Electric & Magnetic fields - inner product

[thehangedman]

I have read that the electric and magnetic fields are always "perpendicular". Is that true? And if so, does that mean the inner product of the two vectors is zero?

E_x * B_x + E_y * B_y + E_z * B_z = 0 ?


Also, is there any special meaning in electrodynamics to the quantity:

| B |^2 - 1/c^2 | E |^2

where | B |^2 = B_x * B_x + B_y * B_y + B_z * B_z
and | E |^2 = E_x * E_x + E_y * E_y + E_z * E_z

Thank you!

 

[Bill_K]

Answer to the first question is no, E and B are perpendicular in a plane wave, but not in general.

Answer to the second question is yes! There are two quantities associated with the electromagnetic field that are invariant under a Lorentz transformation. One is E·B, the other is E² - B². These are Lorentz scalars. Calculate them in one frame and they'll continue to have the same value in every other frame.

 

[chrisbaird]

They are only perpendicular for traveling waves with constant wave fronts, such as plane waves, spherical waves, cylindrical waves, etc. If you place the plates of charged a capacitor near each end of a current-carrying solenoid, the electric field is parallel to the magnetic field.