# Inhomogeneous Electromagnetic Wave Equation

1. Homework Statement

Consider a medium where $$\vec{J_f} = 0$$ and $${\rho_f}=0$$, but there is a polarization $$\vec{P}(\vec{r},t)$$. 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:

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

2. Homework Equations

Maxwell's equations:

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

$$\vec{D}={{\epsilon}_0}\vec{E} - \vec{P}$$ .......(5)

In the notes J and $$/rho$$ 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 $$J = J_f and \rho={\rho}_f$$).

3. 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:

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

But using these exact equations and the identity:

$$\vec{\nabla}\times\vec{\nabla}\times\vec{E} = \vec{\nabla}(\vec{\nabla}\cdot\vec{E}) - {\nabla}^2\vec{E}$$ .......(6),

the homogeneous wave equation is found:

$${\nabla}^2\vec{E} - \frac{1}{c^2}\frac{{\partial}^2\vec{E}}{{\partial}t^2}=0$$ ........(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:

$$\frac{1}{{\epsilon}_0}{\nabla}^2(\vec{D}-\vec{P})-\frac{1}{{{\epsilon}_0}c^2}\frac{{\partial}^2({\vec{D}-\vec{P}})}{{\partial}t^2}$$

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

$$\frac{1}{{\epsilon}_0}{\nabla}^2\vec{D} - \frac{1}{c^2}\frac{{\partial}^2\vec{D}}{{\partial}t^2}-\frac{1}{{\epsilon}_0}{\nabla}^2\vec{P} + \frac{1}{c^2}\frac{{\partial}^2\vec{P}}{{\partial}t^2}$$

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: $${\nabla}^2\vec{P}={\vec{\nabla}}(\vec{\nabla}\cdot\vec{P})$$

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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