(1370)}$$
Step 2: $E_0 = \sqrt{2 \times 1.257 \times 10^{-6} \times 3 \times 10^8 \times 1370}$
$$E_0 = \sqrt{1.032 \times 10^6} \approx 1{,}016 \text{ V/m}$$
Step 3: The corresponding peak magnetic field:
$$B_0 = E_0/c = 1016/(3 \times 10^8) \approx 3.39 \text{ μT}$$
Verify that $1/\sqrt{\mu_0 \epsilon_0}$ gives the speed of light.
Step 1: $\mu_0 \epsilon_0 = (4\pi \times 10^{-7})(8.854 \times 10^{-12})$
$= 1.113 \times 10^{-17}$ s²/m²
Step 2: $c = 1/\sqrt{1.113 \times 10^{-17}} = 1/\sqrt{1.113 \times 10^{-17}}$
$= 2.998 \times 10^8$ m/s ✓
Maxwell calculated this in the 1860s and recognized it as the speed of light — unifying optics with electromagnetism.
Practice Problems
Show Answer Key
1. $\lambda = 3\times10^8/(5\times10^9) = 0.06$ m $= 6$ cm
2. $f = 3\times10^8/(550\times10^{-9}) = 5.45\times10^{14}$ Hz
3. $B_0 = 500/(3\times10^8) = 1.67$ μT. $I = 500^2/(2 \times 4\pi\times10^{-7} \times 3\times10^8) = 332$ W/m²
4. $E_0 = \sqrt{2\mu_0 c I} = \sqrt{2(4\pi\times10^{-7})(3\times10^8)(10^6)} \approx 2.74\times10^4$ V/m
5. $t = d/c = 1.5\times10^{11}/(3\times10^8) = 500$ s $\approx 8.3$ minutes
6. $f = 3\times10^8/(10^{-10}) = 3\times10^{18}$ Hz