Maxwell's Equations: Ampere's Law and Electromagnetic Waves, Assignments of Advanced Physics

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Electromagnetic

waves

Introduction The basic laws of electromagnetism were established in the

nineteenth century. Initially, the electric and magnetic interaction were considered to be independent, each of them governed by its own Gauss-like law. Ampere's law provided the first hint of a linkage, since it acknowledged the fact that moving charges produce magnetic fields. The link between electric and magnetic fields was further clarified by Faraday's induction law. Finally, James Clerck Maxwell, completed the set of laws that fully characterize the electro- magnetic interaction. Maxwell also went a step further: by combining the electromagnetic equations, he was able to show that electric and magnetic fields in free space satisfy a wave equation in which the wave speed happens to be exactly equal to the measured speed of light. The obvious conclusion was that light is nothing but an electromagnetic wave. In this chapter, we will explore this fascinating discovery.

Maxwell’s equations Maxwell's equations in integral form

When Maxwell started his work on electromagnetism, the known field equations were Gauss’ laws for the electric and magnetic fields, Ampère’s law, and Faraday’s induction law. You probably know these equations in integral form. Gauss’ law for the electric field states that

∫^ E^ ⋅^ d S^ =

q

S ε 0

where E is the electric field, q the electric charge enclosed by the surface S and ε 0 = 8.854 × 10 -12^ N -1^ m -2^ C 2 is the vacuum permittivity. Since there are no magnetic charges, Gauss’s law for the magnetic field is

∫^ B^ ⋅^ d S = S

Ampère’s law relates any current I with the magnetic field B it creates:

∫^ B^ ⋅^ d l^ = I L

exactly equal to the current that leaves the volume. Thus the net current is zero. Imagine, however, that the above surface encloses one of the plates of a capacitor. If the net current flowing across the surface is zero, this means that the net charge enclosed by the surface is constant. In other words, the charge on a capacitor plate should remain constant. This, of course, is not true. Maxwell figured out an ad hoc way of fixing this problem. He reasoned that for the case of a closed surface the inconsistency is removed if the right-hand side of Eq. (3) is replaced by μ 0 ( I + d q /d t ), where q is the charge enclosed by the volume. When the line shrinks to a point, we get I + d q /d t = 0, or I = - d q /d t. This is obviously true, for the net current leaving a volume is by definition the rate of change of the charge inside the volume. Using Gauss’ law to express the charge in terms of a surface integral of the electric field, Maxwell proposed that Ampère’s law be modified to

B ⋅ = + ⋅

∫ d^ ∫ 

d

d

l μ 0 I ε 0 E d S

L t S

For the case of a closed surface, Maxwell’s modification of Ampère’s law must be correct. But he also assumed that the new expression would be valid even in cases when the surface is not closed. Of course, this would have to be verified by experiment. For cases where the electric field does not depend on time, such as in DC circuits, the new term added by Maxwell is zero and we recover the standard Ampère’s law. Eqs. (1), (2), (4), and (140) are known, together, as Maxwell equations. They characterize the electromagnetic fields completely. They also lead to the wave equation for the electromagnetic field. However, the wave equation is a differential equation. To show that it follows from Maxwell’s equations, we must first rewrite Maxwell’s equations in differential form.

Maxwell’s equations in differential form

Let us first consider Gauss’ law for the electric field. Suppose that we apply Eq. (1) to the infinitesimal volume element in

Fig. (2).

1 1 1 1 1 1 0 0 0 0 0 11110 1111 1111

0000 0000 0000

111111000000

E ( x ) E ( x + d x )

E ( y + d y )

E ( y )

x

y

x

y

Figure 2 Infinitesimal volume on which we apply Gauss’ law for the electric field

The surface integral over the six faces of the cube becomes

S x^ x y y

z z

E x x E x y z

E y y E y x z

E z z E z x y

q

∫ =^ [^ +^ − ]

• (^) [ + − ]

• (^) [ + − ]

d d d

d d d

d d d =

where q is the charge inside the cube. This charge can be written q = ρ d V = ρ d x d y d z. On the other hand, the terms inside the square brackets can be written in terms of the derivatives of the field, so that we obtain

E

x

E

y

E

z

x + y^ + z =

0

This is Gauss’ law in differential form. We can now define a special vector that simplifies the notation. Our “vector“, denoted as ∇∇∇∇, is defined, in terms of its components, as

where the vector j represents the current density. Its magnitude is the current per unit area at a given point, and its direction indicates the direction of the current.

Wave solutions to

Maxwell equations

Solution to Maxwell’s equations in free space

Let us consider Maxwell’s equations in free space, where the charge density ρ as well as the current density j are zero. Let us assume that the only non-zero component of the electric field lyes along the y -axis, that is, E = (0, E, 0) and that the magnetic field has a single component along the z -axis, B = (0, 0, B ). If you don’t see any reason for making these assumptions, it’s because we are cheating: since the solution to Maxwell’s equations will have this form, we will simplify the math a lot by making the assumptions at the outset. However, our solution is no less rigorous: once we find a solution - by whatever dishonest means - we know it must be the solution. In advanced courses you will learn how to derive the solutions to Maxwell’s equations under the most general conditions.

E

B

Y

X

Z Figure 4 Orientation of the electric and magnetic fields for the proposed solution to Maxwell’s equations in free space. Figure 4 shows our electric and magnetic fields, which due to our choice are perpendicular to each other. Notice that although the electric field only has a component along the y - axis, this component will in general be a function of the three coordinates x , y , and z and the time, that is, E =E ( x , y ,z, t ). Similarly, although the orientation of the electric field

is along the z -axis, this component will in general be a function of x , y , z, and t : B = B ( x,y,z,t ). We will now apply the four Maxwell equations to these fields.

A) GAUSS LAW FOR THE ELECTRIC FIELD

 ⋅ (^ )

E

x y z

E

E

y

since the charge density in free space is zero.

B) GAUSS LAW FOR THE MAGNETIC FIELD

 ⋅ (^ )

B

x y z

B

B

z

C) FARADAY’S LAW

∇ × = = − +

E

u u u

u u

u

x y z

x y

z

x y z

E

E

z

E

y

B

t

where u (^) x , uy and u (^) z are unit vectors along the directions of the three axes. Notice that Faraday’s law is a vector equation. It actually contains three scalar equations, one for each component. Equating the different components (and not including the trivial equation 0 = 0 for the y- direction), we obtain

2 2 0 0

2 2

B

x

B

t

Similarly, by taking the x -derivative of the first equation in (20) and comparing with the t- derivative of the second equation, we find

2 2 0 0

2 2

E

x

E

t

Thus the electric field and the magnetic field both satisfy a wave equation! The wave speed in this equation is c = (μ 0 ε 0 ) -1/2^. Intrigued, Maxwell used the known values of ε 0 and μ 0 and computed the speed c. He found

c =

× ⋅ ×

= ×

− − − − − −

0 0

6 2 12

8

m kg C C N m

m/s

2 1 2

This is exactly equal to the speed of light. The conclusion was unavoidable: light is nothing but electromagnetic waves! If confirmed, Maxwell had made one of the most important scientific discoveries of all times. Because he had introduced an ad hoc modification to the electromagnetic equations, he felt that his result had to be confirmed experimentally. The confirmation was provided by Heinrich Hertz.

Electromagnetic waves are transverse waves

Since the electric and magnetic fields satisfy the same wave equation as the mechanical waves we have discussed so far,

we can immediately write the solutions as traveling or standing waves. For example, one possible solution is E = E (^0) sin ( kx- ω t ). A similar solution would be valid for the magnet- ic field. The two solutions are related, because the fields are coupled. For example, if E = E 0 sin ( kx- ω t ), then from Eqs. (20) we obtain B = B 0 sin ( kx - ω t ), with kE 0 = ω B 0. Using ω = ck , we find

E 0 = cB 0 (24)

Notice that the wave propagates in the x -direction, whereas the E -field is in the y -direction and the B -field is in the z- direction. Of course, there is nothing special about our choice of axis. We could have oriented the axis in any arbitrary way. This means that our result is completely general: in electromagnetic waves, the electric and magnetic fields are perpendicular to the direction of propagation. Therefore, electromagnetic waves are transverse waves. The electric and magnetic fields are also mutually perpendicular, as shown in Fig. 6

111111111111111 111111111111111 111111111111111 111111111111111

000000000000000 000000000000000 000000000000000 000000000000000

1111111111111111 1111111111111111 1111111111111111

0000000000000000 0000000000000000 0000000000000000 B = E/c

E

Y

X

Z Figure 6 Schematic diagram of a linearly polarized electromagnetic wave propagating in the x - direction

The direction of the electric field is conventionally regarded as the direction of polarization. For the waves in our solution, the field is said to be linearly polarized because the electric and magnetic fields are in fixed directions. However, you can show that the fields E = [0, E 0 sin ( kx- ω t), E 0 cos ( kx- ω t )] and B = [0, B 0 cos ( kx- ω t), -B 0 sin ( kx- ω t )]

equations would be valid only for a reference frame at rest relative to the ether. But if there is no ether and Maxwell equations are to be valid for all inertial observers, then all these observers should “see” the same wave equation. The only way to make the wave equation the same in all inertial reference frames is to abandon the Galileo transformations that relate the motion of objects seen by different observers. For example, the well-known (and obvious) addition-of- velocities theorem can no longer be valid. This was absurd and unacceptable to most physicists in the late XIXth century. So they concluded there must be something called ether. Hence they devised experiments (the famous Michelson-Morley series) to measure the speed of the Earth relative to the ether. This velocity was found to be zero. In other words, the Earth was found to be the object in the Universe which was at rest with respect to the ether. This idea would have been attractive to the Church officials who “helped” people like Galileo correct his “mistakes,” but it was unacceptable to the physicists of the early 20th century. So they were forced to abandon the standard transformation laws and embrace the so-called Lorentz transformations, which have the property that they make the wave equation for light look the same in all reference frames. The first to propose this revolution was Albert Einstein in a 1905 paper entitled “On the electrodynamics of moving bodies.” You can now understand the reason for the word “electrodynam- ics” in the first paper on relativity theory: relativity is almost a necessity once we accept Maxwell’s equations.

Energy in the

electromagnetic

field

In your previous physics courses, you found that the energy needed to create an electric field is given by

W d

V = 12 ε (^0) ∫ E^2 V , (^) (25)

where the integral over the entire volume occupied by the electric field. To avoid confusion between the notation for electric field and energy, we use here W for energy. We can therefore define an electric energy density E given by

EE = 1 / 2 ε 0 E^2 (26)

Similarly, the energy needed to set up a magnetic field B is given by

W d

V = (^) ∫

2

B V , (27)

so that the magnetic energy density is given by

EB =

2 μ (^0)

B^2

An electromagnetic wave has both electric and magnetic fields, so that its energy density will be the sum of Eq. (26) and Eq. (28). Thus we obtain

E = EE + EB = 1 / 2 ε 0 E^2 + 1 2 μ (^0) B^2 =^ ε 0 E^2 (29)

where we have used c^2 = 1/(ε 0 μ 0 ) and B = E / c. Notice that for an electromagnetic field the magnetic and electric energies are equal. In a previous chapter, we showed that the intensity of a wave, that is, the energy per unit time passing across a unit surface perpendicular to the direction of propagation, is I = c E. Hence the intensity of an electromag- netic wave is given by I = c ε 0 E^2 (30)

Momentum in the

e le ct r o m a gn e t ic

field

E

B

Y

X

Z

q

Figure 7 A charge under the influence of an electromagnetic wave absorbs a net momentum from the fields. Let us now consider a charge q , initially at rest at some

d

d

W

t

= F ⋅ v = q v ⋅ E = qv Ey y

The average of this over a cycle is

d

d

W

t

= q v Ey y

If we note that B (^) z = E (^) y /c and compare Eq. (34) with Eq. (35), we conclude that

F

p

= = u

d

d

d

t c d

W

t

x

Hence the momentum transfer and the energy transfer and directly proportional. In analogy with the energy case, we can define a momentum density P given by P = E / c u (^) x (38)

Here P is the momentum per unit volume contained in an electromagnetic wave.

Why do fields carry

e n e r g y a n d

momentum?

We have computed the energy density and the linear momentum density in electromagnetic waves. We can also show that there is a corresponding angular momentum density associated with these waves. At this point you may wonder what is the meaning of all this. When the concepts of energy and momentum were introduced in previous physics courses, they were associated with particles having a certain mass. However, now we seem to be associating the energy and the momentum with the fields themselves. We say that the wave “carries” energy and momentum. In the case of elastic waves, this is no problem because the waves involve the motion of masses, so that the energy we are talking about is the standard energy of vibrating particles. But there are no “particles” vibrating in an electromagnetic wave, it’s the fields themselves which oscillate. How can the fields “have” energy and momentum? The whole idea is even more disturbing when you remember that the fields were introduced as a mathematical artifact. The “real” thing was

the force between charges, the fields were viewed only as a practical way to compute those forces. When you derived the “energy” of an electromagnetic field as proportional to the integral of E^2 over the volume, this was little more than a mathematical curiosity. It was always clear that you were talking about the standard potential energy of a system of charges. When electromagnetic waves are considered in their proper relativistic context, however, they must carry energy and momentum. The reason why the total momentum of a system of particles is conserved is Newton’s third law. Because any pair of particles exert equal and opposite forces on each other, the momentum gained by one of them is exactly compensated by the momentum lost by the other one. But Newton’s third law is not valid for charges in motion, because the force (the electric and magnetic fields) cannot travel faster than the speed of light. When a charge is suddenly shaken, it takes some time for a second particle to “feel” this displacement. If Newton’s third law is not valid, then momentum cannot not be conserved unless we assume that the missing momentum is “traveling” with the wave. Hence only the sum p particles + p fields can remain constant. To the extent that we believe that energy and momentum are conserved, we must accept the fact that the fields carry these quantities. At this point, the fields cease to be a mathematical artifact. They are as “real” as the particles which generate them.

Radiation We have so far discussed the solution to Maxwell equations

in free space, where ρ = 0 and j = 0. Of course, a possible solution is also E = 0 and B =0, i.e. , no field at all. In free space, this solution is just as good as the electromagnetic waves we proposed above. The reason why the wave is the right solution is that somewhere in space, maybe far from our “free” space, the charge and/or the current is not zero. Thus E and B cannot be zero at those points. On the other hand, if charge and current are zero everywhere , then the zero- field solution is the right solution and there are no waves.

continuous. Thus the field lines must bend and the field will be transverse at the boundary between the “old” and the “new” Coulomb field. This transverse section is nothing but the electromagnetic wave produced by the sudden motion of the charge. If the charge had been moving at constant velocity, it would produce a standard Coulomb electric field plus a magnetic field. The information that the charge is moving would have infinite time to arrive to the observer, and relativity would not be violated. It is only when the velocity is changed, i.e., when the charge is accelerated, that electromagnetic waves are produced. We can thus state Electromagnetic waves are produced by accelerated charges.

Radiation and

Quantum Mechanics

Our conclusion that charges under acceleration radiate electromagnetic waves has a dramatic impact on our understanding of the atomic structure. Experiments suggest that electrons in atoms orbit around the nucleus in much the same way planets orbit around the Sun. But an electron in orbit around the nucleus is under acceleration (centripetal acceleration) and must radiate electromagnetic waves. Because these waves carry energy away from the electron, the electron should rapidly spiral down to the nucleus: the atom should not be stable! This paradox can only be solved with quantum mechanics. According to quantum theory, the electron can be viewed as a wave like the standing waves in a cord. When the electron is in one of the “normal modes” the wave is stationary and no radiation is emitted. When the electron changes from one normal mode to a different one, the electron wave is suddenly modified and radiates electromagnetic waves during the transition. It is ironic that Maxwell equations, the most impressive achievement of classical physics, brought about the demise of the very fundamentals on which they were built. Today we now that the Galilean transformations have to be replaced

by relativistic transformations and that Quantum Mechanics replaces Newtonian Mechanics. Maxwell equations them- selves are the only “surviving” element of classical physics.

Maxwell’s equations

inside a material

Dipole moment and polarization

The dipole moment of a system of charges is defined as

p = (^) ∑ q (^) i r i

i ,^ (39)

where r i is the position vector of charge i. When the system consists of two charges q and -q , Eq.(39) reduces to the familiar expression p = q a , where a is a vector with its origin at the negative charge and its tip at the positive charge. The polarization P of a material is defined as the dipole moment per unit volume. If a certain substance has N atoms or molecules per unit volume, each with a dipole moment p , the polarization is given by P = N p. In this discussion, we will consider materials that do not have a permanent polarization, but become polarized in the presence of an external electric field. We will also assume that the induced polarization P is parallel to the external electric field E. This is expected for homogeneous media and for cubic crystals.

E P

Figure 10 Microscopic picture of the polarization induced in a material by an external electric field. Let us consider any non-conducting material in the presence of an external electric field.The atoms or molecules will be distorted in such a way that a net charge appears at the