### Lorentz invariants

#### Problem:

In a certain reference frame a static uniform electric field E0 is at an angle θ0 ≠ π/2 to a static, uniform, magnetic induction B0, (B0 = E0/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 E2 - c2B2 and (E∙B)2 are invariant under a Lorentz transformation.
• Details of the calculation:
(a)  E2 - c2B2 and (E∙B)2 are the same in every reference frame.
In K we have E0∙B0 = E0B0cosθ0 = (E02/c)cosθ0.
In K' we have E'∙B' = E'B'cosθ.
Since E02 - c2B02 = 0 in K, we have  E'2 - c2B'2 = 0 in K', therefore E' = cB'.
We then have E'∙B' = (E'2/c)cosθ.
E0∙B0 = E'∙B' -->  E'2 = E02cosθ0/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.