Vector Calculus: Find E & B at General Time t

[P3X-018]

Homework Statement

At time t = 0, the vectors $\textbf{E}$ and $\textbf{B}$ are given by $\textbf{E} = \textbf{E}_0$ and $\textbf{B} = \textbf{B}_0$, where te unit vectors, $\textbf{E}_0$ and $\textbf{B}_0$ are fixed and orthogonal. The equations of motion are

$$\frac{\mathrm{d}\textbf{E}}{\mathrm{d}t} = \textbf{E}_0 + \textbf{B}\times\textbf{E}_0$$

$$\frac{\mathrm{d}\textbf{B}}{\mathrm{d}t} = \textbf{B}_0 + \textbf{E}\times\textbf{B}_0$$

Find $\textbf{E}$ and $\textbf{B}$ at a general time t, showing that after a long time the directions of $\textbf{E}$ and $\textbf{B}$ have almost interchanged.

The Attempt at a Solution

Now this looks like a "simple" coupled differential equations, instead of using the formal method to solve te system, I differentiate with respect to time and get

$$\ddot{\textbf{E}} = \dot{\textbf{B}}\times\textbf{E} = \textbf{B}_0(\textbf{E}_0\cdot\textbf{E})+\textbf{B}_0\times\textbf{E}_0$$

And similar equation for $\ddot{\textbf{B}}$ (replace E's with B's). But I don't know what to use this result for, this is again a coupled differential equations. Is there a better way to solve this problem? I want to find E and B as functions of time.
Any hint would be appreciated.

 

[HallsofIvy]

Homework Statement

At time t = 0, the vectors $\textbf{E}$ and $\textbf{B}$ are given by $\textbf{E} = \textbf{E}_0$ and $\textbf{B} = \textbf{B}_0$, where te unit vectors, $\textbf{E}_0$ and $\textbf{B}_0$ are fixed and orthogonal. The equations of motion are

$$\frac{\mathrm{d}\textbf{E}}{\mathrm{d}t} = \textbf{E}_0 + \textbf{B}\times\textbf{E}_0$$

$$\frac{\mathrm{d}\textbf{B}}{\mathrm{d}t} = \textbf{B}_0 + \textbf{E}\times\textbf{B}_0$$

Find $\textbf{E}$ and $\textbf{B}$ at a general time t, showing that after a long time the directions of $\textbf{E}$ and $\textbf{B}$ have almost interchanged.

The Attempt at a Solution

Now this looks like a "simple" coupled differential equations, instead of using the formal method to solve te system, I differentiate with respect to time and get

$$\ddot{\textbf{E}} = \dot{\textbf{B}}\times\textbf{E} = \textbf{B}_0(\textbf{E}_0\cdot\textbf{E})+\textbf{B}_0\times\textbf{E}_0$$

And similar equation for $\ddot{\textbf{B}}$ (replace E's with B's). But I don't know what to use this result for, this is again a coupled differential equations. Is there a better way to solve this problem? I want to find E and B as functions of time.
Any hint would be appreciated.

Well, you have "uncoupled" B and E- that's a start. The equations for the various components of E are still coupled. Go ahead and solve them by "uncoupling" again. How complicated that is will depend upon the constant vectors E₀ and B₀. If they had one or more components 0, that would simplify things a lot.

 

[P3X-018]

Well, you have "uncoupled" B and E- that's a start. The equations for the various components of E are still coupled. Go ahead and solve them by "uncoupling" again. How complicated that is will depend upon the constant vectors E₀ and B₀. If they had one or more components 0, that would simplify things a lot.

So I assume there is no simple "shortcut" to solve this? Because there is no more information then what I wrote in #0, about the start vectors, so I can't assume some components are 0. This seems more like a diff. equation problem than a vector calculus. Ofcourse this problem is relatively easy to solve when only considering it as a system of coupled differential equations. I thought that there would be some "tricks" to solve this another way.