Charge in Electric Field and Flashing Satellites

a) Write the relativistic equations of motion for a particle of charge 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.

 (i)

b) Show that a particular solution of the equations of motion is given by and show explicitly that the parameter τ used to describe the worldline of the charge q in equation (1) is the proper time along this worldline.

c) Define the acceleration 4-vector for this motion and show that it has constant magnitude. Draw a space-time (x, ct) diagram showing the worldline (1) and the direction of the acceleration vector at three typical points on the worldline (with τ < 0, τ = 0, τ > 0)

d) Suppose an observer moves along the worldline (1), starting at t = 0 and x = mc²/qE. Also, at τ = 0 she leaves behind a satellite that remains at rest at x = mc²/qE. The satellite emits flashes of light at a rate f that is constant in the satellite’s rest frame. Show that only a finite number mfc/qE of flashes ever reach the observer.

e) Some-time after τ = 0 the observer, always moving along the worldline (1), decides to retrieve the satellite. Show that she cannot wait longer than t = 3mc/4qE or τ = (mc/qE) sinh^-1(3/4) to decide to do so.

Hint: To retrieve it at this limiting time, she must “reach down” to the satellite with the speed of light, bring it back at the speed of light, and wait indefinitely long for its return.

SOLUTION

Starting from a 4-vector potential(ϕ, A), we can obtain equations of motion for a charged particle in the electromagnetic field

(1)

By definition

Therefore (1) becomes

(2)

In this case of one-dimensional motion, where there is only an electric field E and momentum p in the x direction, we obtain

(3)

where

(4)

b) To show that (i) is a solution to (3), we write

Now

so

Since 1- tanh² = 1/cosh², we may rewrite (4)

Differentiating,

verifying (3). To show that τ is the proper time for the particle, we must demonstrate that

From (i)

So

as required.

c) Define the 4-momentum as (ε/c, p, 0, 0), where ε is the energy ε = mc²γ and p is the momentum p = mc βγ. The 4-acceleration is given by (1/m)(dp^μ /dτ)

From (i), x² – c²t² = (mc²/qE)², which defines a hyperbola (see Figure 1.1a).

Figure 1.1a

Figure 1.1b

d) From Figure 1.1b, we see that flashes emitted at a constant frequency f will cross the worldline of the particle until point A, where the trajectory of the satellite is above ct = x. To find the number of flashes, we find the time of intersection of x = mc²/qE and x = ct, i.e., t = mc/qE. The number of flashes is therefore ft = mcf/qE.

e) As shown in Figure 1.1c, we need the intersection of

and

Figure 1.1c

where is found from

Thus,

Therefore,

and

.