A Brief Overview of Particle Physics
1. Introduction
This article provides some key ideas and equations for the physics of uncharged and charged particles. Uncharged particles are assumed to be the domain of Newtonian mechanics, and charged particles the domain of Maxwellian electrodynamics. The discussion is relativistically rigorous.
2. Newtonian Paradigms.
Newton’s 2^{nd} law (in one dimension) is
F = d(mv)/dt (2_1)
= ma + v(dm/dt).
According to Special Relativity, m is a function of particle speed:
m = gm_{o}, (2_2)
where
g = 1 / (1 – v^{2}/c^{2})^{1/2}. (2_3)
Thus
dm/dt = g^{3}m_{o}va / c^{2}, (2_4)
and Newton’s 2^{nd} law (in one dimension) is
F = g^{3}m_{o}a. (2_5)
Let us call g^{3}m_{o}a the inertial action force. According to Newton’s 3^{rd} law, the corresponsing inertial reaction force is -g^{3}m_{o}a. And as Newton requires, the sum of every action and reaction force is zero.
Although Newton referred to mv as the "quantity of motion," it is now called momentum. A second important quantity, kinetic energy, was originally written as
T = (mv^{2})/2. (2_6)
With the discovery that mass is a function of speed, it is now more correctly written as
T = (m – m_{o})c^{2}. (2_7)
Newton and his contemporaries evidently thought of a particle’s momentum and kinetic energy as being concentrated right at the particle. The space between particles was assumed to be void, and particles attracted each other gravitationally somewhat mysteriously across the void. He deduced that the law of gravitational attraction between two masses is
F = Gm_{1}m_{2} / R^{2}, (2_8)
where G is a constant and R is the separation of the particles.
In addition to gravitational forces, there are so-called contact forces. An example of this force is when a particle is in contact with and acted on by the end of a stretched or compressed spring. Hooke was the first to state the force exerted by a spring upon a contacted particle:
F = -k (DL). (2_9)
Here k is the "spring constant" and DL is the change in length of the stretched/compressed spring from its relaxed length.
2.1 Solving the Newtonian Equation of Motion.
Newton’s 2^{nd} law (Eq. 2_5) is a differential equation:
F = m_{o} d^{2}x/dt^{2} / (1 – (dx/dt)^{2} / c^{2})^{3/2} . (2.1_1)
One can substitute the appropriate force law for F. For example, in the case of a particle on a spring,
-kx = m_{o} d^{2}x/dt^{2} / (1 – (dx/dt)^{2} / c^{2})^{3/2} . (2.1_2)
(In this "equation of motion," it is assumed that the spring is relaxed when the particle is at x = 0.)
The solution of Eq. 2.1_2 can be mathematically challenging. But the particle’s position vs. time can always readily be computed. For example, let us suppose that initially
x(0) = 0, (2..1_3)
dx/dt(0) = ac, where 0 < a < 1, (2.1_4)
g(0) = (1 – dx/dt(0)^{2}/c^{2})^{-1/2}, (2.1_5)
m(0) = g(0) m_{o}, (2.1_6)
d^{2}x/dt^{2}(0) = 0. (2.1_7)
Then we can approximate that
x(dt) = x(0) + v(0)dt + a(0)dt^{2}/2, (2.1_8)
v(dt) = v(0) + a(0)dt, (2.1_9)
g(dt) = (1 – v(dt)^{2}/c^{2})^{-1/2}, (2.1_10)
m(dt) = g(dt) m_{o}, (2.1_11)
a(dt) = (-k x(t)) / (m_{o} g(dt)^{3}). (2.1_12)
The algorithm can be re-applied for successive time epochs until the desired x(t) has been built up.
Fig. 2.1_1 plots x(t) for v(0) = .0001c. Fig. 2.1_2 plots x(t) for v(0) = .99c. It is noteworthy that the motion is periodic in both cases. But in the relativistic case the acceleration is practically zero except at the turning points, where it has large negative and positive values.
Figure 2.1_1
x(t), v(0) = .0001c
Figure 2.1_2
x(t), v(0) = .99c
2.2 Relativistic Sinusoidal Motion.
In Sect. 2.1 the spring force is a given and the particle motion is computed. In Sect. 3 it will be instructive to assume that particle motion is always sinusoidal, and to compute the agent force required to maintain the oscillations.
Let us stipulate that, regardless of its maximum speed, our uncharged particle motion is always
x(t) = A sin(wt). (2.2_1)
The velocity is then
v(t) = wA cos(wt). (2.2_2)
In non-relativistic cases the maximum speed (at the Origin) is by definition much less than c, and Newton’s 2^{nd} law is practically
F = m_{o}a. (2.2_3)
F is sinusoidal.
In relativistic cases where wA, the maximum speed, is nearly c, the required force is non-sinusoidal. From Eq. 2.1_1 we see that the driving force must be
F = --m_{o}w^{2}A sin(wt) / (1 - w^{2}A^{2} cos^{2}(wt) / c^{2})^{3/2}. (2..2_4)
Fig. 2.2_1 plots F(t) for wA = .0001c. Fig. 2.2_2 plots F(t) for wA = .99c.
Figure 2.2_1
F(t), wA = .0001c
Figure 2.2_2
F(t), wA = .99c
Figs. 2.2_3 and 2.2_4 plot the power, Fv, expended by the driving agent.
Figure 2.2_3
P(t), wA = .0001c
Figure 2.2_4
P(t), wA = .99c
Note that despite the non-sinusoidal shape in the relativistic case, the expended power is conservative in the sense that the total work done per oscillation is zero. All of the work expended by the driving agent during parts of a cycle is recouped during other parts of the cycle.
3. Maxwellian Paradigms.
Among other things, Maxwell revolutionized physics by theorizing that (a) electrically charged particles engender electromagnetic fields in the surrounding "empty" space, and (b) charged particles with certain motions generate radiation.
Lorentz showed what force a charged particle experiences in an electromagnetic field. And Abraham and Lorentz showed that, when da/dt is nonzero, a charged particle may experience a force in its own, time-varying field. This force was dubbed the "radiation reaction" force. It is aptly referred to as a "self" force, and is an idea foreign to Newtonian mechanics. (In Newton’s mechanics a particle cannot exert a force on itself.)
Of course according to Newton, if there is a radiation reaction force then there must be a radiation action force. This force augments the Newtonian inertial action force. The relativistically correct formula for the complete agent force on a charged particle is (in one dimension)
F = g^{3}m_{o(total)}a – (q^{2}g^{4}/6pe_{o}c^{3}) (da/dt + 3g^{2}va^{2}/c^{2}). (3_1)
Note that the mass is written as m_{(total)}. This is because electric charge has inertia similar to but in addition to the inertia of the "mechanical" mass in Newton’s 2^{nd} law. As we shall see, the momentum and kinetic energy of moving charge is not local to the charge.
Although the inertial part of the action force is conservative, it is readily shown that the radiation part is not. That is, given a sinusoidally oscillating charged particle, the power per cycle expended to counteract the radiation reaction force is not zero.
The Poynting vector, S, indicates the flux of field energy per unit time at a point in space. Its time integral over a closed surface in space indicates the flow of field energy into/out of the volume enclosed by the surface. It is readily demonstrated that the Integral of F_{RadAct }v dt over an oscillation period equals the spatial Integral of the Poynting vector over an enclosing spherical surface for the same oscillation period. In other words, in the course of an oscillation the Radiation Action force does a net amount of work, and this work precisely equals the amount of field energy per cycle fluxing out through an enclosing surface.
3.1 The Inertial Action Force.
3.1.1 Electromagnetic Mass.
The formula for the Poynting vector is
S = e_{o}c^{2}E X B (3.1.1_1)
where E and B are the electromagnetic field vectors. S / c^{2} is equal to the momentum density in the field. The total momentum in the field of a spherical shell of charge, with radius R and constant speed v << c, is
m_{o(elecmag)}v = q^{2}v / 6pe_{o}Rc^{2}. (3.1.1_2)
This suggests that we define the charge’s "electromagnetic rest mass to be
m_{o(elecmag)} = q^{2} / 6pe_{o}Rc^{2}. (3.1.1_3)
Note that, unlike the momentum of a Newtonian (uncharged) particle, the electromagnetic momentum is distributed in the electromagnetic field; it is not local to the charge.
To the extent a charged particle has both mechanical and electromagnetic mass, it is customary to write that
m_{o(Total)} = m_{o(elecmag)} + m_{o(mech)}. (3.1.1_4)
In general, for a charged particle moving in one dimension,
F_{InertAct} = g^{3}(m_{o(mech)} + m_{o(elecmag)}) a. (3.1.1_5)
3.1.2 Poincare Mass.
The energy in the electric field of a resting spherical shell of charge, of radius R, is
E = q^{2} / 8pe_{o}R. (3.1.2_1)
This might at first seem to suggest that the electromagnetic mass should equal E/c^{2} = q^{2}/8pe_{o}Rc^{2}, and not as specified in Eq. 3.1.1_3. However, the electromagnetic mass derived from the field momentum (Eq. 3.1.1_3) is greater than q^{2}/8pe_{o}Rc^{2} . Indeed
m_{elecmag}c^{2} = E + q^{2}/24pe_{o}R. (3.1.2_2)
Poincare suggested that the extra energy term in Eq. 3.1.2_2 is ascribable to stresses in the charge. For example, consider a spherical shell of charge, q, with constant radius, R. We wish to increase q to q + dq by bringing a small surface increment of charge in from infinity. We deduce that the final field energy will be (q + dq)^{2} / 8pe_{o}R. But simply bringing dq in from infinity is not enough. In order to end up with a shell of uniform charge density, we must compress the original charge in its surface, making a hole for the newly arrived dq. This increases the stress energy in the surface charge, and Poincare concluded that the total stress energy is q^{2}_{ }/ 24pe_{o}R. The charge’s total energy is accordingly its electric field energy plus its stress energy, and this sum equals the electromagnetic mass times c^{2}:
m_{elecmag}c^{2} = E + Stress Energy. (3.1.2_3)
It is of historic note that some of the best minds in physics were puzzled by the inequality of m_{elecmag} and E/c^{2} until Poincare deduced that every distribution of charge has an internal stress energy. We now see that m_{elecmag}c^{2} is actually the sum of a charge’s field energy plus its stress energy:
m_{elecmag}c^{2} = E + m_{Poincare}c^{2}. (3.1.2_4)
3.1.3 Work and Impulse Expended by the Inertial Action Force.
As the inertial action force does positive work, energy pours out into the electromagnetic field, and a much smaller amount increases the Poincare stresses of the length-contracting charge. And as the force does negative work, energy flows back from the field/Poincare stresses to the driving agent. It is readily demonstrated that, for wA<<c, the changes in field energy, attributable to the inertial action force, are primarily changes in the magnetic field energy. (For wA<<c, the changes in Poincare stress energies are negligible.)
The density of the electric and magnetic field energies are:
u_{E} = e_{o}E^{2}/2, (3.1.3_1)
u_{B} = e_{o}c^{2}B^{2}/2. (3.1.3_2)
In the case of charged particles, the total electromagnetic field energy is generally a mixture of inertial and radiative field energy. In this regard it is useful to distinguish the so-called near fields from the far fields. Given an oscillating charge, the flux of field energy outward and inward through an enclosing spherical surface appears to be practically sinusoidal when the surface radius of the spherical surface is on the order of .01l (where l = 2pc/w). This sinusoidal energy flux transitions to a steady outward pulsing for radii of l and greater.
3.2 The Radiation Action Force.
As indicated in Eq. 3_1, the radiation action part of the total action force is
F_{RadAct} = (-q^{2}g^{4} / 6pe_{o}c^{3} )(da/dt + 3g^{2}va^{2}/c^{2}). (3.2_1)
The rate at which the driving agent does work while applying this force is
P_{Rad} = F_{RadAct} v. (3.2_2)
In the near fields the accompanying changes in field energy are dominated by the inertial field energy flux. But in the far fields the pulsing of radiant field energy is dominant and manifestly evident. Fig. 3.2_1 plots the Poynting vector on the y-axis at a value of y = .01l. (The charge motion is sinusoidal along the x-axis with wA = .0001c.) Energy ... largely inertial field energy ... appears to flux in and out sinusoidally at y = .01l.
At y = .1l (Fig. 3.2_2) the flux is a mix of sinusoidal flow in an out, and outward pulsing. The inertial field energy part is fluxing in and out equally, but the radiant field energy is pulsing outward constantly. At y = 10l (Fig. 3.2_3) the flux is practically all outward pulsing.
Figure 3.2_1
Poynting Vector, y = .01l
Figure 3.2_2
Poynting Vector, y = .1l
Figure 3.2_3
Poynting Vector, y = 10l
In Figs. 3.2_1 thru 3.2_3 the values of wA is .0001c (non-relativistic oscillators). Fig. 3.2_4 shows the case where wA = .9c and y = .1l. Note the pulses of energy ... practically all radiant ... at these values. Fig. 3.2_5 shows the same wA at y = 10l. At these distances the fields have "settled down" and the energy flow looks more like the usual radiant energy flow in the far fields.
Figure 3.2_4
Poynting Vector, y = .1l, wA = .9c
Figure 3.2_5
Poynting Vector, y = 10l, wA = .9c
3.3 Energy Conservation
The works done per cycle (a) by the inertial action force, and (b) by the "Poincare force" (which increases/decreases the Poincare mass) are both zero. Consequently the net work done per cycle is attributable solely to the radiation action force. In particular, the work per cycle, expended by a driving agent, is
(3.3_1)
It is easily shown that this W_{Rad} equates to E_{Rad}, the integral of the Poynting vector over an enclosing spherical surface:
(3.3_2)
For computed values, the agreement is quite good for wA up to mildly relativistic values. Theoretically the net energy per oscillation, fluxing out into infinite space, is exactly the same as the work done per cycle by a driving agent. This fluxing energy is of course radiant energy. Table 3.3_1 lists computed values of W_{Rad} and E_{Rad} for a range of wA.
wA |
W_{Rad} |
E_{Rad} |
.0001c |
.01883239 |
.01883239 |
.1c |
1897506E1 |
1897506E1 |
.5c |
2944740E3 |
2944740E3 |
.9c |
6506420E4 |
6506422E4 |
3.4 Computing the Fields of a Charged Particle.
At any time t, the fields of a charged particle can be solved at every point in space except at the point occupied by the particle itself. The solutions are functions of retarded quantities, which generally means quantities from the particle’s past.
Let us again imagine that we have a particle oscillating along the x-axis. And let us say that we wish to determine E and B at some fixed evaluation point (P_{x}, P_{y}) at time t. The retarded time, t_{r}, is by definition the moment in the past when a light signal, originating at the particle, would arrive at (P_{x}, P_{y}) at time t. The retarded position, x_{r}, is what the particle’s x-coordinate was at time t_{r}. Thus
[(P_{x} – x_{r})^{2} + P_{y}^{2}]^{1/2} = c(t – t_{r}). (3.4._1)
As a function of t, the particle’s motion is
x = A sin(wt). (3.4_2)
The particle was at some x_{r} when a light signal, leaving for (P_{x} , P_{y}) would arrive at time t. We can always approximate what t_{r} is by implementing the following algorithm on a computer:
dtmin = 0
dtmax = 5(P_{x}^{2} + P_{y}^{2})^{1/2}
Do
dt = (dtmin + dtmax)/2
t_{r} = t – dt
x_{r} = A sin(wt_{r})
D_{x} = P_{x} - x_{r}
D_{y} = P_{y}
D = (D_{x}^{2} + D_{y}^{2})^{1/2}
if |c dt – D| < 2^{-30} then exit Do
if |c dt – D| > 0 then dtmax = dt
if |c dt – D| < 0 then dtmin = dt
Loop
v_{r} = wA cos(wt_{r})
a_{r} = -w^{2}A sin(wt_{r})
u_{x} = cD_{x} – v_{r}
u_{y }= cD_{y}
Once the utility vector, u, and the retarded quantities have been computed, the field vectors can be solved from
E(P,t) = (q / 4pe_{o}) [D / (D dot u)^{3}] [u(c^{2} – v_{r}^{2}) + D X (u X a_{r})], (3.4_3)
B(P,t) = (1 / c) (D / D) X E(P, t). (3.4_4)
Note that the field vectors depend only upon retarded positions, velocities and accelerations. da_{r}/dt does not play a role, although the solved fields are a mixture of inertial and radiant field energies. Eqs. 3.4_3 and 3.4_4 are relativistically rigorous.
Although we cannot solve for the fields right at a charge, we can solve for them arbitrarily close to a charge. We can then box the charge in, so to speak, and extrapolate the electric field right to the charge. When this is done, the radiation reaction force is readily discernible as part of the total self electric force that the charge experiences in its own, time-varying fields.
3.5 Multiple Charges and Interactive Lorentz Forces.
Up until now we have considered only the fields and driving forces for a single charge. In those cases the driving agent exerted action forces on the charge, and the charge exerted equal and oppositely directed reaction forces upon the driving agent.
In this section we consider a case where there are two discrete charges. Each experiences a Lorentz force in the other charge’s field, in addition to a driving agent force. For example, let us suppose that the charges are held at rest on the x-axis at
x_{A} = 0 (3.5_1)
x_{B} = 1 meter. (3.5_2)
If the charges are both positive then they repel each other. The constraining agent must counteract the repulsive forces if the charges are to remain at rest.
The situation is more interesting if the charges oscillate in tandem. In this case the driving agent must exert three forces on each charge: (1) an inertial action force; (2) a radiation action force; (3) a counteraction to the force the charge experiences in the other charge’s field. It is worth noting that the third force will not in general be the simple, inverse square Coulombic force of charges at rest, particularly when wA ~ c.
Of course the power expended on each charge will now be the sum of the three forces, multiplied by v = wA cos(wt). We can imagine a surface in space, enclosing both oscillating charges. And we can compute S(P, t) at each point on said surface, and at each instant in an oscillation period. When we integrate S over the surface, and over an oscillation time of 2p/w, we find that the net outward energy flux precisely equals the total work per period done by forces (2) and (3) specified above. In brief, the radiation of the multi-charge system equals the radiations that would be emitted by the two charges when isolated from one another, plus the radiation associated with their interaction.
Table 3.5_1 lists W, the total work per cycle expended to drive two 1-Coulomb charges, and E, the outward energy flux per cycle time through an enclosing surface. The amplitude of oscillation is A = 1 meter in every case, and the separation of the charges is always 1 meter. Values of W and E are given for a range of wA. In every case the agreement is quite precise.
Table 3.5_1
wA |
W_{Rad} |
W_{Int} |
W_{Total} |
E_{Total} |
.1c |
32932174 |
32891920 |
75824095 |
75854094 |
.5c |
5886694564 |
5326231311 |
11212925876 |
11212925873 |
.75c |
31729882912 |
15214865509 |
46944748422 |
46944748402 |
.95c |
342581943526 |
19650079784 |
362232023310 |
363250848855 |
W and E Compared, 2 Charges
4. Conclusions.
Thanks to digital computers, it is possible to solve problems that were intractable in Maxwell’s time. Several insights result:
1. The power to drive an uncharged particle at relativistic oscillations is not sinusoidal, but it is conservative in the sense that the net work done per oscillation is zero.
2. Power expended (a) to drive a charge sinusoidally, and (b) to account for changes in the Poincare mass, is also conservative, but the work done by the driving agent is manifest as energy flow out into and in from the charge’s field.
3. In addition to the conservative power, expended to drive a charge sinusoidally, a driving agent must also do a specific amount of positive work each oscillation. This positive work is manifest as pulses of radiant field energy out into infinite space.
4. When multiple charges are forced to oscillate, the driving agent will generally have to do work counteracting the interactive Lorentz forces, as well as the work that must be done to counteract the radiation reaction forces. The total work, expended per cycle, precisely equals the energy flux per cycle away from the charge and into infinite space.