Independent fields' components in Maxwell's equations

In summary, Maxwell's equations in a source-free, isotropic, linear medium can be rewritten as a set of equations for the transverse and longitudinal field components. By choosing z as the propagation axis, it is possible to express all six field components as functions of just two scalar functions, E_z and H_z. This can be directly proven by combining certain equations to eliminate cross terms and solve for the transverse field components in terms of the longitudinal ones.
  • #1
EmilyRuck
136
6
In a source-free, isotropic, linear medium, Maxwell's equations can be rewritten as follows:

[itex]\nabla \cdot \mathbf{E} = 0[/itex]
[itex]\nabla \cdot \mathbf{H} = 0[/itex]
[itex]\nabla \times \mathbf{E} = -j \omega \mu \mathbf{H}[/itex]
[itex]\nabla \times \mathbf{E} = j \omega \epsilon \mathbf{E}[/itex]

If we are looking for a wave solution, traveling along the [itex]z[/itex] direction, with [itex]k = k_z = \beta[/itex], that means

[itex]\displaystyle \frac{\partial}{\partial z} = e^{-j \beta z}[/itex]

and the above equations (after some steps) become[itex]\displaystyle \frac{\partial E_x}{\partial x} + \frac{\partial E_y}{\partial y} = j \beta E_z[/itex]
[itex]\displaystyle \frac{\partial H_x}{\partial x} + \frac{\partial H_y}{\partial y} = j \beta H_z[/itex]

[itex]E_x = - j \displaystyle \frac{1}{\omega \epsilon} \frac{\partial H_z}{\partial y} + \displaystyle \frac{\beta}{\omega \epsilon} H_y[/itex]
[itex]E_y = - \displaystyle \frac{\beta}{\omega \epsilon} H_x + j \displaystyle \frac{1}{\omega \epsilon} \frac{\partial H_z}{\partial x}[/itex]
[itex]E_z = - j \displaystyle \frac{1}{\omega \epsilon} \frac{\partial H_y}{\partial x} + j \displaystyle \frac{1}{\omega \epsilon} \frac{\partial H_x}{\partial y}[/itex]

[itex]H_x = j \displaystyle \frac{1}{\omega \mu} \frac{\partial E_z}{\partial y} - \displaystyle \frac{\beta}{\omega \mu} E_y[/itex]
[itex]H_y = \displaystyle \frac{\beta}{\omega \mu} E_x - j \displaystyle \frac{1}{\omega \mu} \frac{\partial E_z}{\partial x}[/itex]
[itex]H_z = j \displaystyle \frac{1}{\omega \mu} \frac{\partial E_y}{\partial x} - j \displaystyle \frac{1}{\omega \mu} \frac{\partial E_x}{\partial y}[/itex]

It should be possible to express the transverse field components as functions of the longitudinal field components:

[itex]E_x = E_x (E_z, H_z)[/itex]
[itex]E_y = E_y (E_z, H_z)[/itex]
[itex]H_x = H_x (E_z, H_z)[/itex]
[itex]H_y = H_y (E_z, H_z)[/itex]

Which is equivalent to state that just two scalar functions [itex]E_z = f(x,y)e^{-j \beta z}[/itex] and [itex]H_z = g(x,y)e^{-j \beta z}[/itex] are actually independent. But how could it be proved? It is not evident from the equations I wrote above: they show instead that [itex]E_z, H_z[/itex] appear to be functions of [itex]E_x, E_y, H_x, H_y[/itex].
The only link I could find is http://my.ece.ucsb.edu/York/Bobsclass/201B/W01/potentials.pdf: at the bottom of page 11, it shows that all the fields can be expressed in terms of two scalar functions. But it is not a direct approach, because it uses the Hertz Vector potentials.

Is there a direct approach to prove that all the 6 field components are function of just 2 of them?
 
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  • #2
You chose z as the propagation axis, so the fields along that axis won't behave like those in x and y.
 
  • #3
DuckAmuck said:
You chose z as the propagation axis, so the fields along that axis won't behave like those in x and y.

This is reasonable. But why?
Anyway, there is a direct approach to express [itex]E_x, E_y, H_x, H_y[/itex] as functions of [itex]E_z, H_z[/itex] only.
For example, let's combine the first and the fifth equations in order to cancel [itex]H_y[/itex]: we will obtain

[itex]E_x = -j \displaystyle \frac{1}{k^2 - \beta^2} \left( \omega \mu \frac{\partial H_z}{\partial y} + \beta \frac{\partial E_z}{\partial x} \right)[/itex]

Where [itex]k^2 = \omega^2 \mu \epsilon [/itex]. A similar result for [itex]E_y[/itex] can be obtained from equations 2 and 4. The same is for the magnetic field components.
 

FAQ: Independent fields' components in Maxwell's equations

1. What are independent fields' components in Maxwell's equations?

The independent fields' components in Maxwell's equations refer to the electric and magnetic fields, which are the fundamental quantities used to describe electromagnetic phenomena. These fields are independent of each other, but they are related through Maxwell's equations.

2. What are the four Maxwell's equations?

The four Maxwell's equations are Gauss's law, which relates the electric flux to the enclosed charge; Gauss's law for magnetism, which relates the magnetic flux to the enclosed current; Faraday's law, which states that a changing magnetic field induces an electric field; and Ampere's law, which relates the magnetic field to the current flowing through a closed loop.

3. How are the independent fields' components related in Maxwell's equations?

In Maxwell's equations, the electric and magnetic fields are related through two of the equations: Faraday's law and Ampere's law. These equations show that a changing electric field induces a magnetic field, and a changing magnetic field induces an electric field. This is known as electromagnetic induction.

4. What is the significance of Maxwell's equations in electromagnetism?

Maxwell's equations are the foundation of electromagnetism and are essential for understanding and predicting the behavior of electric and magnetic fields. They have been extensively tested and validated through experiments and are crucial for the development of technology such as electric motors, generators, and communication systems.

5. Are there any other important properties of the independent fields' components in Maxwell's equations?

Yes, there are several other important properties of the independent fields' components in Maxwell's equations. For example, the electric and magnetic fields are transverse waves, meaning they oscillate perpendicular to the direction of propagation. They also have the ability to carry energy and momentum, and they interact with matter to produce various effects, such as the force experienced by charged particles in an electric or magnetic field.

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