242. Maxwell’s Equations: Electromagnetic Waves Predicted and ObservedLearning Objectives
The Scotsman James Clerk Maxwell (1831–1879) is regarded as the greatest theoretical physicist of the 19th century. (See Figure 1.) Although he died young, Maxwell not only formulated a complete electromagnetic theory, represented by Maxwell brought together all the work that had been done by brilliant physicists such as Oersted, Coulomb, Gauss, and Faraday, and added his own insights to develop the overarching theory of electromagnetism. Maxwell’s equations are paraphrased here in words because their mathematical statement is beyond the level of this text. However, the equations illustrate how apparently simple mathematical statements can elegantly unite and express a multitude of concepts—why mathematics is the language of science. Maxwell’s Equations
Maxwell’s equations encompass the major laws of electricity and magnetism. What is not so apparent is the symmetry that Maxwell introduced in his mathematical framework. Especially important is his addition of the hypothesis that changing electric fields create magnetic fields. This is exactly analogous (and symmetric) to Faraday’s law of induction and had been suspected for some time, but fits beautifully into Maxwell’s equations. Symmetry is apparent in nature in a wide range of situations. In contemporary research, symmetry plays a major part in the search for subatomic particles using massive multinational particle accelerators such as the new Large Hadron Collider at CERN. Making Connections: Unification of ForcesMaxwell’s complete and symmetric theory showed that electric and magnetic forces are not separate, but different manifestations of the same thing—the electromagnetic force. This classical unification of forces is one motivation for current attempts to unify the four basic forces in nature—the gravitational, electrical, strong, and weak nuclear forces. Since changing electric fields create relatively weak magnetic fields, they could not be easily detected at the time of Maxwell’s hypothesis. Maxwell realized, however, that oscillating charges, like those in AC circuits, produce changing electric fields. He predicted that these changing fields would propagate from the source like waves generated on a lake by a jumping fish. The waves predicted by Maxwell would consist of oscillating electric and magnetic fields—defined to be an electromagnetic wave (EM wave). Electromagnetic waves would be capable of exerting forces on charges great distances from their source, and they might thus be detectable. Maxwell calculated that electromagnetic waves would propagate at a speed given by the equation $$c=\frac{1}{\sqrt{{\mu}_{0}{\epsilon}_{0}}}.$$When the values for ${\mu}_{0}$ and ${\epsilon}_{0}$ are entered into the equation for $c$, we find that $$c=\frac{1}{\sqrt{(8\text{.}\text{85}\times {\text{10}}^{\text{12}}\phantom{\rule{0.25em}{0ex}}\frac{{\text{C}}^{2}}{\text{N}\cdot {\text{m}}^{2}})(\mathrm{4\pi}\times {\text{10}}^{7}\phantom{\rule{0.25em}{0ex}}\frac{\text{T}\cdot \text{m}}{\text{A}})}}=\text{3}\text{.}\text{00}\times {\text{10}}^{8}\phantom{\rule{0.25em}{0ex}}\text{m/s},$$which is the speed of light. In fact, Maxwell concluded that light is an electromagnetic wave having such wavelengths that it can be detected by the eye. Other wavelengths should exist—it remained to be seen if they did. If so, Maxwell’s theory and remarkable predictions would be verified, the greatest triumph of physics since Newton. Experimental verification came within a few years, but not before Maxwell’s death. Hertz’s ObservationsThe German physicist Heinrich Hertz (1857–1894) was the first to generate and detect certain types of electromagnetic waves in the laboratory. Starting in 1887, he performed a series of experiments that not only confirmed the existence of electromagnetic waves, but also verified that they travel at the speed of light. Hertz used an AC $\text{RLC}$ (resistorinductorcapacitor) circuit that resonates at a known frequency ${f}_{0}=\frac{1}{\mathrm{2\pi}\sqrt{\text{LC}}}$ and connected it to a loop of wire as shown in Figure 2. High voltages induced across the gap in the loop produced sparks that were visible evidence of the current in the circuit and that helped generate electromagnetic waves. Across the laboratory, Hertz had another loop attached to another $\text{RLC}$ circuit, which could be tuned (as the dial on a radio) to the same resonant frequency as the first and could, thus, be made to receive electromagnetic waves. This loop also had a gap across which sparks were generated, giving solid evidence that electromagnetic waves had been received. Hertz also studied the reflection, refraction, and interference patterns of the electromagnetic waves he generated, verifying their wave character. He was able to determine wavelength from the interference patterns, and knowing their frequency, he could calculate the propagation speed using the equation $\upsilon =\mathrm{f\lambda}$ (velocity—or speed—equals frequency times wavelength). Hertz was thus able to prove that electromagnetic waves travel at the speed of light. The SI unit for frequency, the hertz ($\mathrm{1\; Hz}=\text{1 cycle/sec}$), is named in his honor. Section Summary
Problems & ExercisesExercise 1Verify that the correct value for the speed of light $c$ is obtained when numerical values for the permeability and permittivity of free space (${\mu}_{0}$ and ${\epsilon}_{0}$) are entered into the equation $c=\frac{1}{\sqrt{{\mu}_{0}{\epsilon}_{0}}}$. Exercise 2Show that, when SI units for ${\mu}_{0}$ and ${\epsilon}_{0}$ are entered, the units given by the righthand side of the equation in the problem above are m/s.
