How to Express the Angle of Rotation for a Rotated Electric Field

[unscientific]

Homework Statement

Suppose an E-field is rotated by angle ø₂. Express ø₂ in terms of:

Homework Equations

The Attempt at a Solution

I used the rotation matrix, and compared LHS and RHS but it led to nowhere:
$$E'= RE$$
$$\left ( \begin{array}{cc} E_1' sin (ky-ωt+ø_2) \\ 0 \\ E_2' cos (ky-ωt+ø_2) \end{array} \right ) = \left ( \begin{array}{cc} cos ø_2 & 0 & sin ø_2 \\ 0 & 1 & 0 \\ -sin ø_2 & 0 & cos ø_2 \end {array} \right) \left ( \begin{array}{cc} E_1 sin (ky-ωt) \\ 0 \\ E_2 cos (ky-ωt+ø_1) \end{array} \right )$$

 

[vanhees71]

It's a very badly post question, because it's not defined around which axis you should rotate. I guess it's the $y$ axis from what's in the white box. You should send the complete question or ask those who have posed the problem for clarification ;-)).

This said, the behavior of the electric field under rotations is very well defined, because $\vec{E}$ is a vector field (in the sense of 3D Euclidean space),
$$\vec{E}'(t,\vec{r}')=\hat{R} \vec{E}(t,\vec{r})=\hat{R} \vec{E}(t,\hat{R}^{-1} \vec{r}').$$
The emphasis is on field, because you have to consider both the field value and the spatial argument in the transformation rule!

 

[unscientific]

It's a very badly post question, because it's not defined around which axis you should rotate. I guess it's the $y$ axis from what's in the white box. You should send the complete question or ask those who have posed the problem for clarification ;-)).

This said, the behavior of the electric field under rotations is very well defined, because $\vec{E}$ is a vector field (in the sense of 3D Euclidean space),
$$\vec{E}'(t,\vec{r}')=\hat{R} \vec{E}(t,\vec{r})=\hat{R} \vec{E}(t,\hat{R}^{-1} \vec{r}').$$
The emphasis is on field, because you have to consider both the field value and the spatial argument in the transformation rule!

Sorry, I don't really get what you mean. Do you mean a matrix multiplication: $$\vec{E}'(t,\vec{r}')=\hat{R} \vec{E}(t,\vec{r})$$ ?