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Notes for Phys 4620
Last revised 05/11/
These notes are just intended to give an overview of the major equations covered in class.
1 Electrodynamics
At last we consider time–dependent phenomena in electromagnetism. In doing so we find that the electric and magnetic fields are related (through time–dependence). We will arrive at it though the phenomenon of electromagnetic induction and “Faraday’s law” but to discuss this we need to learn more about how current flows in a wire. Yup, real currents.
1.1 Electromotive Force
The current density within a conductor is empirically proportional to the electric field within the conductor: J = σE (1)
which is the general form of “Ohm’s law”. The proportionality factor σ is the conductivity of the material; its reciprocal ρ = 1/σ is the resistivity of the material. Note, under conditions where charge is flowing the electric field can be non-zero within a (imperfect) conductor. When we discuss the total current I flowing from one electrode to another we find that it is proportional to the potential difference V between them:
V = IR (2)
For steady currents the charge density within the conductor is zero; excess charge must again lie on the surface. A model for the motion of electrons which relates their drift velocity to the electric force which they “feel” (and hence the current density to the electric field in the material) gives
J =
nf λq^2 2 mvthermal
E (3)
where n is the number density of molecules, f is the number of free electrons per molecule, λ is the mean free path for the electrons, q is the electron charge, m is their mass and vthermal is their thermal speed. The power delivered to a resistive material by a current flowing through a potential difference V is P = V I = I^2 R (4)
h R
x
B
v
Figure 1: Simple circuit moves through a magnetic field; an emf is set up by the magnetic force on the moving charges.
Electromotive force is a name given to the line integral of the force f which drives the current around a circuit:
E =
f · dl (5)
For electrostatic conditions where a battery supplies the force which drives the current around a circuit, the potential difference between the terminals of the battery is also equal to the emf of the circuit:
V = −
∫ (^) b
a
E · dl = E (6)
E can be viewed as the work done per unit charge by the source.
1.2 Motional emf
A generator exploits the phenomenon of motional emf which can be understood using the ideas already studied in the course since it only involves steady fields and currents (though the circuit is in motion). A very simple generator is shown in Fig. 1. Here there is a uniform magnetic field and a rectangular loop of wire with resistance R moves through it with speed v, as shown. The emf set up in the wire comes from the magnetic force on the charges in the wire and is given by
E =
fmag · dl = vBh
and it is related to the current in the wire by E = IR. The work done on the charges comes from the force which is dragging the loop. The emf can expressed in a nice way using the magnetic flux through the loop:
Φ =
B · da (7)
Then one can show: E = − dΦ dt
The flux rule 8 is more general than our basic example; it is true for wires of arbitrary shape moving in arbitrary directions in non-uniform magnetic fields. The wire can even change its shape as it moves. We can have cases of motional emf where we can’t use the flux rule. The generator shown in example 7.4 (“homopolar generator”) of the text is a good example. In general a conductor which moves through a B field has eddy currents set up in it which dissipate energy.
The the emf induced in 2 due to changes in current in loop 1 is
E 2 = − dΦ 2 dt
= −M
dI 1 dt
It is also true that changes in current in a loop induce an emf in the same loop:
E = −L dI dt
Regarding this formula, we often say that a changing current in a circuit generates a back–emf in the circuit.
1.5 Energy in Magnetic Fields
When we increase the current in a circuit we are doing work against the back emf which is induced. If we start with zero current in a circuit with self-inductance L and over time build it up to a value I, the work done is W = 12 LI^2 (13) One can show that more generally the work done in setting up a volume current distribution J is
W = (^12)
V
(A · J) dτ (14)
This expression can be rewritten as an integral over all space involving only the B field:
W =
2 μ 0
all space
B^2 dτ (15)
1.6 Maxwell’s Equations
The equations governing the E and B fields before Maxwell were thought to be:
(i) ∇ · E =
ρ (ii) ∇ · B = 0
(iii) ∇ × E = −
∂B
∂t (iv) ∇ × B = μ 0 J
As it turns out, these equations are inconsistent. They may be fixed up by replacing the last of these by
∇ × B = μ 0 J + μ 0 0
∂E
∂t The correction gives the important result that a changing electric field will induce a magnetic field. The Maxwell equations are:
∇ · E =
ρ (16)
∇ · B = 0 (17) ∇ × E = −
∂B
∂t
∇ × B = μ 0 J + μ 0 0
∂E
∂t
Together with the force law, F = q(E + v × B)
they give the entire content of classical physics, at least the part dealing with electrical forces. (Newton’s law of gravity is needed for the gravitational force.)
1.7 Magnetic Charge
There is a symmetry about the Maxwell equations that is upset by the fact that there is no magnetic charge (nor a current of moving magnetic charges). If there were, the Maxwell equations would be
(i) ∇ · E =
ρe (iii) ∇ × E = −μ 0 Jm −
∂B
∂t (ii) ∇ · B = μ 0 ρm (iv) ∇ × B = +μ 0 Je + μ 0 0
∂E
∂t
and both kinds of charges would be conserved:
∇ · Jm = − ∂ρm ∂t ∇ · Je = − ∂ρe ∂t But no one has yet found any magnetic charge, and they’ve looked! It turns out that the existence of magnetic charge would explain why charge is quantized.
1.8 Maxwell’s Equations in Matter
The Maxwell equations 16 –19 also hold in matter but there we want to make a distinction between bound and free charges and currents and in doing so we make use of the new fields D and H. We now have to consider time derivatives of the induced charges and currents. To deal with this we need to introduce the polarization current Jp:
Jp =
∂P
∂t
so that in the Maxwell equations we will substitute for ρ:
ρ = ρf + ρb = ρf = −∇ · P
and for J:
J = Jf + Jb + Jp = Jf + ∇ × M +
∂P
∂t With these substitutions and some algebra we get
∇ · D = ρf (21) ∇ · B = 0 (22) ∇ × E = −
∂B
∂t
∇ × H = Jf +
∂D
∂t
These equations are every bit as true as 16 –19; they only use a division of the charge and current into bound and free parts.
but using the Maxwell equations and some vector identities one can express the rhs completely in terms of the fields, and then one gets:
dW dt
d dt
V
0 E^2 +
μ 0
B^2
μ 0
S
(E × B) · da (30)
which is Poynting’s theorem, the work–energy theorem as formulated in EM. An important new quantity is the Poynting vector,
S ≡
μ 0
(E × B) (31)
whose meaning is that S · da is the energy per time crossing the area element da. If we define the electromagnetic energy density as
uem =
0 E^2 +
μ 0
B^2
and uemch as the energy density of the (massive) mechanical particles, then we can write a local version of the Poynting theorem as
∂ ∂t
(umech + uem) = −∇ · S (33)
2.3 Momentum and Electromagnetism
We can get a theorem similar (in spirit, at least) to the Poynting theorem if we consider the net force acting on the particles contained within a volume V. This is the same as the rate of change of the total momentum of these particles. We start with
F =
V
(E + v × B)ρ dτ =
V
(ρE + J × B) dτ
and then use the Maxwell equations and lots of vector identities to rewrite this in terms of the fields alone. The results is very hairy so some definitions are in order to make the result comprehensible. We introduce the Maxwell stress tensor,
Tij ≡ 0
EiEj −
δij E^2
μ 0
BiBj −
δij B^2
which is a beastie with two indices and will be denoted with a double arrow:
T.
When we dot a vector with
T we get a vector out. Thus we write:
(a ·
T )j =
i=x,y,z
aiTij (∇ ·
T )j =
i=x,y,z
∂xi Tij
With this definition the total force on the charges in V can be expressed as
F =
dpmech dt = − 0 μ 0
d dt
V
S dτ +
S
T · da (35)
Here, the first term on the rhs represents the rate of loss of the momentum contained in fields inside V,
pem = μ 0 0
V
S dτ
The second term represents the flow of momentum across the surface of that volume. We can get a differential form of Eq. 35 with the definition of the electromagnetic momentum density:
p
em = μ 0 0 S (36)
so that if p
mech is the density of the momentum of the (massive) particles, we can write
∂ ∂t
(pmech + pem) = ∇ ·
T (37)
and this gives us an interpretation of
T : −Tij is the momentum in the i direction crossing a surface oriented in the j direction per unit area, per unit time (whew!). We also note that S plays a dual role: It represents a flow of em energy but is also related to the linear momentum density.
2.4 Electromagnetic Angular Momentum
Last but not least we note that we could carry out a similar discussion for the angular momentum of mechanical/em system. One finds that the density of angular momentum contained in the em field is
`em = r × p
em = 0 [r × (E × B)] (38)
3 Electromagnetic Waves
3.1 Review of Waves
A wave is a travelling disturbance that maintains a fixed shape and travels at a constant velocity. Except when it isn’t. Through absorption, the size of a wave can diminish; if the medium is dispersive that parts of the wave having different frequencies will travel at different speeds and the wave will distort; and finally in the case of electromagnetic waves, the waves don’t travel through a physical medium at all, they just... travel! Aside from that, it’s a fine definition for a wave. A wave function of the ideal type to be considered has the form
f (z, t) = g(z ± vt) (39)
that is, the variables only appear as the combination z ± vt. A stretched string is perhaps the clearest example of how the wave equation arises. If f (z, t) is the displacement of the bit of the string at z away from the equilibrium position at time t, one can show that f (z, t) satisfies the equation
∂^2 f ∂z^2
v^2
∂^2 f ∂t^2
where
v =
T
μ
T being the string tension and μ being the linear mass density of the string. Eq. 40 is known as the (classical) wave equation and it admits all solutions of the form
f (z, t) = g(z ± vt)
For the one-dimensional two-region problem, the solution is:
AR =
v 1 − v 2 v 2 + v 1
Ai and AT =
2 v 2 v 2 + v 1
AI (47)
˜f (z, t) = ( ˜A cos θ)ei(kz−ωt)^ xˆ + ( ˜A sin θ)ei(kz−ωt)^ yˆ (48)
3.3 EM Waves in Vacuum
The electric and magnetic fields in vacuum also satisfy a wave equation of the form of Eq. 40. If we start with the Maxwell equations in vacuum:
(i) ∇ · E = 0 (iii) ∇ × E = −
∂B
∂t (ii) ∇ · B = 0 (iv) ∇ × B = μ 0 0
∂E
∂t
and apply the curl operator twice to E (or B) we can substitute time derivatives of these fields and arrive at separate wave equations for the E and B fields:
∇^2 E = μ 0 0
∂^2 E
∂t^2 , ∇^2 B = μ 0 0
∂^2 B
∂t^2
We note that each of these contains three wave equations, i.e. one for each component of E and B. The wave equations 49 give the speed of these waves; it is
v = c =
0 μ 0 = 3. 00 × 10 8 m s (50)
so that the speed of light can be deduced from the measured quantities 0 and μ 0.
As the basis for our study of EM waves we consider waves of a definite frequency which travel in the z direction and have no x or y dependence. These are monochromatic plane waves and they have the general form:
E˜(z, t) = E˜ 0 ei(kz−ωt)^ , B˜(z, t) = B˜ 0 ei(kz−ωt)^ (51)
where the amplitudes ˜E 0 and ˜B 0 are complex-valued constant vectors. When these waves are put into the Maxwell equations we find some restrictions on the (vector) amplitudes. First, ( E˜ 0 )z = ( B˜ 0 )z = 0
so that the directions of E and B are always perpendicular to the direction of propagation; EM waves are transverse. Secondly, the electric and magnetic fields are related via
B˜ 0 = k ω (ˆz × E˜ 0 )
More generally, we can consider a plane wave propagating in a general direction by making k into a vector; for a wave propagating in the direction of k, polarized along direction ˆn, the monochromatic plane waves are
E˜(r, t) = E˜ 0 ei(k·r−ωt)^ nˆ (52)
B˜(r, t) = 1 c E˜ 0 ei(k·r−ωt)(kˆ × nˆ) =^1 c kˆ × E˜ (53)
where ˆn · kˆ = 0. The actual E and B fields for the monochromatic plane wave are
E(r, t) = E 0 cos(k · rωt + δ)ˆn (54)
B(r, t) =
c E 0 cos(k · r − ωt + δ)(ˆk × nˆ) (55)
3.4 Energy and momentum in EM Waves
The energy density u, Poynting vector S and momentum density p for a monochromatic plane wave
are all time dependent; the only sensible thing to consider is the time average of these quantities since the wave frequencies we consider are so large. One uses the fact that the average of a cosine- squared factor is 12 and for a wave propagating in the z direction, one finds:
〈u〉 = 12 0 E 02 〈S〉 = 12 c 0 E^20 ˆz 〈p〉 =
2 c 0 E 02 zˆ (56)
The magnitude of the time-averaged Poynting vector is the intensity of the plane wave:
I ≡ 〈S〉 = 12 c 0 E 02 (57)
I is the average power per unit area carried by the wave. Since radiation carries momentum to a surface it exerts a pressure on the surface. For a perfect absorber the radiation pressure (force per unit area) is
I = 12 0 E 02 =
I
c
3.5 EM Waves in a Linear Medium
In a region with no free charge or no free current the Maxwell equations are
(i) ∇ · D = 0 (iii) ∇ × E = −
∂B
∂t (ii) ∇ · B = 0 (iv) ∇ × H =
∂D
∂t
and if in addition it is a linear homogeneous medium then
D = E H =
μ
B
with and μ constant. Then we arrive at wave equations for E and B just as before but now the wave speed is given by
v =
√μ =^
c n
where
n ≡
μ 0 μ 0
is the index of refraction of the material. Often, μ is close to μ 0 so that n ≈
r. All the previous results for vacuum carry over with and μ replacing 0 and μ 0.
k R
k I
k T q R q T q I
1 2
z E I
B I
E T
B T
E R
B R
Figure 4: EM wave incident on interface is polarized in the plane of the page.
the reflected and transmitted amplitudes in terms of the amplitude of the incoming wave. After a lot of work one arrives at Fresnel’s equations,
E˜ 0 R =
α − β α + β
E^ ˜ 0 I , E˜ 0 T =
α + β
E^ ˜ 0 I , (63)
where α ≡ cos θT cos θI
and β ≡ μ 1 n 2 μ 2 n 1 [One can also get a pair of Fresnel equation for the case where the incoming wave is polarized perpendicular to the plane of the page; Griffiths leaves that as a (hard) exercise.] For the in–the–plane polarization case, there is some intermediate angle θB where there is no reflected wave, i.e. it has zero amplitude. This occurs where α = β, which gives:
sin^2 θB = 1 − β^2 (n 1 /n 2 )^2 − β^2
or tan θB ≈ n 2 n 1
where the second approximate condition is for the typical case where μ 1 ≈ μ 2
3.7 EM Waves in Conductors
In a conductor there is a free current Jf that will enter into the Maxwell equations; if the conductor follows Ohm’s law then we can use Jf = σE. There is also a free charge ρf but one can show that in a good conductor the free charge density quickly dissipates and we can take it to be zero. Combining the equations similar to the vacuum case, one gets modified wave equations
∇^2 E = μ
∂^2 E
∂t^2
∂E
∂t , ∇^2 B = μ
∂^2 B
∂t^2
∂B
∂t
which now contain a term with a single time derivative. The equations still admit plane-wave type solutions: E˜(z, t) = ˜E 0 ei(˜kz−ωt)^ , B˜(z, t) = ˜B 0 ei(˜kz−ωt)^ (66)
but now the wave number ˜k is complex; we will use ˜k = k + iκ. Here, k determines the wavelength of wave but κ gives the length over which the wave is attenuated.
We can solve for k and κ and we get
k = ω
μ 2
[√
( (^) σ ω
] 1 / 2
κ = ω
μ 2
[√
( (^) σ ω
] 1 / 2
the distance that it takes the wave to attenuate in amplitude by 1/e is the skin depth d,
d =
k In a conductor the E and B fields are no longer in phase. The B field lags the E field by an amount φ, δB − δE = φ where φ = tan−^1 (κ/k)
3.8 Frequency Dependence of Permittivity
The discussions so far has dealt with monochromatic plane waves where are idealizations; a real wave is a wave packet of finite length. Such a wave can be treated as a sum of harmonic waves (with different wavelengths) by Fourier analysis. But now we need to realize that the speed light in a medium can depend on the frequency of the wave. If there is such a dependence then we say we are dealing with a dispersive medium. In such a medium a travelling wave will not retain its shape. In this situation one must be very careful about what we mean by wave speed. For a harmonic wave with frequency ω and wavenumber k, the wave velocity is given by
v = ω k
a number which in some cases can be bigger than c! For a wave packet one can show that the “envelope” travels at a speed
vg =
dω dk which gives the speed of the information and energy flow, and generally vg is smaller than c.
One can make a very simple model for the way that the interaction of an EM wave with matter can give rise to a dispersion relation. The model has an electron oscillating in one dimension on a spring with natural frequency ω 0 and a damping force proportional to the velocity with damping constant γ. The frequency of the EM wave is ω. After arriving at an expression for the oscillating dipole for the system, we build up a macro- scopic system by assuming that for each molecule in the substance there are fj electrons for which the natural frequency is ωj and the damping constant is γj ; the number density of molecules is N. When we do this, we get complex–valued polarization for the substance:
P˜ = nq
2 m
j
fj ω^2 j − ω^2 − iγjω
E˜ (68)
which gives a complex dielectric constant. The plane–wave type solutions are actually attenuated waves of the form
E˜(z, t) = ˜E 0 e−κz^ ei(kz−ωt)^ (69)
b
a
y
x
z
Figure 6: Rectangular waveguide which extends along the z axis. The sides are a (along x) and b (along y), with a > b.
we get equations for Ex, Ey, Bx and By :
Ex = i (ω/c)^2 − k^2
k ∂Ez ∂x
Ey = i (ω/c)^2 − k^2
k ∂Ez ∂y
− ω ∂Bz ∂x
Bx = i (ω/c)^2 − k^2
k ∂Bz ∂x
ω c^2
∂Ez ∂y
By = i (ω/c)^2 − k^2
k ∂Bz ∂y
ω c^2
∂Ez ∂x
so that it suffices to specify the longitudinal components Ez and Bz , since these equations will then give the other components. The longitudinal components satisfy the equations [ ∂^2 ∂x^2
∂^2
∂y^2
]
Ez = 0 [ ∂^2 ∂x^2
∂^2
∂y^2
]
Bz = 0
We can solve for the case where Ez = 0, and such a solution is a transverse electric (TE) wave. Likewise, the case where Bz = 0 gives a solution which is a transverse magnetic (TM) wave. The case where both Ez and Bz are zero is called a TEM wave, but this cannot occur for a hollow waveguide.
For a rectangular waveguide whose cross-section has sides a (along x) and b (along y), with a ≥ b, as shown in Fig. 6, we solve for the TE modes; a solution using separation of variables does the trick. If we try Bz (x, y) = X(x)Y (y)
applying the boundary conditions gives
Bz (x, y) = B 0 cos(mπx/a) cos(nπy/b)
where k =
(ω/c)^2 − π^2 [(m/a)^2 + (n/b)^2 ]
and n and m are indices for the particular mode (solution), also called the TEmn mode. For mode mn the frequency must be greater than the cutoff frequency,
ωmn = cπ
(m/a)^2 + (n/b)^2
Interestingly enough, the wave velocities for the various modes are greater than c:
v =
ω k
c √ 1 − (ωmn/ω)^2
but the energy is transported at the group velocity, which is
vg = c
1 − (ωmn/ω)^2
which is < c.
4 Potentials and Fields
In this section we will finally find the fields due to time-dependent charge densities and currents, without the fudges (quasi-static approximation) used earlier. To find out what happens in the general case, we will have to make serious use of the scalar and vector potentials (V and A). Note, in introducing time–dependence (and obtaining the full set of Maxwell equations) we actually ignored what happened to the potentials, so that is our first order of business!
4.1 Scalar and Vector Potentials
Back in electrostatics and magnetostatics we had
E = −∇V B = ∇ × A
but in electrodynamics the first of these is no longer true; we only wrote it down because E had zero curl, but that is no longer true. It is true that B has zero divergence, so the second of these still holds: Even with time dependence we still have a vector potential A such that B = ∇ × A. Faraday’s law tells us that it is the quantity E + ∂A/∂t which has zero curl (not E) so that it is this quantity that we must set equal to the (negative) gradient of some scalar function V. This gives V a new meaning and the new relation between E an the potentials is
E = −∇V −
∂A
∂t
One can write down a couple equations connecting V and A which follow from the Maxwell equations, the simpler of which is
∇^2 V +
∂t
(∇ · A) = −
ρ
but they can be simplified with a particular choice for V and A. The freedom to make such a choice is now discussed!
4.2 Gauge Transformations
The potentials V and A are not uniquely determined by the sources. There is always of choice of these functions of space and time which will give the same fields E and B. If we have potentials V and A, it is always possible to find new potentials V ′^ and A′^ which will also work by the formulae
A′^ = A + ∇λ V ′^ = v −
∂λ ∂t
A reasonable guess as to generalizing 79 for the time-dependent case would be to include tr in the argument of the source function inside the integral. That would give :
V (r) =
4 π 0
ρ(r′, tr)
r
dτ ′^ A(r) = μ 0 4 π
J(r′, tr)
r
dτ ′^ (81)
where we mean that ρ(r′, tr) is the charge density at the point r′^ at the time tr. The potentials in 81 (which are the correct ones!) are called the “retarded potentials”. Some comments: In electrostatics we also had expressions for the fields E and B as integrals over the sources (Coulomb’s law and the Biot–Savart law). But taking those integrals and adding tr to the argument would not be correct because those fields would not satisfy the Maxwell equations. Secondly, one can show that the potentials in 81 do satisfy the Maxwell equations and also the Lorentz condition. But the proof takes some care because of the complicated nature of the integrals. The argument tr contains r and r′^ so taking the space derivatives is tricky. Griffiths shows that the V (r, t) given in 81 does satisfy the condition on V in 77. The proof for A would be similar, and he leaves it for you to show that they satisfy the Lorentz condition!
4.4 The Right Equations for E and B
Starting from the retarded potentials we can get the fields, but the result is not so simple. It is:
E(r, t) =
4 π 0
∫ [
ρ(r′, tr)
r^2
ˆrrr +
ρ˙(r′, tr)
cr
ˆrrr −
J˙(r′, tr)
c^2 r
]
dτ ′^ (82)
B(r, t) =
μ 0 4 π
∫ [
J(r′, tr)
r^2
J˙(r′, tr)
cr
]
× rˆrrdτ ′^ (83)
... two equations which are noteworthy for not having been written down explicitly in the literature until the 60’s.
4.5 The Li´enard–Wiechert Potentials
We close the chapter by using the formulae for the retard potentials to get the potentials from a moving point charge. One might think that a moving point charge should be very simple and indeed we will arrive at a definite formula for the potentials but the derivation of these formulae is tricky because of the singular nature of the charge distribution. The first thing to specify is the (time–dependent) location of the charge q. It is given by the function w(t). At a time t we are getting the “news” of the charge’s location at some earlier time tr. (We are assured that there is only one such time, though one needs to think about that!) At that time the charge was at w(tr) so that the time of travel of the news was |r − w(tr)|/c. But the time of travel is the same as t − tr, so that we have
t − tr = |r − w(tr)|/c (84)
which generally is some equation for tr which can be solved. (Note that this is little bit different
from the previous usage of tr where we were given a point r′^ and then we calculated tr = t − r/c
from it.) We then put things into the formula
V (r, t) =
4 π 0
ρ(r′, tr)
r
dτ ′
which is not as easy as it looks! A careless evaluation of the integral just gives V = q/(4π 0 r) (the
static Coulomb potential but evaluated at the retarded position) but that isn’t correct; there are two ways to see why not. The first way is to set up the charge density correctly with a delta function and do the integral carefully. The steps will be given in a separate handout on the web page; an additional factor arises from the complicated argument of the delta function. The second way to see it is to consider the point charge q as the limiting case of a continuous charge distribution and then observing that we get an additional factor from the geometry of points over which the integral is done. Of course, the additional factor is the same in both cases. The upshot is that the potential we get is
V (r, t) =
4 π 0
qc
rc − rrr · v
We also find (using J(r′, tr) = ρ(r′, tr)v(tr)),
A(r, t) =
μ 0 4 π
qcv
(rc − rrr · v)
v c^2 V (r, t) (86)
Equations 85 and 86 are called the Li´enard–Wiechert potentials for a moving point charge. It is worth pointing out that although there is a correction to the naive delayed potentials the correction factor comes from geometry and not from relativity, and in any case the only element put into the general formulae for the delayed potentials was the idea of a delay time for the “news”. (One can say this is really one manifestation of relativity, but that’s a matter of taste)
4.6 Fields of a Moving Point Charge
We’re not done yet. We would like to get the E and B fields from the moving charge considered in the last section. First, we want to find E from the V and A in 85 and 86. This involves taking a gradient of V and a time derivative of A, but from the complicated dependence of the variables on the retarded time these operation require a lot of care and Griffiths exercises this care over 3 full pages of the book! In the end, the result is
E(r, t) = q 4 π 0
r
(rrr · u)^3
[(c^2 − v^2 )u + rrr × (u × a)] (87)
where the vector u is defined as u ≡ cˆrrr−v and r′, v and a are the position, velocity and acceleration
of the charge at the retarded time. The two terms in 87 To get the B field, (carefully) take the curl of A. The result is
B(r, t) =
c
rrr × E(r, t) (88)
To finish things off, we put these into the Lorentz force law to get the total force on moving charge (charge Q, location r, velocity V) from another moving charge (charge q, location r′, velocity v, acceleration a, where these are all evaluated at the retarded time). It is (drum roll):
F = qQ 4 π 0
r
(rrr · u)^3
[(c^2 − v^2 )u + rrr × (u × a)] +
V
c
[
ˆrrr × [(c^2 − v^2 )u + rrr × (u × a)]
]
This equation —in principle— contains all of classical EM. But it would not have been a good idea to start with this equation; Coulomb’s law is easier to handle. It’s so messy that it’s only good for showing off my ability to put equations into LATEX. And it looks good, doesn’t it?
