# Exercises

1. Consider two Cartesian reference frames, and , in the standard configuration. Suppose that moves with constant velocity with respect to along their common -axis. Demonstrate that the Lorentz transformation between coordinates in the two frames can be written

where . Show that the previous transformation is equivalent to a rotation through an angle , in the - plane, in ( , , , ) space.

2. Show that, in the standard configuration, two successive Lorentz transformations with velocities and are equivalent to a single Lorentz transformation with velocity

3. Let and be the displacement vectors of some particle in the Cartesian reference frames and , respectively. Suppose that frame moves with velocity with respect to frame . Demonstrate that a general Lorentz transformation takes the form

 (1982)

where . If and are the particle's velocities in the two reference frames, respectively, demonstrate that a general velocity transformation is written

4. Let be the Earth's approximately constant orbital speed. Demonstrate that the direction of starlight incident at right-angles to the Earth's instantaneous direction of motion appears slightly shifted in the Earth's instantaneous rest frame by an angle . This effect is known as the abberation of starlight. Estimate the magnitude of (in arc seconds).

5. Let and be the electric and magnetic field, respectively, in some Cartesian reference frame . Likewise, let and be the electric and magnetic field, respectively, in some other Cartesian frame , which moves with velocity with respect to . Demonstrate that the general transformation of fields takes the form

where .

6. A particle of rest mass and charge moves relativistically in a uniform magnetic field of strength . Show that the particle's trajectory is a helix aligned along the direction of the field, and that the particle drifts parallel to the field at a uniform velocity, and gyrates in the plane perpendicular to the field with constant angular velocity

Here, , and is the particle's (constant) speed.

7. Let and . Prove the following statements, assuming that and are not both zero.
1. At any given event, is perpendicular to either in all frames of reference, or in none. Moreover, each of the three relations , , and holds in all frames or in none.

2. If then the field is said to be null. For a null field, is perpendicular to , and , in all frames.
3. If and then there are infinitely many frames (with a common relative direction of motion) in which or , according as or , and none other. Precisely one of these frames moves in the direction , its velocity being or , respectively.

4. If then there are infinitely many frames (with a common direction of motion) in which is parallel to , and none other. Precisely one of these moves in the direction , its velocity being given by the smaller root of the quadratic equation , where , and . In order for to be real we require . Demonstrate that this is always the case.

8. In the rest frame of a conducting medium, the current density satisfies Ohm's law , where is the conductivity, and primes denote quantities in the rest frame.
1. Taking into account the possibility of convection currents, as well as conduction currents, show that the covariant generalization of Ohm's law is

where is the 4-velocity of the medium, the 4-current, and the electromagnetic field tensor.
2. Show that if the medium has a velocity with respect to some inertial frame then the 3-vector current in that frame is

where is the charge density observed in the inertial frame.

9. Consider the relativistically covariant form of Maxwell's equations in the presence of magnetic monopoles. Demonstrate that it is possible to define a proper-4-current

and a pseudo-4-current

where and are the flux and density of electric charges, respectively, whereas and are the flux and density of magnetic monopoles, respectively. Show that the conservation laws for electric charges and magnetic monopoles take the form

respectively. Finally, if is the electromagnetic field tensor, and its dual, show that Maxwell's equations are equivalent to

10. Prove that the electromagnetic energy tensor satisfies the following two identities:

and

where

11. A charge moves in simple harmonic motion along the axis, such that its retarded position is .
1. Show that the instantaneous power radiated per unit solid angle is

where , and is a standard spherical polar coordinate.
2. By time averaging, show that the average power radiated per unit solid angle is

3. Sketch the angular distribution of the radiation for non-relativistic and ultra-relativistic motion.

12. The trajectory of a relativistic particle of charge and rest mass in a uniform magnetic field is a helix aligned with the field. Let the pitch angle of the helix be (so, corresponds to circular motion). By arguments similar to those used for synchrotron radiation, show that an observer far from the charge would detect radiation with a fundamental frequency

where , and that the spectrum would extend up to frequencies of order