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Sorry if this is a naive question but I've been struggling in trying to proof this for a week. Consider an electromagnetic wave with wave vector $\vec{k}=k\hat{n}$, the Maxwell stress tensor can be written as $T_{ij}=-un_in_j$, where $u$ is the energy density of the electromagnetic wave and it's independent of the wave polarization.

For simplicity, let's consider a monochromatic wave with arbitrary polarization. In the orthonormal base ($\hat{e_1},\hat{e_2},\hat{n}$), the fields can be expressed as:

$\vec{E}=(E_1\hat{e_1}+E_2\hat{e_2})e^{i(\vec{k}\cdot\vec{r}-\omega t)}$

with $\vec{B}=(\hat{n}\times\vec{E})/c$, so:

$\vec{B}=\frac{1}{c}(E_1\hat{e_2}-E_2\hat{e_1})e^{i(\vec{k}\cdot\vec{r}-\omega t)}$

The average energy density is defined as:

$u=\frac{\epsilon_0}{4}(\vec{E}\cdot\vec{E}^*+c^2\vec{B}\cdot\vec{B}^*)$

Using the above relations, we get:

$u=\frac{\epsilon_0}{2}(E_1^2+E_2^2)$

On the other hand, the Maxwell stress tensor is defined as:

$T_{ij}=\epsilon_0[E_iE_j+c^2B_iB_j-\frac{1}{2}\delta_{ij}(E^2+c^2B^2)]$

However, I have no idea how to express the Maxwell stress tensor as required, since I can't get rid of the polarization or can't properly express the components in the correct base. Thus my main question is how to properly express an reduce the tensor components (one example will be enough to get the idea).

For example, let's consider the term

$T_{xy}=\epsilon_0(E_xE_y+c^2B_xB_y)=\frac{\epsilon_0 (E_1^2+E_2^2)}{2} e^{2i(\vec{k}\cdot\vec{r}-\omega t)}\left (\frac{(E_1\hat{e_1}(\hat{x})+E_2\hat{e_2}(\hat{x}))((E_1\hat{e_1}(\hat{y})+E_2\hat{e_2}(\hat{y}))}{(E_1^2+E_2^2)}-\frac{((E_1\hat{e_2}(\hat{x})-E_2\hat{e_1}(\hat{x}))((E_1\hat{e_2}(\hat{y})-E_2\hat{e_1}(\hat{y}))}{(E_1^2+E_2^2)}\right )$

where I don't understand how to express the polarization base in terms of the Cartesian components, or how to get rid of the exponential (since we don't have a complex conjugate product).

If I try the particular case where I align the ($\hat{e_1},\hat{e_2},\hat{n}$) to ($\hat{x},\hat{y},\hat{z}$), I can find the non-zero components as:

$T_{xx}=u\frac{(E_1^2-E_2^2)}{(E_1^2+E_2^2)} \ \ , \ \ $ $T_{xy}=u\frac{(E_1E_2)}{(E_1^2+E_2^2)} \ \ , \ \ $ $T_{yy}=u\frac{(E_2^2-E_1^2)}{(E_1^2+E_2^2)} \ \ , \ \ $ $T_{zz}=u$

where only the $T_{zz}$ satisfies the required answer, while the others don't, so I don't really know how else I can proceed.

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  • $\begingroup$ $u=\frac{\epsilon_0}{4}(\vec{E}\cdot\vec{E}^*+c^2\vec{B}\cdot\vec{B}^*)$ should be $u=\frac{\epsilon_0}{2}(\vec{E}\cdot\vec{E}^*+c^2\vec{B}\cdot\vec{B}^*)$ $\endgroup$ – my2cts Mar 3 at 12:20
  • $\begingroup$ @my2cts For the energy density, you're right, the $1/2$ factor comes from taking the average but it's pointless if I'm not taking the average too of the Maxwell tensor. On the other hand, why did you change the sign in the last term so it vanishes? I copied the formula for the Maxwell stress tensor from Jackson's and it is the one in my original question. $\endgroup$ – Charlie Mar 4 at 3:51
  • $\begingroup$ Really? I checked other books on there's still a positive sign in the last term. For example, here it has the same expression: en.wikipedia.org/wiki/Maxwell_stress_tensor $\endgroup$ – Charlie Mar 4 at 18:38
  • $\begingroup$ A second error is that E and B are complex valued. You start out with the correct notation but then continue, following Jackson I presume, as if the fields are real valued. The stress tensor is correct. $\endgroup$ – my2cts Mar 5 at 0:26

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