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Topic 12 of 15

Maxwell's Equations & EM Waves

James Clerk Maxwell unified electricity, magnetism, and optics in four elegant equations. His crucial addition — the displacement current — predicted that changing electric fields produce magnetic fields, completing a symmetry between $\vec{E}$ and $\vec{B}$. The immediate consequence was the existence of electromagnetic waves traveling at $c = 1/\sqrt{\varepsilon_0\mu_0} \approx 3\times10^8$ m/s — the speed of light. This was one of the most profound predictions in the history of science.

Key Concepts

Displacement Current

Maxwell added a term to Ampère's law: a changing electric field acts as a current source for

\vec{B}

. The displacement current density is

J_D = \varepsilon_0\,\partial E/\partial t

. This is why a magnetic field exists in the gap of a charging capacitor even though no real current flows there.

Maxwell's Four Equations

(1) Gauss's law:

\oint\vec{E}\cdot d\vec{A} = q_{\text{enc}}/\varepsilon_0

. (2) No magnetic monopoles:

\oint\vec{B}\cdot d\vec{A} = 0

. (3) Faraday's law:

\oint\vec{E}\cdot d\vec{l} = -d\Phi_B/dt

. (4) Ampère-Maxwell:

\oint\vec{B}\cdot d\vec{l} = \mu_0(I_{\text{enc}} + \varepsilon_0\,d\Phi_E/dt)

. Together they contain all of classical electromagnetism.

Electromagnetic Waves

Maxwell's equations predict self-sustaining waves where oscillating

\vec{E}

produces oscillating

\vec{B}

and vice versa. The waves are transverse (

\vec{E}\perp\vec{B}\perp\hat{v}

), travel at

c = 1/\sqrt{\varepsilon_0\mu_0}

in vacuum, and the amplitudes are related by

E_0 = cB_0

Poynting Vector and Intensity

The Poynting vector

\vec{S} = (1/\mu_0)\vec{E}\times\vec{B}

gives the instantaneous power per unit area carried by the EM wave. Its time-averaged magnitude is the intensity:

I = E_0^2/(2\mu_0 c) = E_0 B_0/(2\mu_0)

Radiation Pressure

EM waves carry momentum. For a perfectly absorbing surface, the radiation pressure is

P = I/c

. For a perfectly reflecting surface,

P = 2I/c

. This is the principle behind solar sails.

Key Equations

Speed of light (from EM constants)

c = \frac{1}{\sqrt{\varepsilon_0\mu_0}} \approx 3.00\times10^8 \text{ m/s}

Predicted by Maxwell's equations from measured electric and magnetic constants — immediately identified with the speed of light.

E and B amplitudes

E_0 = cB_0

In an EM wave, the electric and magnetic amplitudes are related by the speed of light. They oscillate in phase.

Wave intensity

I = \frac{E_0^2}{2\mu_0 c}

Time-averaged power per unit area of an EM wave. Also written I = E₀B₀/(2μ₀).

Radiation pressure (absorption)

P_{\text{rad}} = \frac{I}{c}

Radiation pressure on a perfectly absorbing surface. For perfect reflection: P = 2I/c.

Worked Example

Electric and Magnetic Amplitudes of an EM Wave

Problem

An EM wave in vacuum has intensity $I = 1000$ W/m². Find (a) the amplitude of the electric field $E_0$ and (b) the amplitude of the magnetic field $B_0$ .

Solution

(a) Solve $I = E_0^2/(2\mu_0 c)$ for $E_0$ . Note that $2\mu_0 c = 2(4\pi\times10^{-7})(3\times10^8) \approx 754\,\Omega$ :

E_0 = \sqrt{2\mu_0 c\cdot I} = \sqrt{754\times1000} = \sqrt{7.54\times10^5} \approx 868\text{ V/m}

(b) Magnetic amplitude from $B_0 = E_0/c$ :

B_0 = \frac{E_0}{c} = \frac{868}{3.00\times10^8} = 2.89\times10^{-6}\text{ T} = 2.89\,\mu\text{T}

Answer E₀ ≈ 868 V/m; B₀ ≈ 2.89 μT.

Practice

Exercises

7 problems

1 of 7

An EM wave has electric field amplitude $E_0 = 600$ V/m. What is the magnetic field amplitude $B_0$ (in $\mu$ T)?

Your answer

μT

2 of 7

An EM wave has $E_0 = 800$ V/m. What is its intensity (in W/m²)? Use $2\mu_0 c \approx 754\,\Omega$ .

Your answer

W/m²

3 of 7

A radio station broadcasts at $f = 100$ MHz. What is the wavelength (in m) of its EM waves?

Your answer

4 of 7

Find the electric field amplitude $E_0$ (in V/m) of an EM wave with intensity $I = 500$ W/m². Use $2\mu_0 c \approx 754\,\Omega$ .

Your answer

V/m

5 of 7

A laser beam has intensity $I = 1000$ W/m² and strikes a perfectly absorbing surface. What is the radiation pressure (in $\mu$ Pa)?

Your answer

μPa

6 of 7

An EM wave has wavelength $\lambda = 0.50$ m. What is its frequency (in MHz)?

Your answer

MHz

7 of 7

A small antenna radiates $P = 100$ W isotropically (equally in all directions). What is the intensity (in W/m²) at a distance $r = 5.0$ m?

Your answer

W/m²

Key Takeaways

Maxwell's displacement current completed the symmetry: just as a changing $\vec{B}$ induces $\vec{E}$ (Faraday), a changing $\vec{E}$ induces $\vec{B}$ (Maxwell).
EM waves are self-sustaining oscillations of $\vec{E}$ and $\vec{B}$ propagating at $c = 3\times10^8$ m/s; they require no medium.
In a wave, $\vec{E}$ , $\vec{B}$ , and the direction of propagation are mutually perpendicular, and $E_0 = cB_0$ .
Intensity $I = E_0^2/(2\mu_0 c)$ : to double the intensity you need $\sqrt{2}$ times the field amplitude.
The full EM spectrum — radio, microwave, infrared, visible, UV, X-ray, gamma — are all EM waves differing only in frequency.