maxwell's equations and electromagnetic waves

32
UNIT 14 EQUAnONS GNETIC WAVES' Structure 14.1 Introduction Objectives 14.2 axw well's Equations Asymmetry in the Fundamental Laws of Electromagnetism Generalisationof AmpEre's Law: Displacement Current Putting Maxwell's Equations Together 14.3 Electromagnetic Waves Tl~e Wave Quation Plane Wave Propagation in Empty Space 14.4 Maxwell's Equations and Plane Wave Propagation in Dielectric Media Maxwell's Equations in Dielectric Media Plane Wave Propagation in Dielectrics 14.5 Energy Carried by Electromagnetic Waves: Poynting's Theorem 14.6 Summary 14.7 Terminal Questions 14.8 Solutions and Answers 14.1 INTRODUCTION At this point of the course, you how the four fundamental laws that govern electric and magnetic phenomena, nameIy Gauss' law for electric fields, Gauss' law for magnetic fields, ArnpCre's law and Faraday's law. All these laws together explain the electric and magnetic interactions that make matter act as it does. Recall that in the introduction to this block we had promised to assemble all that you had learnt in this course into a single set of equations, called Maxwell's equations. This is precisely what we will do in this unit. ' MaxweIl's equations govern the behaviour of electric and magnetic fields everywhere and describe all electromagnetic phenomena. For example, they help us explain why a compass needle points north, why light bends when it enterswater, why thunderstorms occur, why we see aurora in the polar regions, and many other natural phenomena. These equations also form the basis for the operation of a large number of devices in use today, e.g., electric motors, television transmitters and receivers, microwave ovens, telephones, computers, radars, cyclotrons, etc. Understanding Maxwell's equations is, indeed, a truly rewarding experience. Apart from explaining this wealth of phenomena, Maxwell's equations lead us to a fundamental insight into the nature of light and other electromagnetic radiations. You will also share this insight when you study about the nature of electromagnetic waves and their propagation in this unit. In the next unit we shall discuss the reflection and refraction of electromagnetic waves and some of their technological applications. Study guide To be able to get the most out of this unit you should be well aware of the materials of Blocks 1 to 3. Further, it would help you to review Block 2 of the cou'w on 'Oscillations and Waves' (PHE- 02). It will also be useful to keep the Block 1 of the physiq course Mathematical Methods in Physics-I (PHE- 04) at hand for the sake of reference. We have put the complex derivations of some important results in an

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Page 1: Maxwell's equations and electromagnetic waves

UNIT 14 EQUAnONS GNETIC WAVES'

Structure

14.1 Introduction Objectives

14.2 axw well's Equations Asymmetry in the Fundamental Laws of Electromagnetism Generalisation of AmpEre's Law: Displacement Current Putting Maxwell's Equations Together

14.3 Electromagnetic Waves Tl~e Wave Quation Plane Wave Propagation in Empty Space

14.4 Maxwell's Equations and Plane Wave Propagation in Dielectric Media Maxwell's Equations in Dielectric Media Plane Wave Propagation in Dielectrics

14.5 Energy Carried by Electromagnetic Waves: Poynting's Theorem

14.6 Summary

14.7 Terminal Questions

14.8 Solutions and Answers

14.1 INTRODUCTION

At this point of the course, you h o w the four fundamental laws that govern electric and magnetic phenomena, nameIy Gauss' law for electric fields, Gauss' law for magnetic fields, ArnpCre's law and Faraday's law. All these laws together explain the electric and magnetic interactions that make matter act as it does. Recall that in the introduction to this block we had promised to assemble all that you had learnt in this course into a single set of equations, called Maxwell's equations. This is precisely what we will do in this unit.

' MaxweIl's equations govern the behaviour of electric and magnetic fields everywhere and describe all electromagnetic phenomena. For example, they help us explain why a compass needle points north, why light bends when it enterswater, why thunderstorms occur, why we see aurora in the polar regions, and many other natural phenomena. These equations also form the basis for the operation of a large number of devices in use today, e.g., electric motors, television transmitters and receivers, microwave ovens, telephones, computers, radars, cyclotrons, etc. Understanding Maxwell's equations is, indeed, a truly rewarding experience.

Apart from explaining this wealth of phenomena, Maxwell's equations lead us to a fundamental insight into the nature of light and other electromagnetic radiations. You will also share this insight when you study about the nature of electromagnetic waves and their propagation in this unit. In the next unit we shall discuss the reflection and refraction of electromagnetic waves and some of their technological applications.

Study guide

To be able to get the most out of this unit you should be well aware of the materials of Blocks 1 to 3. Further, it would help you to review Block 2 of the cou'w on 'Oscillations and Waves' (PHE-02). It will also be useful to keep the Block 1 of the physiq course Mathematical Methods in Physics-I (PHE-04) at hand for the sake of reference. We have put the complex derivations of some important results in an

Page 2: Maxwell's equations and electromagnetic waves

to this unit. You may like to study the relevant sections in the Appendix whenever we refer to them in the text. This will lead to a clarity of perception and also show you how the results are based on logical mathematical reasoning. You will not he examined for the ~apnte~al give11 ha the Appendix. This is a big unit, However, you should be able to complete Sec. 14.2 in a couple of hours and devote the remaining 6 h to Secs. 14.3 to 14.5.

Objective

After studying this unit you should be able to

explain the symmetry considerations which led to Maxivell's equations

write Maxwell's equations in charge-free and current-free regions, regions containing charges and currents, and in dielectric media . derive electromagnetic wave equations from Maxwell's equations . explain the nature of electroinagnetic waves

apply Maxwell's equations and the electromagnetic wave equations (with plane wave solutions) in vacuum and dielectric media

compute the Poynting vector and the energy carried by electromagnetic waves .

14.2 WELL'S EQUATIONS

Recall all the laws governing electric and magnctic phenomena that you have studied in this course. Which ones amongst them can be thought of as fundamental? Let us try to list these fundamental laws in a table.

Table 14.1: A tentative list &the fuadnmenbl laws governing dccMc and n~agnetic phenomma

S.No.

1 ~ a u s s ' law for the magnctic field

1 1.

Rraday's law of electromogentic induction

Law

Magnctic field lincs closc on themselves, they do not begin or end at my point. (This in~plies tllnt an isolated

Gauss' law for the electric fiold

d@8 I t& ~ , d , - --.,- Changing magnetic field gives dl

rise to electric field I or

Whnt the law snys Mothen~nticnl statement

Charges give rise to electric field; olcctriic field lines begin and end on charges

You may be wondering why we have not put Coulomb's law and the law of Biot and Savart in the table. As you have already studied in Blocks 1 and 3, these laws may be viewed as fundamental only for stationary or slowly moving charges. In fact, Coulomb's law is equivalent to Gauss' law when the flux is not changing, i.e., dis/at = 0. Similarly, Biot-Savart's law follows from Gauss' law for magnetic field and AmpCre's law. Thus, both Coulomb's law and Biot-Savart's law can be obiained from a combination of two of the four laws listed in Table 14.1. All the other-equations that you have studied in this course apply to special situations and are incorporated in the four equations given in Table 14.1.

4.

M n x w ~ * e Equations .nd agnetic Waves

&E. d S - 2 €0

or

V . E - t o

(14.1)

Amfire's law (for steady currenii only) Electric current gives rise to

magnetic field

qC dl - or

V x B - W J

Page 3: Maxwell's equations and electromagnetic waves

Let us now examine the four laws assembled in Table 14.1, together. Do you see some similarities in them? Notice that the left hand sides of the first bvo laws (Eqs. , 14.1 and 14.2) and the last two laws (Eqs. 14.3 and 1 4 4 , respectively, are completely identical except for the interchanging of E and B. In the first two laws we have surface integrals respectively, over closed surfaces. Similarly, in the last two laws we have line integrals around closed loops. These pairs of equations differ only in the interchange of E and B on the left-hand side. So we call say that the left sides of the equations in Table 14.1 are symmetrical in E and B, in pairs.

What about the right-hand side of these laws? These do not seem symmetrical at all. What is the asymmetry in these laws? k t ' u s find out.

14.2.1 Asymmetry in the Fundameanbl Laws of Els&mmagnetkm We can identify two kinds of asymmetry in the right side of these laws. The right-side of Gauss' law for electric fields bas a charge q enclosed by a surface. But Gaus' law for magnetism has zero on the right side. This asymmetry. arises from the following fact: isolated electric charges exist in nature, but there is no evidence so far that isolated magnetic charges exist. Therefore, the enclosed magnetic charge on the right-side of the second law is zero. If and when magnetic monopoles are discovered the right side of this law would be non-zero for any surface enclosing a net magnetic charge. In the same way, the term yo i ( = yo dq/dt )repaese~ting the flow of electric charges appears on the right side of Eq. (14.4). But no similar term (representing a current of magnetic monopoles) appears on the right side of Eq. (14.3). This is one kind of asymmetry in these laws which would be resolved if we knew for sure that magnetic monopoles existed. Current theories of elementary particles suggesting the existence of magnetic monopoles l~ave prornpted an earnest search for them.

There is another asymmetry in these laws. On the right side of Faraday's law (Eq. 14.3) we have the term - dQ?B/dt. Recall that we interpreted this law by saying that changing magnetic fields produce an electric field. We find no similar term in Am@reYs law. Are we missing something? From symmetry considerations, could we suggest the following?

Changing electric fields produce a magnetic field.

This was the line of thought followed by Maxwell. Showing remarkable insight into the symmetry of electric and magnetic phenomena, be introduced the concept of induced magnetic fields and displacement current. Thus, he generalised Am@rels law to arrive at the symmetrical counterpart of Faraday's law. k t us see how this

-was done.

14.22 Generalisation of Amgreys ILnw : Dtplacement Current Let us reconsider Ampere's law for steady cutrents. For mathematical convenience we use its differendal form.

where J is the current density associated with the electric current i. You know that

i - J . dS. Let us see if we can use Eq. (14Ja) for fields that vary with time. If

we take the divergence of both sides of Eq. (145a), we get

v. (vxp) = k ( V - J ) (14.5b)

F~! vecwr h l d ~ , itaa be The left-hand side of Eq. (14.5b) is zero (see the margin remark). This gives us that showuthal V . ( V x A ) - 0 V*J - 0

This equation is true only in the special case of steady currents. Recall the continuity equation that follows from the conservation of electric charge. It tells us that

Page 4: Maxwell's equations and electromagnetic waves

or in integral form Maxwell's Equatiola.. Eledmmagnetic Wu\ -

that is, the divergence or net outflow of J from some region is the rate of decrease of charge contained in the region. It is only for steady currents that V . J = 0. Thus, a term is missing from Eq. (14.5a) for time-varying fields. This term should

be the time derivative of some vector field so that for static fields, the equation would reduce to (14.5a). Perhaps, Maxwell's most important contribution was the determination of this missing term. Maxwell modified Eq. (14.5a) by

aE adding a term po ee -to Eq. (14.5a); which was rewritten in the differential form as

at

The integral form of Eq. (14.6a) can be obtained by integrating both sides of the equation over some open surface S and applying Stokes' theorem. We have given the derivation in the margin alongside. Thus, we obtain the generalisation of Ampdre's law as carried out by Maxwell.

Notice that in writing this equation the minus sign in Faraday's law is replaced by a plus sign. This is dictated by experiment and considerations of symmetry. The factor EO is inserted to express the equation in SI units. Eqs. (14.6a and b) tell us that there are two ways of setting up a magnetic field:

1) by an electric current and

2) by a changing electric field

Eqs. (14.6a and b) give the generalised differential and integral forms of Ampt5re's law. These are also termed the ArnpBre-Maxwell's law. Remember that Maxwell did not derive this law from any empirical considerations. He was motivated by symmetry considerations and he deduced the additional term by requiring Am$re's law to be consistent with the conservation law of electric charge. Since Maxwell's time many experiments including direct measurement of the magnetic field associated with a huge capacitor, have confirmed this remarkable insight of Maxwell. Let us now study Eq. (14.6b) closely and understand its meaning. You can verify that the term EO dCDE/dt has the dimensions of a current. Let us examine this term further.

Displacement current

Although the changing electric flux is not an electric current, it has the same effect as a current in producing magnetic fields. For this reason Maxwell called this term the displacement current and the name has stuck. It is given as

where 4 ( - EO 5) is termed the displawrncnt current density. The word

'displacement' does not have any physical meaning. But the word 'current' is relevant in the sense that the effect of the displacement current cannot be distinguished from that of a real current in producing magnetic fields. So, we can say that a magnetic field can be set up by a conduction current i or by a displacement current id, Thus, we can express Anpire-a ax well's law as

Integrating both sides of &.(14.60) over PO open surface S we get

Q V X B ) . ~ - W & J . ~ S

+ w r o ~ s & ~ . d ~

Applying Stokes' law to the LHS and noting that i = J S ~ . d s

and* - & E , ~ s

we obtain

~ Q E $ l . d - poi+,Ao~-- dl

Page 5: Maxwell's equations and electromagnetic waves

To better understand the role of the displacement current, let us consider a parallel plate capacitor and detMnhe the displacement cumnt in a cqeuit containing the capacitor.

Example 1: Displacement cunrent In a circuit containhg a parallel-plate capacitor.

Let a parallel plate capacitor be charged by a constant current i as shown in Fig. 14.1. Suppose the plates are large in comparison with their separation.

Fig. 14.1 I

Then, there will be an E field only between the plates and to a good approximation, it will be uniform over most of the area of the plates. Under these conditions we shall use Eq(14.n) to determine id for the parallel plate capacitor. Let us apply Eq. (14.7b) to a circuit around the wire leading to the capacitor. Let us construct two "balloon-liken surfacess, and S2 as shown in Fig. 14.1. Note that the wire and hence the free current i penetrates the surface &and the related contour C encircles the wire. The second surface Sz encloses one of the capacitor plates, No free current penetrates this surface but the contour C again encircles the wire. For the surface S, and the related contour C, the right-side of Eq. (14.7%) gives

The displacement current term is zero since E is constant id a wire carrying a constant current. Hence for S,, Eq(14.n) becomes

For the surface S2 with the same contour C,

J J.dS - 0 4

since no free current penetrates S,. However, as free charge is being stored and removed from each capacitor plate, a time varying field E will develop between the plates. The lines of flux of this field will penetrate Sz, thus

Page 6: Maxwell's equations and electromagnetic waves

since the contour C is same for both S1 and S2, it should require that i I id. To k I m ~ ' r Equqtions mud alpl la tic waves

SRQW this, we apply Gauss' law to a pill box volume enclosing a surface of area S which surrounds the positive plate:

where q idthechargo on the plate and S is its area ( - n R~ ) . ~ h u s

and

Hence for the circuit

for either S1 or S2 when both conduction'and displacement currents are taken into account.

Note that the contours bounding the two surfaces are chosen to be the same. If the displacement current term were not present in Eq. (14.7b), we would have had an inconsistency: choosing the two surfaces having the same contour would yield d'fferent results (po i for S1 and zero for S,). Therefore, we can think of the

acement current as 'completing the circuit': where conduction current stops 47 flo ing,'displacemedt current takes over to complete the circuit. This is another way of thinking about continuity equation for charge, i.e,, Eq. (14.5~).

This example of a parallel plate capacitor shows concretely the necessity for the displacement current term in the fourth Maxwell's equation. To get an idea of the importance of displacement current we would like you to work out the bllowing SAQ.

SAQ 1 spend 5 min Obtain the ratio of conduction and displacement current densities

aE J - a E, Jd I w, E = Eo sin a t in copper at a frequency of 1 MHz. Repeat at

for Teflon at 1 MHz. For copper 6 = EO, p = p,-,, a = 5.8 x 10' ~m-' . For Teflon,

e = 2 . 1 ~ 6 , ~ = poando = 3 x 1 0 - ~ ~ m - ' .

It was indeed Maxwell's genius to ;ecogniw that Ampere's law should be modified to reflect the symmetry suggested by ~arau!~'s law. To honour Maxwell, the four complete laws of electromagnetism are given the name Maxwell's equations. Maxwell's equations belong to the category of the fundamental laws of nature. As you have seen, they are not derived from any fundamental precepts by logical reasoning and mathematical calculations. Fundamental laws of nature are generalisations of our h o d e d g e and they are discovered, found or ascertained. We will end this section with a brief oveiview of these equations.

Page 7: Maxwell's equations and electromagnetic waves

14.2.3 Potting Maxwell's Qua tions Together Let us first list these equations.

Table 14.2 : Maxwell's muations

Ampire's law: Electric current and changing electric field give rise to v B - Clo J + Eo - dcPE magnetic field 1 ( l')l$cB.d - w+weo- dl

I Gauss' law: Magnetic field lines close on themselves, no magnetic charges exist

This set of equations, first published by Maxwell in 1864, governs the behaviour of electric and magnetic fields everywhere. These are written for the fields in vacuum, in the presence of electric charges and electric currents. Notice that the lack of symmetry in these equations, with respect to E and B is entirely due to the absence of magnetic charge and its corresponding current. In charge-free regions, the terms containing q and i (or p and 3) are zero and Maxwell's equations take the following form:

Integral from

4 E . a - ' go

dl

Table 14.3: Maxwell's equations in vacuum with no source charges or currents

Differential form

- ' v . E - WI

aB V x E - --

at

S.No.

1.

2.

V . B - 0

What the Equation says

Gauss' law: Charges give rise to electric field, eledric field lines begin and end on charges

Faraday's law: Changing magnetic field gives rise to electric field

4 B . d ~ - 0

Differential Form

You can see that in charge-free regions the symmetry is complete; the electric and magnetic fields appear on an equal footing. The constants eo and appear in Amptre-Maxwell's law due to our choice of units. You could be wondering about the discrepancy in sign. The difference of signs in Eqs. (14.9) and (14.11) or in Eqs. (14.13) and (14.15) is actually due to symmetry: it reflects the complementary way in which electric and magnetic fields give rise to each other.

Integral Form

V . E - 0

aB V x E - -- at

What do Maxwell's equations tell us? In a nutshell, the first two equations (in Table 14.2) tell us that an electric field is set up in two ways: by electric charges and by a variable magnetic field; the last two equations tell us that a magnetic field ,has no sources (there are no magnetic charges) and it is set up by electric currents and a variable electric field. Maxwell's equations also indicate that a variable magnetic field cannot exist without an electric field, and a variable electric field, without a magnetic field. This is why the two fields are not regarded as separate. An electromagaetic field is a single entity. In this manner, Maxwell succeeded in formulating mathematically a unified theory of electricity and magnetism.

The consequences of Maxwell's formulation are legion - all of electrical and radio engineiring is contained in these equations. M e r , the presence of the displacement current term in Eq. (14.15) alongwith Eq. (14.13) implies the existence of . ' electromagnetic waves. This forms the discussion of the next section. But before proceeding further you should apply Maxwell'ls equations to time-varying fields.

J S ~ . d S - 0

dt

(14.12)

(14.13)

Page 8: Maxwell's equations and electromagnetic waves

--

SAQ 2

Maxwell's Eqwtions end F3cdmmrgnefic Waves

Under what conditions do the following time-varying electric and magnetic fields satisfy Maxwell's equations (Eqs. 14,12 to 14.15) ?

Spetrd A 18 mitt

E = kEos in(y-v t ) A

B = iBosin(y-vt)

where Eo and Bo are constants.

14.3 ELECTROMAGNETIC WAVES

As we have said earlier, one of the great successes of Maxwell's equations was that they predicted the existence of electromagnetic waves. They were subsequently discovered by Hertz in 1887. Moreover, at high enough frequencies, these waves were shown to be light waves. Now we know that radiowaves, infrared, visible, ultraviolet, X-rays and gamma rays are all electromagnetic waves differing only in frequency. In this section we will see liow Maxwell's equations lead to the prediction of electromagnetic waves. From Maxwell's equations, we will derive an equation which is just the wave equation and understand its physical meaning.

14.3.1 The Wave Quzlticpn We shall first derive the wave equation from Maxwell's equations in a region of space where there is no charge or current (Eqs, (14.12 to 14.15) given in Table 14.3). As ybu can see, these equations are coupled, first order, partial differential equations. But we can uncouple these equations. For the sake of mathematical convenience, we will use their differential form. Taking the curl of Eqs. (14.13) and (14.15) we get,

and

We now make use of the following vector identity for any vector field F:

V X ( V X F ) = v ( v . F ) - V ' F

Thus, on using Eq. (14.15) alongwith this vector identity, Eq. (14.16a) yields

Since V . E - 0 from Eq. (14.12), we get

Similarly, Eq. (14.16b) alongwith Eq. (14.13) yields

Thus, from Maxwell's equations, we get two uncoupled second order partial differential equations for the time-varying E and B fields in vacuum in the absence of charge or current:

Page 9: Maxwell's equations and electromagnetic waves

Electromagnetism

You should gain complete familiarity with this kind of a mathematical manipulation. So we are giving an exercise for you. Follow the same procedure and obtain second order uncoupled partial differential equations from Maxwell's equations in the presence of electric current only, (with the free charge in the region being zero).

Spend SAQ 3 10min Show that the uncoupled partial differential equations for the E and B components

of an electromagnetic field in charge-free material media are given by

Here p = 0, J = a E and E ~ , p0 have been replaked by the E, p of the medium, in Maxwell's equations (Eqs. 14.8 to 14.11).

Let us now consider Eqs. (14.17a and b) in detail. Recall the classical wave equation

which describes a wave travelling with speed v. You must havestudied this equation in the Physics Courses PHE-02 and PHE- 05. Comparing this with Eqs. (14.17a) and (14.1%) tells us that the latter are wave equations and imply changing electric and magnetic fields which propagate like waves in space. Recall from Maxwell's equations that a changing electric field gives rise to a magnetic field, which itself may be changing with time. Taken together Eqs. (14.17a) and (14.1%) suggest the possibility of self-sustaining travelling electromagneticfields in which a change in the electric field continually gives rise to the changirtg magnetic field, and vice-versa. Thus, electromagnetic waves can be thought of as structures consisturg of electric and hragneticfields that travelji-eely through empty space. Comparing Eqs. (14.17a and b) with Eq. (14.19) gives us the speed of the electromagnetic waves. It is

This is just the speed of light in vacuum! The implication is extremely exciting: light is an electromagnetic wave. This conclusion would not surprise you today. But imagine what a triumph it was in Maxwell's times. Do you recall where EO and came in the theory in the first place? They appeared as constants in Coulomb's law and Biot-Savart law. We can measure them in exper'iments involving charged pith balls, batteries, and wires-xperirnents which have ;<Thing to do with light. And yet, in Maxwell's theory these two are related in a beautifully simple manner to the speed of light! Notice also the crucial role of the displacement current term in AmpBreMaxwell's law. Without this term, the wave equation would not have emerged. Thus, according to Maxwell's equations, empty space supports the propagation of electromagnetic waves at a speed given by Eq. (14.20). We would like you to understand clearly the nature of these waves.

Nature of Electromagnetic Waves

We have suggested above that an electromagnetic wave is constituted of s time-varying electric and.magnetic fields. How do we visualise such a travelling

electromagnetic wave? To do so, let us consider an electromagnetic wave travelling

Page 10: Maxwell's equations and electromagnetic waves

through a region in empty space (i.e., charge-free and current-free region). As the Maxwell'e Equations and Elcdmmagnetic Wavcs

wave passes over it, the magnetic flux through the region will change and according to Faraday's law, induced electric fields will appear in that region. These induced electric fields are, in fact, the electric component of the travelling electromagnetic wave. But as the wave moves through the same region, the flux of the induced electric field will also change in time. The changing electric flux will induce a magnetic field as per Ampire-Maxwell's law. This induced magnetic field is simply the magnetic component of the electromagnetic wave. Thus, Faraday's and ~rn~ire-Maxwell 's law describean induced field (E or B field) that arises from the other changing field. The other field, in turn, arises from the change in the first field. So we have a self-perpetuating electromagnetic wave whose E and B fields exist and change without the need for charged matter. In this way, Maxwell's equations teach us that a beam of sunlight is a configuration of changing electric and magnetic fields travelling through space. The same is true for radio waves, microwaves, infrared rays, ultraviolet rays, X-rays and y-rays.

In relation to electromagnetic waves, we would like to stress on one important point. It is not enough that an electromagnetic field satisfies the wave equations (14.17a and b). It must also satisfy Maxwell's equations. Altbough the wave equations are a necessary consequence of Maxwell's equations, the converse is not true. Thus, in solving wave equations, you must take special care to see whether the solutions satisfy Maxwell's equations. Only then would they represent an electromagnetic wave.

As an example, consider the electromag~letic field of SAQ 2. You can verify that it vE0 satisfies Maxwell's equations provided Eo - vBo and 3, = -T, where c is the C

speed of light given by c - I/=. Together these require that v = * c and Bo c - Eo. You can also verify that E and B of SAQ 2 satisfy the wave equations (14.17a and b). This is an example of a plane electromagnetic wave.

fig kt.% 'libc wave described by the E rad B Belds ol SAQ 2 Is shown at two different ht6 Raarbcr thnl w*(hlng varies with x or z; whtcvtr is happening at a point ou Uley.uis b h r p p n b tvtqwberr 00

tb@p*rpmdicular pknc t k ~ u g h the point As time pas- Ulc enlirr pUan slida to the dpbt bEuu~ ~ - ~ t h a ~ - ~ ~ ~ u e a t ~ ~ ~ ~ a ~ d t r ~ t u i t b d a l ~ ~ d t , ~ w ~ * d d ) - Y A L L O ~ U W & W ~ have a plnlme wave h v e l l h g with a w a s b u t spctd v In the J dirrcli01.1.

Page 11: Maxwell's equations and electromagnetic waves

Electromapetism Did you notice that we have introduced a new term: plane electroinagnetic wave? You may well ask: What is a plane electromagnetic wave? The term plane is meant to indicate that the field vectors E and B at each point in space lie in a plane, with the planes at any two different points being parallel to each other (study Fig. 14.3). Hereafter in our discussion we shall be mainly concerned with plane electromagnetic waves, as these are found to be very useful in various areas of physics, engineering and technology. As you have seen, the electromagnetic field given in SAQ 2 is a specific example of a plane electromagnetic wave. Our interest now is to find the general plane wave solutions of the wave equations (14.17a and b) in empty space, which also satisfy Maxwell's equations.

r' 14.3.2 Plane Wave Propagation in Empty Space z

Let us consider a scalar function of the form A ( k . r - vt ). Now k . r -,d, where d ~ig. 14.d Plane is a constant, is the equatioqof a plane whose normal is in the dirsction electromagnetic wave. Therefore, the functionA ( k . r - vt )represents a plane wave for k . r = d. We can

show that E and B fields of this form satisfy the wave equations. For this, let us express these fields in terms of their scalar components

and

We can show that these scalar components, and hence the E and B fields satisfy Eqs. (14.17a and 14.1%). The scalar components Em Ey, E,, Bm By, B, are of the form A ( k . r - vt ). Then ths fields given by Eqs. (14.21a and b) represent waves propagating in the direction k at a speed v.

A

Let q = k . r - vt. We can takeA to stand for either of the components Ex, E,, E,, B, B,,, B,. We now use the chain rule to express the differentiation ofA with respect to x,y,z and t in terms of q:

And

a2A a2j4 Similarly, you can detenniiie - and - and show that ay2 a2

Why don't you try proving this result in the following SAQ?

Spend SAQ 4 10 min

Prove Eq. (14.22a). (You will have to use the result that l2 + 2 + n2 - 1, where 1, m, n are the direction cosines of k)

Similarly, we can show that

Page 12: Maxwell's equations and electromagnetic waves

. . . a2a 2 a ' a - T = V -

(14.22b) at q2

Compating Eqs. (14.2%) and (14.22b) yields the wave equation for A:

We can express this result for the three components of the electric field :

A A A

Since x, y, z are constant vectors, we can write that

In the same way, we can arrive at a similar result for the B field:

A

Thus, wehave shown that fields of the form E = E ( k . r - vt )and B - B ( k . r - vt ) satisfy the wave equation. The speed of the propagation of

1 wave is given by v = ------ - c, where c is the speed of light. G So far we have established that fields of the general form

A

~ ( 2 . r - v t ) a n d ~ ( k , r - v t ' )

represent plane wave solutions of the wave equations in empty space, We must also make sure that they satisfy Maxwell's equations, Let us ascertain th? conditions under which these solutions satisfy Maxwell's equations.This will give us an idea of the properties of the electromagnetic waves. We can show that if we insert these solutions in Maxwell's equations (14.12 to 14.15), we obtain the following conditions:

MaxweU's Equations and. Electromagnetic Waves

with

1 c p - (14.2&)

G If we take the magnitudes of either Eq. (14.24b) or (14.24d), we get the relationship between electric,and magnetic field strengths:

Page 13: Maxwell's equations and electromagnetic waves

Eleetromgnetism The mathematical procedure for deriving these equations is given in the Appendix. Thus, the plane electromagnetic waves given by Eqs. (14.21a and b) satisfy Maxwell's equations yielding a set of algebraic equations (14.24a to f) in the case where there are no sources. These equations also reveal the properties of plane electromagnetic waves. Eqs. (14.24b and d) together tell us that E and B are perpendicular to each other. You can verify these properties by taking the scalar product E . B which is zero. And Eqs. (14.24a and c) show that E and B are perpendicular to k, i.e., the direction of propagation of the electromagnetic wave (Fig. 14.4). An electromagnetic wave is, thus, a transverse wave. h general E and B can have any functional dependence on r and t. Such a plane electromagnetic wave is shown in Fig. 14.5. To sum up, we have learnt that a plane electromagnetic wave in empty space has the following properties.

Fig.144: Eldmmqndic waves ue trrmversc.

Properties of plane electromagnetic waves in empty space

The field pattern travels with speed c. I 2. At every point in the wave at any instant of time, the electric and magnetic field

strengths are related. by Eq. (14.24b).

13. The electric field and the magnetic field are perpendicular to one another and also to I ille direction of wave propagation, is . , the wave is transverse.

A particularly important type of plane electromagnetic wave solution used in many areas of physics is the sinusoidal wave of a given frequency. Let us briefly consider such waves.

~onochrom'atic sinusoidal electromagnetic plane waves

Of all possible wave forms you have studied, waves of the form

would surely be the most familiar. Such waves are called monochromatic sinusoidal electromagnetic plane waves. (Monochromatic means single colour. Since, frequency corresponds to colour, especially for light, sinusoidal waves of a single frequency are called monochromatic). The argument of the cosine function ( k . r - wt ) represents the phase of the wave. In view of Euler's formula

ill e - cos 8 + i sin 8

the sinusoidal waves given by Eqs. (14.25a and b) can be written as

where Re (A) denotes the real part of A. Suppose we introduce the complex function

F = ~ ~ i ( k . r - m t ' )

Then the actual (real) wave is the real part of the complex wave

If we know g, it is a simple matter to find F. The advantage of using the complex notation is that exponential functions are much easier to manipulate than sine and cosine Wctions. Therefore, from now onwar* we shall express E and B fields using the exponential notation: I I

Page 14: Maxwell's equations and electromagnetic waves

In general, Eo and Bo are complex vector amplitudes independent of r and t. However, when dealing with actual waves, we shall take the real parts of Eqs. (14.25~ and d). The complexity of this notation should not bother you. For ekample, you can see that the E and B fields of SAQ 2 are special cases of Eqs. (14.25a and

w b) where k is parallel to the y direction, v - -and Eo ( k, w ) I Eo ; a d

A k

B, ( k, w ) I Bo x. The advantage of dealing with waves of thk form is two-fold:

1) Any wave train can be regarded as a linear superposition of waves (for different values of w and k) of the above forms.

2) For monochromatic plane electromagnetic waves given by Eqs. (1'4.25a and b) the operations in Maxwell's equations assume a particularly simple form:

In general, we summarise this property by the equivalence relations I v*[ } - i k * ( ) (14.27a)

The symbol * means any of the operations (grad, div, curl) of V upon either a scalar or a vector quantity. With this equivalence we can rewrite Maxwell's equations for monochromatic plane waves for empty space.

k . E o = 0 (14.28a)

kxEo - wBo (14.28b)

k . B o - 0 (14.28~)

As we have already said, it is now well known that radiowaves, microwaves, blackbody radiation, light waves, X-rays and gamma rays are all electromagnetic radiations. These constitute what is called as the electrornngnetic spectrum. .What distinguishes each of these radiations is their frequency or wavelength (see Fig.14.6).

Fig. 1461 The dccttwnrlplclic spcdrum

C

id 10s I@ 101 16 10.' itY 10s I@ 10d 106 '1~7 I@

10 OH^

id, 101 16 id I@ I@ I@ id I@ I@ 100 '1011 ion 1013 IOU 10B loM 1017

1 l l I l l ~ r 1 1 1 1 1

7 - m ~ ~ .- i;

i ~ l i I 1 1 I I ~ ~ .

d i o . Wio . 5

miayavw

e r I I

3 -5 a

t i " h d , r .C

4 I (

!j x

Page 15: Maxwell's equations and electromagnetic waves

Elcctromngaetlam Can you determine the wavelength of the plane em wave of Eq. 14.25a7 Give it a try.

Spend SAQ 5 5 min Show that the wavelength of the plane monochromatic sinusoidal electromagnetic

wave (Eq. 14.Za) is given by

2~ L = - (14.29a)' k

C and hence the frequency is given by f a h (14.29b)

The quantity k is called the wave number as, there are k/2n wavelengths per unit aisthnce, and k is called the wave vector.

You may now like to stop for a while and recapitulate what you have studied so far in this section: In empty space (i.e., charge-free and current- free region) Maxwell's equations predict electromagnetic waves. Maxwell's equations model a travelling electromagnetic wave as being constituted of time- varying electric and magnetic fields. The electric and magnetic fields are not produced by any external sources, but are mutually induced. Thus, an induced E (B) field arises from the changing B (E) field. The B (E) field itself arises from the changing E (B) 'field resulting in a self- perpetuating electromagnetic wave which travels with the speed of light. The E and B fields in an electromagnetic wave must satisfy the wave equations as well as Maxwell's equations. Plane electromagnetic waves of the form A ( k . r - vt ) are of special interest in physics. Such waves have the property that their E and B field vectors at each point in space lie in a plane with the pkanes at N o different points being parallel to each other. They are perpendicular to each other and also to the direction in which the wave travels. Thus electromagnetic waves are transverse waves. Even amongst plane electromagnetic waves, sinusoidal waves of single frequency [ A ( k, w ) e

- i ( w f - k . r ) ] are used very frequently in physics for many reasons.

We will now illustrate these ideas with an example. - -

Example 2

Consider an electromagnetic wave in empty space whose electric field is given by

Determine the direction of propagation, the wave number, the frequency and the magnetic field of the wave.

solution

We will make use of the properties of E, B and k to determine these details of the wave. To find the direction of propagation of the wave, let us examine the argument of the exponential.

$ompaging 10' t + Bz with ot - k . r we a n see th$ the direction of propagation is k = - z. This direction also satisfies the property k . E = 0. We can find the wave number from the relation

Page 16: Maxwell's equations and electromagnetic waves

w lo8 The frequency of the wave isf = - = - HZ 1.67 x lo7 Hz. The magnetic 23c 23c field is of a form similar to the E field:

To evaluate Bo, we make use of the properties k . B = 0 and k x E = c B. This gives the direction of B:

A A A A A A

B = k x E = - z x x ,= -y The magnitude of Bo can be found from the relation ( E ) / I B I = c, which yields Bo = 6 0 / ~ . Thus

Be---yexp - i 10 t + - z tesla C O A [ ( ) ]

TO ensure that you have grasped the ideas of this subsection, you must now work out an SAQ.

SAQ 6

a) The electric field given by

L \ I 1

represents the E field of a plane electromagnetic wave in a charge-free and current free region. Calculate the associated magnetic field. Find the wavelength and frequency of the wave.

A

b) Consider two electromagnetic waves travelling in the y and - j directions, respectively, described by

Show that when the two waves are both present in empty space, with the resultant electric and magnctic fields

E - El+E2, B = B1+&

they still satisfy Maxwell's equations.

Now that you have understood plane electromagnetic wave propagation in empty space, we would like to discuss another related aspect: propagation of plane waves in dielectric media. It is important to study this aspect as it helps us understand many phenomena, viz. propagation of light in glass and water; and of X-rays and gamma rays in human body, and so on. But in order to study electromagnetic wave propagation in dielectric media, we must first write Maxwell's equations for the same and derive the wave equation. This is what we shall study in the next section.

14.4 MAXWELL'S EQUATIONS AND PLANE WAVE PROPAGATION IN DIELECTRIC MEDIA

Ma.well's Equations mcl Electromagnetic Waves

Spend 15 min

As listed in Table 14.2, Maxwell's equations are complete and correct. However, when you are working with materials, there is a more pertinent way to write these equations. So let us now write Maxwell's equations inside matter. We are especially interested in dielectrics. At this Stage you will frnd it usefrl to look up the relevant equations given for dielectric nlatt$al,, in the relevant units of Blocks 2 and 3.

Page 17: Maxwell's equations and electromagnetic waves

Electromagnetism

You know that P is simply the electric dipole moment per unit volume. For the nonstatic case, any change in the electric polarization involves a flow of charge which yields a polarization current. The corresponding polarization current density

aE Jp , which arises because pp changes with time, is given by -, i.e., at

alp Jp = -

at

Similarly, the magnetization M in a dielectric results in a bound current

JM = V X M (14.30~)

Magnetization, as you know, is just the magnetic dipole moment per unit volume. You must note, however, that the polarization current Jp has nothing to do with the bound current J,. The bound current is associated with the magnetization of the material and involves the spin and orbital motion of electrons. By contrast, Jp is the reslilt of linear motion of charge when the electric polarization changes. If, for example, P points to the right and is increasing, then each positive charge moves to the right and each negative charge to the left, resulting in the polarization current. Keeping this discussion in mind, we can write the total charge density a's

P = Pf+Pp p r - V * p (14.31a)

and the total current density as

Here pr is the free charge density and Jf is the free current density. Substituting these expressions for p and J in Maxwell's equations (Eqs. 14.8 to 14.11) we obtain Maxfiell's equations for material media:

. We can recast Eqs. (14.32) to (14.35) in a form similar to Eqs. (14.8) to (14.11). For this we introduce auxiliary fields

D = E , E + P (14.36a)

14.4.1 Maxwel19s Equations in Dielmectris Mdta Recall that dielectric materials are subject to electric polarization and magnetization. In such materials there is ad accumulation of 'bound' charge and current. You have learnt that in a dielectric, for the static case, an electric polarization P results in an accumulation of bound charge

Then we can rewrite Eqs. (14.32) and (14.35) as follows:

Page 18: Maxwell's equations and electromagnetic waves

In an-isotropic (linear) dielectric medium, P is parallel to E ( P = co X, E ) and M is parallel to W ( M - X, M ). For such materials we can write Eqs. (14.36a and b) as

and

Here

x is the electric susceptibility, and X , is the magnetic susceptibility. In general E

and are frequency dependent. If the medium is hoinogeneous, E and p are constants, i.e., they do not vary from point to point. Given this information you can write the Maxwell's equations inside a dielectric yourself.

Spend SAQ 7 10 min

Show that Maxwell's equations for an isotropic (linear) dielectric media take the following forms:

Differential form

V . E - 5 Integral form

J S ~ . d S f (14.39)

, Eqs. (14.39 to 14.42), given Eqs. (14.36a and b) and Eqs. (14.38a and b) are the fundamental laws of electromagnetism inside dielectrics. We can now consider plane wave propagation in dielectric media which are linear and homogeneous.

14.4.2 Plane Wave Propagation in Dielectrics Let us first consider regions inside matter where there is no free charge or free current. Maxwell's equations for such regions insidk a linear medium become

V , E - 0

You can see that these equations are similar to Eqs. (14.12 to 14.15). Once again you can follow the procedure adopted in Sec. 14.3.2 and derive the wave equation for electromagnetic waves propagating through charge-free and current-free linear homogeneous media.

and

Maxwell's Equations mad Electromagnetic Waves

Page 19: Maxwell's equations and electromagnetic waves

Electromagnetism

SAQ 8

Spend Prove Eqs. (14:43a and b) 5 min

Thus, we find that in a linear homogeneous medium, electromagnetic waves propagate at a speed

Now a well known result from optics tells us that the speed of light in a transparent medium is reduced by a factor of n:

where n is the index of refraction. It follows that n is related to the electric and magnetic properties of materials by the equation

For a dielectric p = and n = = K where K is the dielectric eonstoat of

the material.

You should realise that to be able to relate the expression of the index of refraction of a material with its electric and magnetic properties was another triumph of Maxwell's equations. After all, if the index of refraction as measured optically could be calculated so easily from the dielectric constant measured electrically, it would be a convincing piece of evidence for Maxwell's identification of light with electromagnetic radiation.

We can once again write plane wave solutions similar to Eqs. (14.21a and b)or (14.25a and b) for the wave equations (14.43a and b) for a linear, homogeneous media with no free charge and no free current. The only difference is that the . waves travel with a speed v given by Eq. (14.44) :

E - ( - v t ) ; B = ~ ( i ( . r , - v t )

with the conditions that

You can verify that monochromatic sinusoidal plafie waves of the form

E ( r , t ) = E o ( k , o ) e x p ( - i w t + i k . r ) (14.46a)

satisfy the wave equation and Maxwell's equations inside charge-free and current-free linear, homogeneous dielectric media, with the conditions that

k.Eo = 0 k .Bo P 0

Page 20: Maxwell's equations and electromagnetic waves

These equations show that the plane wave solutions in a dielectric having no free charge and no free current resemble the'plane wave solutio~s in vacuum: B is perpendicular to E and the wave travels in the direction of k which is perpendicular to both E and B. What is the difference? The speed at which the wave travels in the dielectric is different from c, the speed of light in vacuum, by a constant factor, termed the index of refraction (in optics).

.Thus, if we compare a wave in a dielectric with a wave of the same frequency in vacuum, the wavelength in the dielectric will be less than the vacuum wavelength by a factor [ l/n ), since frequency wavelength = wave speed.

Now in the last section of this unit we shall consider another interesting aspect of electromagnetic waves. Consider the following situation. When you sit outdoors on a cold winter morning in bright sunshine, you feel warm after a while. Why does this happen? Obviously, it is the energy carried by the sunlight which gets transferred to you and gives you this pleasant sensation. You already know that waves transport energy from one region of the space to another. In this section, we will determine the amount of energy carried by electromagnetic waves across the space. '

14.5 ENERGY CARRIED BY ELECTRO GNETP1IC . WAVES : POYNTING'S THEOREM

Recall that in Unit 4 , we expressed the work necessary to assemble a continuous static charge distribution (against the Coulomb repulsion of like charges) as

We can rewrite this expression in terms of the electric field E. We first use Gauss' law to express p in terms of E:

Thus

Now we use the following vector identity

V . ( E V ) = ( V . E ) V + E . ( V V )

Since E - - VV, we get

Applying Gauss' divergence theorem to the first term, we obtain

Now suppose we enlarge the volume to include all the cha&z-Any extra volume will not contribute to WE, since p - 0 in that voluhe. But as we enlarge the

volume, the integral 012 can only increase, since the integrand is positive. Therefore, the surface integral must decrease so that the sum remains the same. If we integrate over a11 space, the surface integral goes to zero, and we are left with

Maxwell's Eguatiors and Elfftaromagnetic Waves

Page 21: Maxwell's equations and electromagnetic waves

Electromagnetism where E is the resulting electric field. Likewise, we have in Unit 13 shown that the work required to get currents going (against the back emf) is (Eq. 13.24)

where B is the resulting magnetic field. Evidently, the total energy stored in a current and charge distribution can be expressed in terms of electric and magnetic field produced by this distribution as

We would now like to derive Eq. (14.48~) more generally, keeping in view the energy conservation law.

Suppose some charge and current configuration produces fields E and B at time t. Suppose the charges move around. We would like to know: Wow much work, dW, is done by the electromagnetic forces on these charges in the small time interval dt? According to the Lorentz force law, the work done on an element of charge dq is

Now, dq - p dVand p v = J, so the total power delivered on all the cl~arges i11

some volume V is given by

Let us express Eq. (14.49) in terms of the fields alone, using Amphe-Maxwell's law to eliminate J:

A well known vector identity gives us the result

Combining this result with Faraday's law, V x E = - - at 7 it follows that

We can also write

so that

Putting this into Eq. (14.49) we get

Now applying divergence theorem to the second term on the RHS, we hav.e

where S is the surface bouding the volume

Page 22: Maxwell's equations and electromagnetic waves

Eq. (14.51) is the mathematical statement of Poynting's theorem; it expresses conservation iof energy in electromagnetism. The first integral on the right represents the total energy stored in the fields, WEB(Eq. 14.48~). The second term represents the rate at which energy is carried out of V, across its boundary surface, by the electromagnetic fields. Poynting's theorem says, then, that the rate of work

- done on the charges by the electromagnetic force is equal to the decrease in energy stored in thefield, minw the energy which flowed out through the surface.

The quantity

is called Boynting's.vectos; it represents the energy flux density - that is, S.da is the energy per unit time transported by the fields across a surface da. We can state Poynting's theorem more compactly in terms of S and WEB:

Of course, the work W. done on the charges will increase their mechanical energy (kinetic, potential, or whatever). If we let U, denote the mechanical energy density, so that

and use UEB for the energy density of the fields,

i then i

i and hence I

This is the differential version of Poynting's theorem. Compare this with the continuity equation expressing conservation of charge:

The charge density is replaced by the total energy density (mechanical plus electromagnetic), while the current density is replaced by the Poynting vector, Thus, the Poynting vector S describes the flow of energy in the same way that J describes the flow of charge.

Let us now.recapitulate what you have studied in this unit.

'\ Maxwell's ]Equations and Electromagnetic Waves

-

m Maxwell's equations constitute the fundamental set of differential equations describing electric and magnetic fields. These equations in their integral and differential forms are tabulated below for different situations.

Page 23: Maxwell's equations and electromagnetic waves

i

Maxwell's equations In vacuum in charge-he and current-free regions V . E - o

aB V X E - -- at

V . B - 0

aE V x B - P o E o Z

Maxwell's equations in vacuum In the presence of charges and currents

Maxwell's equations allow wave solutionsfor electric and magnetic fields in vacuum and in dielectric media. These equations also model electromagnetic waves, as constituted of time-varying, selfperpetuating, electric and magnetic fielak, which are mutually perpendicular and perpendicular to the direction of propagation. Ehctromagnetic waves ate transverse waves. The electromagnetic wavespropagate in vacuum at the speed of light. For

- i ( u t - k . r ) monochromatic sinusoidal electromagnetic waves of the form Eo e , Bo e-i(u~-k.r) , Maxwell's equations in charge-free and current-free empty space simplify to yield the following set of equations.

k.Eo - 0 k.Bo - 0

' k x E o = oB, 0 k x B 0 = -,Eo C

J , E . ~ s - o

d@E % ~ . m = -- dl

J , ~ . d s - 0

d @ ~ $ c ~ . d -

P V . E - - €0

dB V X E m -- at

V . B - o

V X B - p,, ( ~ t ~ ~ $1

Maxwell's equations inside isotmplr, linear dlelectdc media

Such electromagnetic waves are described by their angular freq~ency o), the speed of propagation c (in empty space), the wave number k, the wavelength A.

4 J s ~ . d s .. - €0 -

~ O B $ c ~ . d ~ = -- dt

J s ~ . d s = o

t j C B . d l = p,,,i+weo- ~ Q E dt -

V . E - B e

aB V X E = - - at

V . B - 0

V x B - p ( ~ , t ~ $ )

In a linear dielectric medium, Maxwell's equations allow w.ave solutions for

4 - ; d@E $. .dl - -- dt

J'l3.d~ - 0

$ c ~ . d l = p i t p c - d@fi dl

1 ' C electric and mainetic fields which propagate at a! speed v - - - - where fi !n' n is the refractive index of the medium. .. .

Page 24: Maxwell's equations and electromagnetic waves

.1 The Poynting vector S = - ( E x B ) represents the energy flux density, i.e.,

Po the energy per unit time transported by electromagnetic fields across a surface. Poynting's theorem expresses the conservation of energy in electromagnetism. The work done on the charges by the electromagnetic force is equal to the decrease in energy stored in the field, minus the energy which flowed out through the surface.

IN& QUESTIONS Spend 30 min

1. A plane electromagnetic sinusoidal wave is characte~ised by the following parameters: the wave is travelling in the direction - x ; its frequency is 100

n megahertz (MHz, lo8 cps); the electric field is perpendicular to the z direction. Write down the expressions for the E and B fields that specify this wave.

2. Show that the electromagnetic field described by

E - E, ices kx cos 6 cos wt

B = Bo ( coskr sin ky - ; sin kx cos ky ) sin a t . will satisfy Maxwell's equations in charge-free and current-free empty space if Eo = f i c ~ ~ a n d o = f l c k .

3. The electric field of an electromagnetic wave in vacuum is given by

where E is in volts per meter, t is in seconds, and x is in meters.

i ) Determine the frequency v.

ii) Determine the wavelength h.

iii) Determine the direction of propagation of the wave. .I.

' iv) Determine the direction of the magnetic field. '

14.8 SOLUTIONS AND ANSWERS

Self-Assessment Questions (SAQs)

1. The conduction current density is

J = a E - a ~ ~ s i n o t ~ m - ~

The displacement current density is

where w = 2 x f and f - lo6 Hz. The ratio of the magnitudes of these.current densities is

For copper at 1 MHz

Maxwell's Equations and Eiectromagnetic Waves

Page 25: Maxwell's equations and electromagnetic waves

For T~,ilon at 1 h1H7

Thus, for copper the conduction current dominates the displacement current. But for Teflon which is a reasonably k ~ o d insulator at 1 MHz, the conduction I

current can be neglected in comparisonbith the displacement current.

2. Substituting the E and B fields in the Maxwell's Eqs. (14.12) to (14.15), we get

p a c d A a A

= ( I - t j - + k - ) . k E , s i n ( y - v t ) a x a y a t .

a - -[E0sin ( y - v t ) ] az - 9

Thus, Eq. (14.12) is an identity.

a~ A - = - ivBOcos(y-vf) at

Eq. (14.13) gives the condition that

Eo = Bov

a iii) V . B = -Basin ( y- v t )

ax

1 aE iv) V x B = --

C2 at

where

~ a B , s i n ( y - v t ) A V x B = -k = - k B o c o s ( y - v t )

8~

vE0 Eq. (14.15) gives the condition that Bo = C

These two conditions, Eo = Bo v and Bo = ' tugether mquire Lhat 7

1 and Eo Boc l 1

11 3. Maxwell's equations for charge-free matserial media are '1

Page 26: Maxwell's equations and electromagnetic waves

V x B = p J + E - ( Z ) where J - a E.

Once again taking the curl of the second equation we can write

The LHS is simplified as follows

v x ( V X E ) - v ( v . E ) - $ E

= - V ~ E sinceV.E I 0

Thus, we get

which is Eq. (14.18a).

Similarly, taking the curl of curl B, we get / a V x ( V x B ) - p ( V x u E + e - V x E )

at

The LHS yields

V X ( V X B ) v ( v . B ) - V ~ B .." .? " - -9% [.: V . B - 0 ]

.- Thus, we have

Maxwell's Equations and Eketromrgnctic Waves

Page 27: Maxwell's equations and electromagnetic waves

which is Eq. (14.18b).

Similarly,

a2A d2A A A 2 7 = - ( z * k ) az a

now a2A a2A a2A

d 2 ~ A A 2 A A 2 A - - ~ [ ( r . k ) + ( y . k ) + z . g ) i

A A A A A A drl

A

But x . k , y . k and z . kare the direction cosines of k Using the result that A A 2 A A 2 A A 2 a2A

( k ) + ( y . k ) + ( z . k ) = l , w e g e t $ ~ = 7whichisEq.(14.22a). %

5. For mathematical convenience, we choose the direction of propagation to be in the x-direction. The expression of Eq. (14.25a) simplifies to

Now the cosine function completes one complete cycle when x advances by the wavelength h. Thus, we must have

This condition .is satisfied only if

Since wave speed - (frequency) x (wavelength),

C *

we have frequency f = -

6.a) The associated magnetic field can be obtained from Eqs. (14.24 a to f). Using Eq. (14.24b) we have

1 A - -kx1000f exp 9 c A A

2 y - z k 2;-l and k = - whence k - - 100 Ik l * 6

Thus the associated magnetic field is 58

Page 28: Maxwell's equations and electromagnetic waves

firirdl'e Equ.tiwr and EkctroM@I& W ~ V &

This is consistent with Eq. (14.240 since

The wavelength of the wave is

c 3 x l d m s - ' = l$s-l The frequency of the wave is f - - - A 300m

b) The resultant electric and magnetic fields can be expressed as follows:

E - El+&

and B - B1+& - - 2 illo ms 2*Y sin

h ': sinA-sinB = 2cos-

2 sin -

We have to show that the resultant E and B fields satisy Maxwell's equations:

and

T ~ U S , V X E -- dy yields Eo = ow

Similarly, V x B - - - dE yields the following c2 at

dE - 2 n c -z2EO- asin 2"" sin - at A A A

Page 29: Maxwell's equations and electromagnetic waves

This is the same as Eq. (14.24f).

Thus, the resultant E and B fields of the plane electromagnetic wave. satisfy '

Maxwell's equations. Such a wave is called a standing wave.

7.(i) From Eq. (14.32) we have

V . ( e 0 E + P ) = pf

or V . D = pf

or E V . E = pf (Since D - e E from Eq. (14.38a))

To cast it in ihe integral form we integrate both the sides in a volume V

In the RHS d the equation$ pf dV - q, the charge enclosed by the volume. v

To the LHS we apply Gauss's divergence theorem and obtain

where S bounds the volume V. Eqs. (14.40) and (14.41) are the same as Eqs. (14.33) and.(14.34).

ii) V x B -

aD V x H - dl + C\O JI psing Eqs. (14.36a and b)]

aD V x H - --+ Jp at

Using Eqs. (14.38a and b) we can write

which is Eq, (14.42).

To cast this in the integral form we integrate both the sides on an open surfiw S .

The surface integrals on the RHS are related to the flux and the current

Applying Stokes' theorem to the LHS we get

where the open surface S is bounded b; the tontour C.

Page 30: Maxwell's equations and electromagnetic waves

8. Once 'again we take the curl of V x E:

Taking the curl of V x B:

. a V x ( V x B ) - p e - ( V x E ) at

Terminal Questions A

1. Since the wave is travelling in the ; direction, E field wilibe normal to x. It is given that E is perpendicular to z. Therefore, E isin the y direction. The expression for the E field is thus

where . w - 2 n f - 2 ~ c x 1 0 ~ ~ z

and 21c k -- - h c

The corresponding B field is given by

2.' Maxwell's equations in charge-free and current-free empty space are

V,E - 0 V . B = 0

aB iii) V x E P -- at

Page 31: Maxwell's equations and electromagnetic waves

1 dE iv) V x B - -- . C2 at

2 W or. --r - 2c2 k

and W Eo = -Bo = ~ Z C B , k 3. Comparing with the expression E Eo cos ( la - ot )

i) The frequency of the wave is

v - 1o8HZ(~: 0 - 2 3 % ~ )

2n 2n.(3)13m ii) The wavelength is h, = - I k 2 3d

A

iii) The direction of propagation is in the + x direction.

iv) Since E is along + j and k is along + ; the B field will be$long + z direction.

Page 32: Maxwell's equations and electromagnetic waves

APPENDIX: H LATIONSHIP BETWEEN E,

You have studied in the unit that plane waves of the form ~ ( & . r - c t ) and ~ ( k . r - c t )

satisfy the wave equations in empty space. We will now insert these wave forms into Maxwell's equations for c h a r g ~ t e e and current-free empty space, and obtain the relationships between E, B and k For this purpose, we will consider each of Maxwell's equations separately.

I. V . E - 0 A A A

Let E - xE,+yE,,+zE,

Thus, we can write Eq. (A.la) as

(A. la)

(A. lb)

(A. lc)

Here Em E , E, are functions of ( & . r - ct ). Let us make the substitution A

q - k.r-ct

Using the chain rule, we can write

aE'%-dE'(r.;) - =- a~ at, a~ (See Sec. 14.3.2)

aE aE aE, aE, A A

~imiluly, - - $ ( &. ; ) and - aY - - ( k . z )

az al Thus,

or

A a A A A V.E E k.-(E,x+E,y+E,z) (.; ~ e $ ~ p r o d ~ i s c o m m u t a t i v e a n d q X, y, z are constant vectors. )

Thus, Maxwell's first equation becomes

aE This implies that i; is perpendicular to - We can show that this also implies that k al* A

is perpendicular to E. For this we assume that E iq along k, i.e., E - f ( q ). Then

and

unless f ( q ) is a constant. put for waves f ( q ) is not copstant, Therefore, E can have no component along k. Thus E is perpendicular to k, that is

Using a similar method as in (I), you can show that

Maxwcll'~ Eqartiws nnd Eke(rornalplctk Waves