Appendix 14G: Moving Point Charge and Lienard

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Appendix 14G: Moving Point Charge and Lienard–Wiechert Potentials

In Appendix 1B, we wrote down the scalar wave equation (1B.30) and its solution (1B.31) in terms of the integral of the charge density at the retarded time [ρ_V]. For the free space medium,

(14G.1)

[ρ_{V}] = ρ_{V} (t - \frac{R_{S P}}{c}) .

$[ρ_{V}] = ρ_{V} (t - \frac{R_{S P}}{c}) .$

Assuming that a point charge Q is defined as an integral of the volume charge density in the limit the volume shrinks to a point, the solution for the scalar potential can be written as

(14G.2)

Φ_{P} (r, t) = \frac{Q (t - \frac{R_{S P}}{c})}{4 π ε_{0} R_{S P}},

$Φ_{P} (r, t) = \frac{Q (t - \frac{R_{S P}}{c})}{4 π ε_{0} R_{S P}},$

where

(14G.3)

R_{S P} = |r - r^{'}| .

$R_{S P} = |r - r^{'}| .$

In the above, S is the source point where Q(t) is permanently located and P is the field point. This cannot be an isolated, time-varying, single charge because of the conservation of charge requirement and can be one of the charges in a dipole source. A single charge is used here mainly to introduce the concept of retarded time. Even if Q is not moving, we have to use the value of the charge at a retarded time t_r:

(14G.4)

t_{r} = (t - \frac{R_{S P}}{c}) .

$t_{r} = (t - \frac{R_{S P}}{c}) .$

Let us now consider a point charge moving with a velocity v along a specified trajectory as shown in Figure 14G.1

FIGURE 14G.1
Geometry for calculation of Lienard–Wiechert potentials due to a moving point charge Q. S is the retarded position of the charge. R_SP is the directed line segment from the retarded position to the field point.

We wish to compute the scalar and vector potentials at a field point P due to this moving source. Note that the retarded time t_r is governed by the equation

(14G.5)

|r - r^{'} (t_{r})| = R_{S P} (t_{r}) = c (t - t_{r}) .

$|r - r^{'} (t_{r})| = R_{S P} (t_{r}) = c (t - t_{r}) .$

This is a classical problem, solved in many textbooks [1,2], and the solution is given under Lienard–Wiechert potentials, stated below:

(14G.6)

Φ_{P} (r, t) = \frac{1}{4 π ε_{0}} \frac{Q}{R_{S P} (1 - \frac{{\hat{R}}_{S P} \cdot v}{c})}

$Φ_{P} (r, t) = \frac{1}{4 π ε_{0}} \frac{Q}{R_{S P} (1 - \frac{{\hat{R}}_{S P} \cdot v}{c})}$

(14G.7)

A_{P} (r, t) = \frac{μ_{0}}{4 π} \frac{Q v}{R_{S P} (1 - \frac{{\hat{R}}_{S P} \cdot v}{c})} = \frac{v}{c^{2}} Φ_{P} (r, t) .

$A_{P} (r, t) = \frac{μ_{0}}{4 π} \frac{Q v}{R_{S P} (1 - \frac{{\hat{R}}_{S P} \cdot v}{c})} = \frac{v}{c^{2}} Φ_{P} (r, t) .$

The electric and B fields are then obtained using (1.7) and (1.8):

(14G.8)

E_{P} (r, t) = \frac{Q}{4 π ε_{0}} \frac{R_{S P}}{{(R_{S P} \cdot u)}^{3}} [u (c^{2} - v^{2}) + R_{S P} \times (u \times a)],

$E_{P} (r, t) = \frac{Q}{4 π ε_{0}} \frac{R_{S P}}{{(R_{S P} \cdot u)}^{3}} [u (c^{2} - v^{2}) + R_{S P} \times (u \times a)],$

where

(14G.9)

u = (c {\hat{R}}_{S P} - v)

$u = (c {\hat{R}}_{S P} - v)$

and $a = \dot{v}$ $a = \dot{v}$ is the acceleration at the retarded time.

The B field is given by

(14G.10)

B = \frac{1}{c} {\hat{R}}_{S P} \times E .

$B = \frac{1}{c} {\hat{R}}_{S P} \times E .$

The first term in (14G.8) is called the generalized Coulomb field and the second term is called the radiation field [1]. In arriving at these results, there are several subtle points to be considered as explained in detail in [1,2]:

Concept of the extended particle.

You probably have come across several places in the book where we treated electron as a point charge. On the other hand, the electron in some other context is treated in nuclear physics as a distributed charge in a spherical volume of radius r₀ given by

(14G.11) $r_{0} = \frac{e^{2}}{4 π ε_{0} m c^{2}},$ $r_{0} = \frac{e^{2}}{4 π ε_{0} m c^{2}},$

where e is the charge and m is the mass of the electron. In the relativity theory, a rigid dimension for an object is not correct in view of the Lorentz contraction of the length of a moving object. Let us consider the point charge Q as the total charge due to a volume charge density, in the sense

(14G.12) $Q =_{lim V \to 0,} \underset{V}{∭} ρ_{V} d V .$ $Q =_{lim V \to 0,} \underset{V}{∭} ρ_{V} d V .$

The numerator N in the solution given in (1.23) for the retarded potential,

(14G.13) $N = \underset{V}{∭} [ρ_{V}] d V^{'},$ $N = \underset{V}{∭} [ρ_{V}] d V^{'},$

does not represent the total charge Q, when Q is moving with a velocity v. Note that the different source points in the volume V have different retarded times t_r. Thus, Q_extd is the new total charge of the extended particle [1]:

(14G.14) $Q_{e x t d} = \underset{V}{∭} ρ_{V} (r^{'}, t_{r}) d V^{'} = \frac{Q}{(1 - \frac{{\hat{R}}_{S P} \cdot v}{c})} .$ $Q_{e x t d} = \underset{V}{∭} ρ_{V} (r^{'}, t_{r}) d V^{'} = \frac{Q}{(1 - \frac{{\hat{R}}_{S P} \cdot v}{c})} .$

One can interpret (14G.14) as evaluating (14G.13) as though the volume element is distorted due to the Lorentz contraction in the direction of the charge motion.
The differentiation in space and time involved in obtaining the fields from the potentials requires careful evaluations, as given in [1].

Let us now consider the power radiated by the moving charge. The relevant electric field in the radiation zone is the second term in (14G.8) and the radiated power density [1] is

(14G.15) $S_{r a d} = \frac{1}{μ_{0} c} E_{r a d}^{2} {\hat{R}}_{S P} .$ $S_{r a d} = \frac{1}{μ_{0} c} E_{r a d}^{2} {\hat{R}}_{S P} .$

The radiated power can be shown to be [1,2]

(14G.16a) $P_{r a d} = \frac{1}{4 π ε_{0}} \frac{2}{3} \frac{Q^{2}}{c^{3}} γ^{6} (a^{2} - {|\frac{v}{c} \times a|}^{2}),$ $P_{r a d} = \frac{1}{4 π ε_{0}} \frac{2}{3} \frac{Q^{2}}{c^{3}} γ^{6} (a^{2} - {|\frac{v}{c} \times a|}^{2}),$

where

(14G.16b) $γ = \frac{1}{\sqrt{1 - v^{2} / c^{2}}} .$ $γ = \frac{1}{\sqrt{1 - v^{2} / c^{2}}} .$

For nonrelativistic velocities, and when the second term is negligible compared to the first on the right side of (14G.16a), the radiated power is given by the Larmor formula

(14G.17) $P_{r a d} = \frac{1}{4 π ε_{0}} \frac{2}{3} \frac{Q^{2}}{c^{3}} a^{2} .$ $P_{r a d} = \frac{1}{4 π ε_{0}} \frac{2}{3} \frac{Q^{2}}{c^{3}} a^{2} .$

This will be the power radiated by the charge Q, which is oscillating (the trajectory of the particle is harmonic) at the instant its velocity is zero and the acceleration is high.

One can discuss several interesting radiation phenomena [1,2], like (1) braking radiation, or bremsstrahlung (v and a are collinear), and (2) synchrotron radiation (circular motion), based on the analysis of a point charge moving in a specified trajectory, the subject of this appendix.

References

1.Griffiths, D. J., Introduction to Electrodynamics, Prentice-Hall, Englewood Cliffs, NJ, 1981.
2.Panofsky, W. K. H. and Phillips, M., Classical Electricity and Magnetism, 2nd edition, Dover Publications, Mineola, NY, 2005.

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Table of Contents for Appendix 14G: Moving Point Charge and Lienard–Wiechert Potentials

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Table of Contents for
Appendix 14G: Moving Point Charge and Lienard–Wiechert Potentials