Electric and Magnetic field Transformation

[fys iks!]

Homework Statement

Considering a rotation around the z-axis, show that the Electric and Magnetic fields fields transform as ordinary vectors.

Homework Equations

The Attempt at a Solution

Could someone please clarify for me I'm not sure what they mean by "transform as an ordinary vector"? If you Google the problem statement the first link to a pdf shows the problem.

Thanks for your time.

 

[grey_earl]

Are you too busy to include a link to the pdf? Sorry, man. I'm too busy to help you.

Seriously, if R(φ) is your rotation matrix describing the rotation around the z axis by an angle φ, then E' = R E and the same for the B field.

 

[fys iks!]

ok i can see that, but what would the rotational matrix look like if is about the z axis?

 

[grey_earl]

Just google for "Rotation matrix z axis", the first link will provide you with an answer ;)

 

[paranormal]

I would first clarify the meaning of "transform as an ordinary vector" in this context. In physics, vectors are quantities that have both magnitude and direction and follow certain transformation rules under rotations. Therefore, transforming as an "ordinary vector" would mean that the electric and magnetic fields follow the standard transformation rules for vectors.

To show this, we can consider a rotation around the z-axis, which can be described by the following rotation matrix:

R = [cosθ -sinθ 0
sinθ cosθ 0
0 0 1]

Where θ is the angle of rotation. Now, let's consider a vector in the x-y plane, represented by the vector [Ex, Ey, 0], where Ex and Ey are the components of the electric field in the x and y directions, respectively. After the rotation, this vector will be transformed to [Ex', Ey', 0]. Similarly, the magnetic field vector [Bx, By, 0] will be transformed to [Bx', By', 0].

Using the rotation matrix, we can write the transformed electric field as:

[Ex' Ey' 0] = [cosθ -sinθ 0
sinθ cosθ 0
0 0 1] [Ex Ey 0]

= [Ex cosθ - Ey sinθ, Ex sinθ + Ey cosθ, 0]

Similarly, the transformed magnetic field can be written as:

[Bx' By' 0] = [cosθ -sinθ 0
sinθ cosθ 0
0 0 1] [Bx By 0]

= [Bx cosθ - By sinθ, Bx sinθ + By cosθ, 0]

We can see that the transformed electric and magnetic fields have the same form as the original fields, with the components being multiplied by the appropriate rotation matrix elements. This shows that the electric and magnetic fields transform as ordinary vectors under rotations around the z-axis.

In conclusion, the electric and magnetic fields transform as ordinary vectors means that they follow the standard transformation rules for vectors under rotations. This can be demonstrated by considering a rotation around the z-axis and showing that the transformed fields have the same form as the original fields, with the components being multiplied by the appropriate rotation matrix elements.