In a certain reference frame a static uniform electric field **E**_{0} is at an angle
θ_{0 }≠ π/2 to a static,
uniform, magnetic induction **B**_{0}, (**B**_{0 }=** E**_{0}/c), in the
SI system of units.

(a) Determine the magnitude of **E** and **B** in a
reference frame in which the angle between **E **and **B**
is θ.

(b) Is there a reference frame for which **E** and **B** are
perpendicular to each other? If so, what is it?

Solution:

- Concepts:

The Lorentz invariance of the electromagnetic field-strength tensor - Reasoning:

In SI units**E**^{2 }- c^{2}**B**^{2}and (**E∙B**)^{2}are invariant under a Lorentz transformation. - Details of the calculation:

(a)**E**^{2 }- c^{2}**B**^{2}and (**E∙B**)^{2}are the same in every reference frame.

In K we have**E**_{0}**∙B**_{0}_{0}B_{0}cosθ_{0}= (E_{0}^{2}/c)cosθ_{0}.

In K' we have**E**'**∙B**'

Since**E**_{0}^{2 }- c^{2}**B**_{0}^{2}= 0 in K, we have**E**'^{2 }- c^{2}**B**'^{2}= 0 in K', therefore E' = cB'.

We then have**E**'**∙B**'^{2}/c)cosθ.

**E**_{0}**∙B**_{0}**E**'**∙B**' --> E'^{2}= E_{0}^{2}cosθ_{0}/cosθ.

(b) If E and B are perpendicular to each other, then**E∙B**= 0. But**E∙B**is invariant and not zero in K. There is no such frame.