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Rotation of Electric Field

  1. Jan 10, 2014 #1
    1. The problem statement, all variables and given/known data

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

    r8ce8k.png

    2. Relevant equations



    3. The attempt at a solution

    I used the rotation matrix, and compared LHS and RHS but it led to nowhere:
    [tex] E'= RE [/tex]
    [tex]\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 )[/tex]
     
  2. jcsd
  3. Jan 10, 2014 #2

    vanhees71

    User Avatar
    Science Advisor
    2016 Award

    It's a very badly post question, because it's not defined around which axis you should rotate. I guess it's the [itex]y[/itex] 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 [itex]\vec{E}[/itex] is a vector field (in the sense of 3D Euclidean space),
    [tex]\vec{E}'(t,\vec{r}')=\hat{R} \vec{E}(t,\vec{r})=\hat{R} \vec{E}(t,\hat{R}^{-1} \vec{r}').[/tex]
    The emphasis is on field, because you have to consider both the field value and the spatial argument in the transformation rule!
     
  4. Jan 11, 2014 #3
    Sorry, I don't really get what you mean. Do you mean a matrix multiplication: [tex]\vec{E}'(t,\vec{r}')=\hat{R} \vec{E}(t,\vec{r})[/tex] ?
     
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