Inhomogeneous Electromagnetic Wave Equation

In summary, the conversation discusses using Maxwell's macroscopic equations to show that the electric field in a medium with zero current and charge density, but with a given polarization, satisfies the non-homogeneous wave equation. The conversation also mentions simplifying the equations by dropping the subscript "f" for J and \rho, and using the identity \vec{\nabla}\times\vec{\nabla}\times\vec{E} = \vec{\nabla}(\vec{\nabla}\cdot\vec{E}) - {\nabla}^2\vec{E} to derive the homogeneous wave equation. The speaker is unsure of their approach and asks for clarification.
  • #1
Warr
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Homework Statement



Consider a medium where [tex]\vec{J_f} = 0[/tex] and [tex]{\rho_f}=0[/tex], but there is a polarization [tex]\vec{P}(\vec{r},t)[/tex]. This polarization is a given function, and not simply proportional to the electric field.

Starting from Maxwell's macroscopic equations, show that the electric field in this medium satisfies the non-homogeneous wave equation:

[tex]{\nabla}^2\vec{E} - \frac{1}{c^2}\frac{{\partial}^2\vec{E}}{{\partial}t^2} = -\frac{1}{{\epsilon}_0}{\vec{\nabla}}(\vec{\nabla}\cdot\vec{P})+{{\mu}_0}\frac{{\partial}^2\vec{P}}{{\partial}t^2}[/tex]

Homework Equations



Maxwell's equations:

[tex]\vec{\nabla}\times\vec{E}=-{\mu}\frac{{\partial}\vec{H}}{{\partial}t}[/tex] ...(1)
[tex]\vec{\nabla}\cdot\vec{E}=\frac{\rho}{\epsilon}[/tex] ...(2)
[tex]\vec{\nabla}\times\vec{H}=\vec{J}+{\epsilon}\frac{{\partial}\vec{E}}{{\partial}t}[/tex] ...(3)
[tex]\vec{\nabla}\cdot\vec{H}=0[/tex] ...(4)

[tex]\vec{D}={{\epsilon}_0}\vec{E} - \vec{P}[/tex] ...(5)

In the notes J and [tex]/rho[/tex] did not have the f subscript, but there was a note saying it was dropped for simplicity. In the problem, the f subscripts were there. I am going to assume they are applying directly to the symbols in the Maxwell equations (that [tex]J = J_f and \rho={\rho}_f[/tex]).

The Attempt at a Solution



My first confusion lies in the fact that when you simplify the Maxwell equations using the two knowns above, you end up with:

[tex]\vec{\nabla}\times\vec{E}=-{\mu}\frac{{\partial}\vec{H}}{{\partial}t}[/tex] ...(1.b)
[tex]\vec{\nabla}\cdot\vec{E}=0[/tex] ...(2.b)
[tex]\vec{\nabla}\times\vec{H}={\epsilon}\frac{{\partial}\vec{E}}{{\partial}t}[/tex] ...(3.b)
[tex]\vec{\nabla}\cdot\vec{H}=0[/tex] ...(4.b)

But using these exact equations and the identity:

[tex]\vec{\nabla}\times\vec{\nabla}\times\vec{E} = \vec{\nabla}(\vec{\nabla}\cdot\vec{E}) - {\nabla}^2\vec{E}[/tex] ...(6),

the homogeneous wave equation is found:

[tex]{\nabla}^2\vec{E} - \frac{1}{c^2}\frac{{\partial}^2\vec{E}}{{\partial}t^2}=0[/tex] ...(7)

So I am kind of confused what I am supposed to be doing here. Did I mess up the assumption and reduce the maxwell equations too much? If not, am I trying to equate the RHS of the inhomogeneous equation to 0? I tried rearrange equation (5) for E, then plug it into the LHS of (1) and got:

[tex]\frac{1}{{\epsilon}_0}{\nabla}^2(\vec{D}-\vec{P})-\frac{1}{{{\epsilon}_0}c^2}\frac{{\partial}^2({\vec{D}-\vec{P}})}{{\partial}t^2}[/tex]

I separated out the terms with D and terms with E in them, using the fact that the operators are linear. And therefore had:

[tex]\frac{1}{{\epsilon}_0}{\nabla}^2\vec{D} - \frac{1}{c^2}\frac{{\partial}^2\vec{D}}{{\partial}t^2}-\frac{1}{{\epsilon}_0}{\nabla}^2\vec{P} + \frac{1}{c^2}\frac{{\partial}^2\vec{P}}{{\partial}t^2}[/tex]and then in the notes, the parts with D equaled 0. It seemed to be a consequence of equation (7), and would just make is so that I had to satisfy the condition: [tex]{\nabla}^2\vec{P}={\vec{\nabla}}(\vec{\nabla}\cdot\vec{P})[/tex]

but I couldn't get that to work out either, using the earlier vector calc identity (equation 6). So basically I am either approaching this wrong, or have made a bad assumption.
 
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  • #2
Still wondering about this problem..Haven't really worked on it, since I'm kinda stumped.

EDIT: Figured out that one of my initial equations had a slight error. Problem solved!
 
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1. What is the inhomogeneous electromagnetic wave equation?

The inhomogeneous electromagnetic wave equation is a partial differential equation that describes the behavior of electromagnetic waves in a medium with varying properties. It relates the electric and magnetic fields of the wave to the sources that are creating the wave.

2. How is the inhomogeneous electromagnetic wave equation different from the homogeneous wave equation?

The homogeneous wave equation assumes that there are no sources present in the medium, while the inhomogeneous wave equation takes into account the effects of sources on the behavior of electromagnetic waves. This makes it a more general and realistic equation for describing electromagnetic wave propagation.

3. What is the significance of the inhomogeneous electromagnetic wave equation in physics?

The inhomogeneous electromagnetic wave equation is a fundamental equation in physics that governs the behavior of electromagnetic waves. It is used in many fields, including optics, electromagnetism, and quantum mechanics, to understand and predict the behavior of electromagnetic waves.

4. How is the inhomogeneous electromagnetic wave equation solved?

The inhomogeneous electromagnetic wave equation can be solved using a variety of methods, such as separation of variables, Fourier transforms, or Green's functions. The specific method used depends on the boundary conditions and sources present in the problem.

5. What are some real-world applications of the inhomogeneous electromagnetic wave equation?

The inhomogeneous electromagnetic wave equation has many practical applications, such as in radio and television broadcasting, radar technology, and medical imaging. It is also used in the design of optical devices, such as lenses and mirrors, and in the study of electromagnetic phenomena, such as diffraction and interference.

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