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

In summary, the conversation discusses the use of a rotation matrix to determine the angle ø2 by which an E-field is rotated. The behavior of the electric field under rotations is well-defined and can be expressed as \vec{E}'(t,\vec{r}')=\hat{R} \vec{E}(t,\hat{R}^{-1} \vec{r}'), where the emphasis is on the field values and the spatial argument in the transformation rule. However, the specific axis of rotation is not defined in the question, making it difficult to determine the exact angle of rotation. Additional clarification is needed to solve the problem.
  • #1
unscientific
1,734
13

Homework Statement



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

r8ce8k.png


Homework Equations


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]
 
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  • #2
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!
 
  • #3
vanhees71 said:
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!

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] ?
 

1. What is the rotation of an electric field?

The rotation of an electric field refers to the change in direction and magnitude of the electric field vector as a function of position in space. It is a measure of how much the electric field changes as you move around in a given area.

2. How is the rotation of an electric field related to electromagnetic waves?

The rotation of an electric field is an essential component of electromagnetic waves. As the electric field rotates, it generates a magnetic field, and the two fields together propagate through space as an electromagnetic wave.

3. Can the rotation of an electric field be controlled or manipulated?

Yes, the rotation of an electric field can be controlled and manipulated through the use of various devices and materials such as capacitors, inductors, and conductors. By altering the properties of these components, the direction and magnitude of the electric field can be changed.

4. How does the rotation of an electric field affect the behavior of charged particles?

The rotation of an electric field can affect the motion of charged particles, causing them to accelerate or change direction. This phenomenon is the basis for many technologies such as particle accelerators and electric motors.

5. Is the rotation of an electric field affected by the presence of other electric or magnetic fields?

Yes, the rotation of an electric field can be affected by the presence of other electric or magnetic fields. This is known as the principle of superposition, where the combined effect of multiple fields on a charged particle is equal to the sum of the effects of each individual field.

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