Electrodynamics and Relativity

Most problems have moved to:

Additional Problems:    

Problem:

(a)  Write the relativistic equation of motion for a particle of charge q and mass m in an electromagnetic field. Consider these equations for the special case of motion in the x-direction only, in a Lorentz frame that has a constant electric field E pointing in the positive x-direction.

(b)  Show that a particular solution of the equations of motion is given by ,  and show explicitly that the parameter t used to describe the world-line of the charge q is the proper time along this world-line.

Problem:

(a)  Write down the relativistic generalization of Newton’s equations for the motion of a charged particle in an electromagnetic field. (Use a convention where the indices a go from 0 to 3, the metric tensor g_ab is g₀₀=1, g_ii=-1, i=1,2,3 and the coordinate 4-vector is x^a=(x⁰,r) with x⁰=ct.)

(b)  Solve the relativistic Newton’s equations for a particle of charge q and mass m moving in constant E and B fields which are parallel to each other and parallel to the z-axis. Use the initial conditions at t=0, p¹(0)=A=constant, p²(0)=p³(0)=0, x¹(0)=0, and choose x²(0) and x³(0) to obtain the simplest form for the trajectory equations. Also, choose the constants of integration so that the proper time t=0 when t=0. You may give your answers in terms of the proper time, but state the relation between t and t.

Problem:

(a)  A fast electron (kinetic energy = 5´10^-17 Joule) enters a region of space containing a uniform electric field E=1000V/m. The field is parallel to the electron's motion and in a direction such as to decelerate it.   How far does the electron travel before it is brought to rest?

(b)  Work the same problem, except this time assume that the direction of the electric field, E, is initially perpendicular to the direction of motion of the electron.   Find both the work done by the fields on the electron and the equation for the electron's path.

Problem:
At t=0 a particle with mass m and negative charge q leaves the origin with a relativistic velocity It moves in a region with    When does it return to the origin?  What is its maximum distance from the origin? Neglect radiation.