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In an electromagnetic wave, is the maximum amplitude of the electric field the same as that of the magnetic field? I ask this question which may seem silly because on all the diagrams representing an electromagnetic wave these two amplitudes seem to be equal.

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    $\begingroup$ $\hat H=\hat E$ equal in what system of units? $\endgroup$
    – hyportnex
    Aug 3, 2020 at 14:49

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First, In the unit systems used for everyday commerce, engineering and undergraduate physics, magnetic and electric field have different units, so the two amplitudes can't be equal, any more than one meter can be equal to one kilogram.

There do exist alternative unit systems that allow comparing electric and magnetic fields directly.

But even if you chose such a unit system, the ratio between them would depend on the medium the wave travels through. Each medium having an impedance that determines what ratio of E and H fields produce a travelling wave in that medium.

For example, the impedance of free space is about 376.7 ohms. That means that for a travelling wave in free space, the ratio $Z_0=\frac{|{\bf E}|}{|{\bf H}|}$ is 376.7 ohms. This impedance is related to the other important electromagnetic properties by

$$Z_0 = \frac{E}{H} = \mu_0 c_0 = \sqrt{\frac{\mu_0}{\varepsilon_0}} = \frac{1}{\varepsilon_0 c_0}$$

But in other materials, with $\varepsilon\ne\varepsilon_0$ and/or $\mu \ne \mu_0$, the material impedance is different, and thus the ratio $\frac{|{\bf E}|}{|{\bf H}|}$ is different.

You'll also find that this impedance difference between materials is very important to determining the reflection characteristics at a boundary between materials.

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    $\begingroup$ Of course 1 kg is not equal to 1 m. 1 kg is equal to $7.42 \times 10^{-28}$ m. $\endgroup$ Aug 4, 2020 at 1:13
  • $\begingroup$ @MichaelSeifert, I have reworded to more explicitly acknowledge the possibility of using a unit system that allows comparing the fields. $\endgroup$
    – The Photon
    Aug 4, 2020 at 1:43
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E and B are equal in free space in Gaussian or Natural units. Their numerical values are not equal in SI units because different units are used for E and B. It's like saying a plot of ground 5,280 feet by 1 mile is not square.

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The phrase "because on all the diagrams representing an electromagnetic wave these two amplitudes seem to be equal." contains an answer. Is there any device to measure amplitude of an electromagnetic wave? No. One should try to solve Maxwell equations and try to find solutions. The solutions provide (perpendicular) oscillation of both magnetic field as well as electric field. The mathematics does not provide/guarantee propagation of waves. Is there any device which observed a single em wave contains both electric field wave as well magnetic field wave. No. Mathematically oscillations of both fields have same amplitude, when we go for graphical representation for oscillations. But, it is not for wave propagations. However, if one field oscillates without propagation, and another field propagates as wave, then both of them have same amplitude (mathematically).

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