How to show that Electric and Magnetic fields are transverse

In summary: I don't know, I'm just asking.I'm not sure, but you might be able to get an equation for it if you try.In summary, the electric and magnetic fields are transverse if and only if the dot product of the plane waves' wave vectors is zero.
  • #1
leonardthecow
36
0

Homework Statement


This isn't necessarily a problem, but a question I have about a certain step taken in showing that the electric and magnetic fields are transverse.

In Jackson, Griffiths, and my professor's written notes, each claims the following. Considering plane wave solutions of the form $$ \textbf{E}(\vec{x}, t) = Re[\vec{E_0}e^{-i(\vec{k} \cdot \vec{x} - \omega t)}] \\ \textbf{B}(\vec{x}, t) = Re[\vec{B_0}e^{-i(\vec{k} \cdot \vec{x} - \omega t)}]$$ since the Maxwell equations demand that the divergences of both E and B are zero, this in turn demands that $$\vec{k} \cdot \textbf{E} = 0 \\ \vec{k} \cdot \textbf{B} = 0.$$

Homework Equations



See above, plus the fact that ##\vec{E_0}## and ##\vec{B_0}## are complex functions.

The Attempt at a Solution


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This has to just be my missing something stupid; I just don't see how the plane wave solutions and the Maxwell equations imply that condition (where the wave vector dotted into the E and B fields is zero). Even doing the divergence out for, say, the x component of the E field, you would have something like $$ (\nabla \cdot \textbf{E})_x = \partial_x ({E_0}_xe^{-i(k_x x - \omega t)}) = \partial_x {E_0}_x - ik_x {E_0}_xe^{-i(k_x x - \omega t)}$$ which, combined with the other components would give you $$ \nabla \cdot \vec{E_0} - i\vec{k} \cdot \textbf{E} = 0 $$ which clearly isn't what any of the textbooks are saying is the case. Is it just that the divergence of the complex function ##\vec{E_0}## is zero? If so, why is that the case? Where am I going wrong here? Thanks!
 
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  • #2
##\vec E_0## is an amplitude, a constant for the plane waves you describe.
 
  • #3
Ah okay, I buy that, thanks! Related question though; ##\vec{E_0}## is defined as $$\vec{E_0}=\textbf{A}_1 + i\textbf{A}_2,$$ where ##\textbf{A}_2## and ##\textbf{A}_2## are in ##\mathbb{R}^3##. In a later proof, my professor makes the claim that $$\vec{k} \cdot \textbf{A}_1 = \vec{k} \cdot \textbf{A}_2 = 0.$$ Now, just by simple substitution into ##\vec{k} \cdot \vec{E_0} = 0## would this not imply only that ##\vec{k} \cdot \textbf{A}_1 = - \vec{k} \cdot \textbf{A}_2##? I don't see why we would assume that both dot products are individually zero.
 
  • #4
Well, if ##\vec k## is real, then ##
\vec{k} \cdot \vec{E_0} = \vec{k} \cdot ( \textbf{A}_1 + i\textbf{A}_2) = 0 + i 0## implies ##\vec{k} \cdot \textbf{A}_1 = \vec{k} \cdot \textbf{A}_2 = 0 ## and you are in business. Is ##\vec k_0## real ? why (or: why not) ?
 

FAQ: How to show that Electric and Magnetic fields are transverse

1. How do you define transverse electric and magnetic fields?

Transverse electric and magnetic fields are defined as waves that propagate perpendicular to each other and to the direction of energy flow. This means that the electric and magnetic fields oscillate in a plane perpendicular to the direction of propagation.

2. What evidence is there to show that electric and magnetic fields are transverse?

There are several pieces of evidence that support the transverse nature of electric and magnetic fields. For example, the polarization of electromagnetic waves, which is the orientation of the electric field, can only occur in a transverse manner. Additionally, the behavior of electromagnetic waves in different media, such as reflection and refraction, also supports their transverse nature.

3. Can you demonstrate the transverse nature of electric and magnetic fields?

Yes, the transverse nature of electric and magnetic fields can be demonstrated through a variety of experiments. One common experiment involves using a polarizing filter and a laser pointer to show the perpendicular orientation of the electric and magnetic fields in an electromagnetic wave. Another demonstration involves using a slinky to simulate the propagation of a transverse wave.

4. How do transverse electric and magnetic fields differ from longitudinal fields?

Transverse electric and magnetic fields differ from longitudinal fields in their orientation and propagation. Transverse fields oscillate perpendicular to the direction of propagation, while longitudinal fields oscillate in the same direction as the wave's propagation. Additionally, longitudinal fields do not exhibit polarization, and their behavior in different media is different from that of transverse fields.

5. What are the implications of electric and magnetic fields being transverse?

The transverse nature of electric and magnetic fields has significant implications in the study and application of electromagnetism. It allows for the development of technologies such as radio, television, and wireless communication. It also plays a crucial role in understanding the behavior of light and other electromagnetic radiation.

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