Physics II · 120 h · Topic 11 of 15

Maxwell's equations

Maxwell combined two constants measured in electrostatics and magnetism experiments, with no connection to optics, and it gave him exactly the speed of light. That coincidence was the proof that light is an electromagnetic wave.

The four equations Displacement current c Refractive index Transmission line

01The four equations

Everything that appeared in the previous units —Gauss, the absence of monopoles, Faraday and Ampère— fits into four equations. Maxwell did not discover all of them: he gathered them, and in doing so he noticed that one was missing a term.

\[ \oint \vec{E}\cdot d\vec{A} = \frac{q_{enc}}{\varepsilon_0} \qquad\qquad \oint \vec{B}\cdot d\vec{A} = 0 \] \[ \oint \vec{E}\cdot d\vec{l} = -\frac{d\Phi_B}{dt} \qquad\qquad \oint \vec{B}\cdot d\vec{l} = \mu_0 I_{enc} + \mu_0\varepsilon_0\frac{d\Phi_E}{dt} \] Gauss’s law for electricity, Gauss’s law for magnetism, Faraday and Ampère–Maxwell. The last term of the fourth —the displacement current— is Maxwell’s addition, and everything else follows from it.

Equation	It says	Practical consequence
Gauss (electric)	Charges are sources of \( E \)	The Faraday cage
Gauss (magnetic)	There are no magnetic charges	The lines of \( B \) always close on themselves
Faraday	A varying \( B \) creates \( E \)	Generators, transformers, induction
Ampère–Maxwell	Currents and a varying \( E \) create \( B \)	The very existence of the electromagnetic wave

02The displacement current

Maxwell’s reasoning is one of the most elegant deductions in physics, and it can be understood with a charging capacitor. Ampère’s law says that the circulation of \( B \) around a loop depends on the current that passes through any surface resting on that loop.

If a flat surface that cuts the wire is chosen, the enclosed current is \( I \). But if a bag-shaped surface that passes between the plates of the capacitor is chosen, no current passes through it: there are no moving charges there. The same law gives two different results, and that means it was incomplete.

\[ I_d = \varepsilon_0\frac{d\Phi_E}{dt} \] Between the plates there is no conduction current but there is a growing electric field, and that growth produces a magnetic field exactly as if there were a current. The contradiction disappears and, incidentally, the universe becomes symmetric: a varying \( E \) creates \( B \), and a varying \( B \) creates \( E \).

From that symmetry the wave follows immediately: a changing electric field generates a magnetic one which, by changing, generates an electric one, and so on indefinitely, moving away from the source without needing any medium. Maxwell combined the four equations and found that the speed of that disturbance was:

\[ c = \frac{1}{\sqrt{\varepsilon_0\mu_0}} = 299\,792\,458\ \mathrm{m/s} \] Two constants measured in electrostatics and magnetism experiments, having nothing to do with light, and when they are combined exactly the speed of light appears. That coincidence was the evidence that light is an electromagnetic wave.

03Propagation in a medium

Inside a material, the permittivity and the permeability change, and with them the speed:

\[ v = \frac{c}{\sqrt{\varepsilon_r\mu_r}} = \frac{c}{n} \qquad\qquad \lambda = \frac{v}{f} \qquad\qquad Z = \sqrt{\frac{\mu}{\varepsilon}} \] \( n \) is the refractive index and \( Z \) the wave impedance of the medium, which in vacuum is 377 Ω. In almost all nonmagnetic materials \( \mu_r = 1 \), so the speed is set by the permittivity alone.

Lab · propagation in a medium

Choose the medium and the frequency. The speed, the wavelength and the impedance appear, and below them the figure that decides a design: beyond what length a trace stops being a wire and has to be treated as a transmission line.

Frequency = 100 MHz

When a trace stops being a wire

As long as the trace is much shorter than the wavelength, the signal arrives “at the same time” at both ends and it can be treated as a node: that is what every lumped-element circuit assumes. When its length approaches a tenth of \( \lambda \), the ends stop being in phase, reflections appear and it has to be designed as a transmission line, with its characteristic impedance and its termination. In FR-4 the signal travels at about 15 cm per nanosecond, and from that number comes all of fast digital design.

04The spectrum

All electromagnetic waves are the same phenomenon and differ only in frequency. The names are historical: they reflect how they were discovered or what they are used for, not any physical difference.

Band	Frequency	\( \lambda \) in vacuum	In electronics
Mains power	50 Hz	6000 km	Never radiates: everything is near field
Medium wave	1 MHz	300 m	Large antennas, broadcasting
VHF and UHF	100 MHz to 1 GHz	3 m to 30 cm	Television, mobile radio
Microwaves	2.4 GHz	12.5 cm	Wi-Fi, Bluetooth, ovens
Infrared	\( 3\times10^{13} \) Hz	10 µm	Remote control, thermography
Visible	\( 5\times10^{14} \) Hz	0.6 µm	Fiber optics, LEDs, photodiodes

The first row explains why a 50 Hz installation does not radiate: with a wavelength of 6000 km, any wire in a house is electrically tiny. And the fourth, why Wi-Fi and the microwave oven share a band: 2.45 GHz is a frequency released for industrial use, and the two coexist there.

05In the lab

Activity 1 · Measuring c with a microwave oven

Heat a chocolate bar without the turntable and measure the distance between the melted spots, which are the antinodes of the standing wave. Multiply by two to obtain \( \lambda \) and by the 2.45 GHz of the oven. You arrive at \( c \) with an error of a few percent.

Activity 2 · Speed in a cable

Send a pulse down a long coaxial cable with the end open and use the oscilloscope to measure the round-trip time of the reflection. Calculate the propagation speed and from it obtain the permittivity of the dielectric.

Activity 3 · Critical length

With a pulse generator with fast edges, observe the signal at the end of unterminated traces or cables of increasing length. Identify beyond what length the reflection steps appear and compare it with \( \lambda/10 \).

06Common mistakes

Believing that Maxwell discovered the four equations. He gathered them and added to one of them the missing term.
Thinking that the displacement current is a current. No charges are moving: it is a varying electric field.
Assuming that the wave needs a medium. It propagates in vacuum; that was the revolutionary part.
Using \( c \) inside a material. There the speed is \( c/n \), and in FR-4 it is almost half.
Treating a long trace as a node. Beyond a tenth of \( \lambda \) it has to be designed as a line.
Believing that the bands of the spectrum are different phenomena. It is the same one, at a different frequency.

07Self-assessment

What did Maxwell add to Ampère’s law?

The displacement current: a varying electric field produces a magnetic field just like a current does.

Why was that correction necessary?

Because without it the law gave two different results depending on which surface was chosen in a charging capacitor.

Where does the speed of light come from?

From \( 1/\sqrt{\varepsilon_0\mu_0} \), two constants measured with no connection to optics.

What is the wave impedance of vacuum?

377 ohms, which is \( \sqrt{\mu_0/\varepsilon_0} \).

At what speed does a signal travel along an FR-4 trace?

About 15 cm per nanosecond: close to half the speed in vacuum.

When does a trace have to be treated as a transmission line?

When its length approaches a tenth of the wavelength of the fastest signal it carries.

08Further reading

Raymond A. Serway and John W. Jewett. Physics for Scientists and Engineers with Modern Physics. 10th ed. (Spanish edition, “Física para ciencias e ingeniería”), Cengage, 2019. Maxwell’s equations brought together and the derivation of the electromagnetic wave.
Edward M. Purcell and David J. Morin. Electricity and Magnetism. 3rd ed. (Spanish edition, “Electricidad y magnetismo”), Cambridge University Press, 2016. The complete path to Maxwell, with the capacitor argument developed with care.
Howard Johnson and Martin Graham. High-Speed Digital Design: A Handbook of Black Magic. Prentice Hall, 1993. What happens when a trace stops being a wire, written for those who design boards and not antennas.

← Previous topic

Magnetic properties of matter

Next topic →

Wave motion