EM waves - wave equation derivation

In summary, the wave equation for a plane EM wave can be derived using the maxwell equation and the change in E-field over a rectangular region. The calculation of the flux phi_B can be done by integrating B(x) over the region. Although the equation depends on the spatial variation of E, the change in B is ignored.
  • #1
Nick89
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Hi,

Something has been bothering me about deriving the wave equation for a plane EM wave. We were showed this derivation in class and had to reproduce it but something is not making sense to me...

The derivation is as follows:

Suppose you have a plane EM wave (in a vaccuum) traveling in the x-direction. The E-field is in the y-direction and the B-field in the z-direction.

Take a rectangular loop in the xy-plane with length [itex]dx[/itex] and height [itex]l[/itex].
Using the maxwell equation:
[tex]\oint \vec{E} \cdot \vec{dl} = - \frac{ d \phi_B}{dt}[/tex]
[tex]E(x+dx,t)l - E(x,t)l = -\frac{d}{dt} B(x,t) l dx[/tex]
[tex]\frac{E(x+dx,t) - E(x,t)}{dx} = - \frac{d}{dt} B(x,t)[/tex]
[tex]\frac{ \partial E(x,t)}{\partial x} = - \frac{ \partial B(x,t)}{\partial t}[/tex]

And similar for B.

Now what bothers me is the calculation of [itex]\phi_B[/itex]... I am told that we are assuming B to be constant over the infinitesemal distance dx. However, we are not assuming the same for E? We are using the change in E-field explicitly to arrive at the partial derivative for E, but we ignore it for B, isn't that strange?

Also, similarly, since we are advancing x with dx, shouldn't we advance t with dt also?
I guess ignoring this is a safe 'approximation' to make since dt = dx / c which is even more small...

What is going on here? Why can we do this?
 
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  • #2
About [tex]\phi_B[/tex] you don't have to compute a spatial variation of [tex]B[/tex], so you can take an average (on space) value of [tex]B[/tex]; about dt, you can take it so small that the fields don't vary appreciably during that time; even if a time derivative of fields compare in the equation, its derivation don't use the fact that time varies:
you compute [tex] \oint \vec{E} \cdot \vec{dl} [/tex] and [tex]\phi_B[/tex] at a fixed instant of time, then you write that the first equals the -time variation of the second.
 
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  • #3
I still don't understand why we can ignore the change in B over the distance dx, but we cannot ignore the change in E over the same distance dx..?
 
  • #4
Nick89 said:
I still don't understand why we can ignore the change in B over the distance dx, but we cannot ignore the change in E over the same distance dx..?

If you wouldn't ignore the spatial variation of B, what would you write instead of B(x)? To compute the flux [tex] \phi_B[/tex] you should in that case integrate B(x) over the rectangular region; you know from the average value theorem of math analysis that such an integral equals the area of the region multiplied by the function you want to integrate, computed in an opportune average point P inside that region; call it B(P). This represent an average value of B(x) inside the region. When dx goes to zero, that average point P becomes a point with coordinate x.
 
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  • #5
Yeah that makes sense, but in other words, we are ignoring the change of B over this distance, since otherwise B(x) would not be an average value, it would be something like B(x + 1/2 dx), but since dx is infinitesemal this doesn't make sense...

All I'm trying to say is, why can we safely ignore the change in B when we cannot ignore (we even explicitely use it) the same change in E?
 
  • #6
Nick89 said:
Yeah that makes sense, but in other words, we are ignoring the change of B over this distance, since otherwise B(x) would not be an average value, it would be something like B(x + 1/2 dx), but since dx is infinitesemal this doesn't make sense...

All I'm trying to say is, why can we safely ignore the change in B when we cannot ignore (we even explicitely use it) the same change in E?

Because the equation explicitely depends on the spatial variation of E in the little rectangle, but not on the spatial variation of B. It's quite te same as if you want to compute the force on an object and its kinetic energy in a small interval of time dt: for the force you need to know acceleration, so you need to know how the velocity v varies in that interval of time; for kinetic energy you are not interested in how velocity varies in dt, so you can take an average value of it in the interval dt.
Hope to have clarified, because I don't know how to explain it better than this way. :smile:
 

1. What is the wave equation for electromagnetic waves?

The wave equation for electromagnetic waves is given by:
2E = με ∂2E/∂t2
where E is the electric field, μ is the permeability of the medium, ε is the permittivity of the medium, and t is time.

2. How is the wave equation for electromagnetic waves derived?

The wave equation for electromagnetic waves can be derived from Maxwell's equations, specifically the curl of the electric field equation. By taking the second time derivative of this equation and substituting in the curl of the magnetic field equation, the wave equation can be obtained.

3. What is the significance of the wave equation for electromagnetic waves?

The wave equation for electromagnetic waves is significant because it describes the behavior and propagation of electromagnetic waves in a given medium. It allows us to understand how these waves travel and how they are affected by different properties of the medium.

4. Can the wave equation for electromagnetic waves be applied to all types of waves?

Yes, the wave equation for electromagnetic waves is a general equation that can be applied to all types of waves, as long as the appropriate values for μ and ε are used for the specific medium in which the wave is propagating.

5. How does the wave equation for electromagnetic waves relate to the speed of light?

The wave equation for electromagnetic waves can be used to calculate the speed of light in a vacuum, which is approximately 3.00 x 108 m/s. This is because the wave equation relates the electric and magnetic fields and their propagation speed to the permeability and permittivity of free space, which are both fundamental constants in physics.

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