1)
a) The electric field before the Lorentz boost is
and
Using the given Lorentz transformations for a boost along direction, we get
Hence after the boost
b) Before the boost
because the magnetic field is zero. After the boost
c) Before the boost
After the boost
Using the definition of we get
2) Before the boost the plane wave solution are
The plane wave solutions for an observer moving along the direction will also be of the form
because we can see from the given Lorentz transformations that for a boost along x-direction y and z don't mix, but the electric and magnetic fields along y and z direction do change. But since initially
They also remain zero after the boost. The third thing we have to convince oureslves of is that palne waves still remain plane wave, this is because palne waves are translationally invariant along the direction of their propagation, therefore shifting the origin along x-direction does not chnage their form.
a) Using Lorentz transformation the new electric field is
Use and write the new field in terms of new coordinates
Comparing equation (21) with equation (18) we get
b) Again comparing equation (18) and (21) we get
Using
c) Comparing equation (18) and (21) it is quite clear that
Now
d)
From equation (26) we get
and
Hence, the speed of light is same in both the frames. This should have been expected as the fact that speed of light is same in every inertial frame is one of the postulates of special relativity.
Exercise 3. (12p) (Lorentz boosts) The Maxwell equations (7) are invariant under Lorentz transformations. This implies...
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Light, radiant heat (infrared radiation), X rays, and radio waves are all examples of traveling electromagnetic waves. Electromagnetic waves comprise combinations of electric and magnetic fields that are mutually compatible in the sense that the changes in one generate the other. The simplest form of a traveling electromagnetic wave is a plane wave. For a wave traveling in the x direction whose electric field is in the y direction, the electric and magnetic fields are given by Ē = E,...
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