The Mathematical Theory of Electricity and Magnetism (5th ed, 1927) — part 32 of 39

Thus we may say, that the ratio of units C is identical with the velocity of propagation of electromagnetic waves, and this again is identical with the velocity of light.

Equations for a Uniform Isotropic Conductor.

583. In an isotropic conductor the current (u, v, w) is proportional at every point to the electric force (X, Y, Z). We are supposing u, v, w to be measured in electromagnetic units. The values of the components of electric force, measured in electromagnetic units, are GX, GY, GZ, these being of course the forces acting on an electromagnetic unit of electrical charge. Thus

by Ohm's Law,

GX . u = , etc.

T

where r is the specific resistance measured in electromagnetic units. If we further put 4nrf= KX, etc., equations (529) become

/4tt(7 K d\ v By 3/3 /K._-

\ r G dtj dy dz

and two similar equations.

On replacing equations (529) by these, the equations of § 574 become the general equations of an isotropic conducting medium.

If we differentiate the three equations of the system (546) with respect to x, y, z and add, we obtain

4t7C Kd(dX dY dZ

t G dtj \dx dy dz

From equation (530) we have

BX dY dZ_4,7rp dx dy dz K '

)-a

582-583 a] Isotropic Conductor 527

so that our equation becomes

dp_ _4tt02 dt~ ~K^9'

If p0 is the value of p at time = 0, the solution of this equation is

_4ttC2

p = Poe K* \

shewing that p falls away exponentially, no matter what electric or magnetic fields may be acting. This equation is identical with that already obtained in § 396, the factor G2 simply corresponding to a change of units. Thus inside a conducting medium any initial charge will rapidly disappear, and we may suppose that

BX BY dZ=Q. =Q

dec dy dz ' "

583 a. Multiply both sides of equation (546) by //, and differentiate with respect to the time. We find

G df- + t dt ~dy\di) dz\di

The right-hand member of this equation may by equations (528) be replaced by

dy \d% dy J dz \ dz dxj

or G

dec \dcc dy dz] \

and this is equal to GV2X, in virtue of the relation

ox dy dz Thus the equation becomes, on dividing through by G,

Kp, d*X 4,7rp, dX

G2 dt2 r dt

= V2X.

This equation involves X only, and so is the differential equation satisfied by X when electromagnetic waves are propagated in a conductor. Naturally Y, Z satisfy similar equations, and equations (528) shew that a, b, c or a, /3, 7 again satisfy similar equations. Thus X, Y, Z, a, /3, 7 all satisfy the same differential equation, namely

djX Inr&dX

dt2 + Kr dt

where a stands for G/^/Kp,. The complete solution of this equation has been given by Riemann*.

• Die vartielle Differ entialgleichung en der Math. Physik, 4th edu. (1901), n. p. 399.

528 Displacement Currents [en. xvn

We may notice that in a dielectric, t = oo , so that the second term dis- appears. The equation then reduces, as it ought, to equation (534) already obtained in § 577. In many problems, the second term is more important than the first. When the first term is omitted, the equation reduces to the well-known equation of conduction of heat, already obtained in § 535 (equation (480)).

To form an estimate of the relative importance of the two terms on the

left, let us examine the case of an alternating current in which the time-

d factor is elpt. We may as usual replace -r- by ip, and the equation becomes

dt

-p* +

4?r(72 Kt

tpW = a2V2%.

The neglect of the first term, which is of course the same thing as neglecting the displacement-current, is clearly permissible if 4>ttG2/Ktp is numerically large. When this ratio is not large, the error produced by the neglect of the first term will be greatest in problems in which t is large (conductors of high resistance) and in which p is large (rapidly changing fields). On substituting numerical values it will be found that in problems of conduction through metals, the neglect of the factor Rrpf^-rrC2 produces a quite inappreciable error unless p is comparable with 1015 — i.e. unless we are dealing with oscillating fields of which the frequency is comparable with that of light-waves. Thus the effect of the displacement-current in metals has been inappreciable in the problems so far discussed, so that the neglect of this effect may be regarded as justifiable. The matter stands differently as regards the problems to be discussed in the next chapter, in which the oscillations of the field are identical with those of light- waves.

Units.

584. We may at this stage sum up all that has been said about the different systems of electrical units.

There are three different systems of units to be considered, of which two are theoretical systems, the electrostatic and the electromagnetic, while the third is the practical system. We shall begin by discussing the two theoretical systems and their relation to one another.

585. In the Electrostatic System the fundamental unit is the unit of electric charge, this being defined as a charge such that two such charges at unit distance apart in air exert unit force upon one another. There will, of course, be different systems of electrostatic units corresponding to different units of length, mass and time, but the only system which need be considered

583 0-585] Units 529

is that in which these units are taken to be the centimetre, gramme and second respectively.

In the Electromagnetic System the fundamental unit is the unit mag- netic pole, this being defined to be such that two such poles at unit distance apart in air exert unit force upon one another. Again the only system which need be considered is that in which the units of length, mass and time are the centimetre, gramme and second.

From the unit of electric charge can be derived other units — e.g. of electric force, of electric potential, of electric current, etc. — in which to measure quantities which occur in electric phenomena. These units will of course also be electrostatic units, being derived from the fundamental electrostatic unit.

So also from the unit magnetic pole can be derived other units — e.g. of magnetic force, of magnetic potential, of strength of a magnetic shell, etc. — in which to measure quantities which occur in magnetic phenomena. These units will belong to the electromagnetic system.

If electric phenomena were entirely dissociated from magnetic phenomena, the two entirely different sets of units would be necessary, and there could be no connection between them. But the discovery of the connection between electric currents and magnetic forces enables us at once to form a connection between the two sets of units. It enables us to measure electric quantities — e.g. the strength of a current — in electromagnetic units, and conversely we can measure magnetic quantities in electrostatic units.

We find, for instance, that a magnetic shell of unit strength (in electro- magnetic measure) produces the same field as a current of certain strength. We accordingly take the strength of this current to be unity in electro- magnetic measure, and so obtain an electromagnetic unit of electric current. We find, as a matter of experiment, that this unit is not the same as the electrostatic unit of current, and therefore denote its measure in electro- static units of current by G. This is the same as taking the electromagnetic unit of charge to be G times the electrostatic unit, for current is measured in either system of units as a charge of electricity per unit time.

In the same way we can proceed to connect the other units in the two systems. For instance, the electromagnetic unit of electric intensity will be the intensity in a field in which an electromagnetic unit of charge experiences a force of one dyne. An electrostatic unit of charge in the same field would of course experience a force of I/O dynes, so that the electrostatic measure of the intensity in this field would be 1/(7. Thus the electromagnetic unit of intensity is 1/(7 times the electrostatic. The following table of the ratios of the units can be constructed in this way:

j. 34

530

Displacement Currents

Ratios of Units.

[CH. XVII

Charge of Electricity. Electromotive Force. Electric Intensity. Potential.

Electric Polarisation. Capacity. Current.

Resistance of a conductor. Strength of magnetic pole. Magnetic Intensity. „ Induction.

Inductive Capacity. Magnetic Permeability.

One electromag. unit = C electrostat. units.

» j> I » "

» = i/o ,, „

;> >) / >' »

— fl

= C2 — c

J) )) ■*-/^' » »»

J> » == -'•/o „ },

_ n

:> » ^ j; )>

» >> ^ I/O ,, ,,

= C2 = I/O2

586. The value of C, as we have said, is equal to about 3 x 1010 in c.G.s. units. If units other than the centimetre, gramme and second are taken, the value of C will be different. Since we have seen that G represents a velocity, it is easy to obtain its value in any system of units.

For instance a velocity 3 x 1010 in c.G.s. units = 671 x 108 miles per hour, so that if miles and hours are taken as units the value of C will be 6'71 x 108.

Practical Units.

587. The practical system of units is derived from the electromagnetic system, each practical unit differing only from the corresponding electro- magnetic unit by a certain power of ten, the power being selected so as to make the unit of convenient size. The actual measures of the practical units are as follows :

For legal and commercial purposes, the units are defined in terms of material standards. Thus according to the resolutions of the International Conference of 1908 the legal (Inter- national) ohm is defined to be the resistance offered to a steady current by a uniform

Quantity Charge of Electricity Electromotive Force"!

Name of Unit Coulomb

Measure in electromag. units

10"1

Measure in

electrostatic units

(Taking C = 3xl010)

3xl09

Electric Intensity > Potential J

Volt

108

FUTT

Capacity

Farad Microfarad

io-9 io-15

9 x 1011 9 xlO5

Current

Ampere

IO"1

3 x 109

Resistance

Ohm

IO9

1

9 x 10"

585-588] Units 531

column of mercury of length 106-300 cms., the temperature being 0°C, and the mass being 14*4521 grammes, this resistance being equal, as nearly as can be determined by experiment, to 109 electromagnetic units. Similarly the legal (International) ampere is defined to be the current which, when passed through a solution of silver nitrate in water, deposits silver at the rate of •00111800 grammes per second.

Physical Dimensions of Units.

588. As explained in § 18, all the electric and magnetic units will have apparent dimensions in mass, length and time. These are shewn in the following table:

Electrostatic

Electromagnetic

Charge of Electricity

e

MLT-1

M%L*

Density „

P

M^IT^T-1

H%ir*

Electromotive Force

E

j|/i jjk T-x

M^L^T~2

Electric Intensity

B(X,

r,z)

M^zr^T-1

M^L^T-'1

Potential

V

M^L^T-1

M$L%T-2

Electric Polarisation

P(f,

9>h)

M^L'^T-1

J/U-*

Capacity

G

L

L-lT2

Current

i

M^L^T-*

M^L^T-1

Current per unit area

(u,v,

w)

M^L~^T~2

M2L~* T-1

Resistance

R

L~lT

LT-i

Specific resistance

T

T

£277-1

Strength of magnetic pole

i m

m^l\

M% L%- T-1

Magnetic Force

H{a,

/3,y)

m}lSt-2

M% L~^ t-1

„ Induction

B (a,

b,c)

m}l~%

M^IT^T-1

Inductive Capacity

K

1

L~2T2

Magnetic Permeability

M

£-2 1*2

1

34—2

CHAPTER XVIII

THE ELECTROMAGNETIC THEORY OF LIGHT Velocity of Light in Different Media.

589. It has been seen that, on the electromagnetic theory of light, the propagation of waves of light in vacuo ought to take place with a velocity equal, within limits of experimental error, to the actual observed velocity of light. A further test can be applied to the theory by examining whether the observed and calculated velocities are in agreement in other media.

According to the electromagnetic theory, if V is the velocity in any medium, and V0 the velocity in vacuo, we ought to have the relation

V J_ / 1 n~VAV VI>0' where K0, /ia refer to free space.

For free space and all media which will be considered, we may take /* = 1. Also if v is the refractive index for a plane wave. of light passing from free space to any medium, we have from optical theory the relation

V0

so that, according to the electromagnetic theory, the refractive index of any medium ought to be connected with its inductive capacity by the relation

"VZ:

One difficulty appears at once. According to this equation there ought to be a single definite refractive index for each medium, whereas the phenomenon of dispersion shews that the refractive index of any medium varies with the wave-length of the light. It is easy to trace this difficulty to its source. The phenomenon of dispersion is supposed to arise from the periodic motion of charged electrons associated with the molecules of the medium (cf. § 610, below), whereas the theoretical value which has been obtained for the velocity of light has been deduced on the supposition that there are no moving charges at any point of the dielectric (cf. § 577). A correction to the value just obtained for v will be needed to represent the effect of the motion of charged electrons in the medium. When this motion is infinitely slow, the correction disappears, so that our equation ought to give the true value of v in the limiting case of light, or other electromagnetic waves, of infinite wave- length. It is impossible to deal experimentally with waves of infinite wave-

589, 590] Velocity of Light in Different Media

533

length, but the following tables* shew that as the wave-length increases, the refractive index v approximates to *JKjKQ.

Water. Ethyl Alcohol.

v£"

V80 = 8-94.

\f K~

Wave-length (cms.)

v (observed)

65

8-88

8-8

8-89

5-7

8-79

3-7

8-10

1-75

7-82

0-8

8-97

0-4

9-50

•0001 26t

1-32

•0000589J

1-33

Wave-length (cms.)

v (observed)

65

5-7

0-8

0-4

•0000589+

4-89

3-4

2-57

2-24

1-36

590. For gases there is quite good agreement between theory and experiment, in spite of the failure of the theory to take all the facts into account.

In the following table, the values of ^J -j^ are mean values taken from

the table already given on p. 132 of the inductive capacities of gases. The values of v refer to sodium light.

Gas

Mean sj — v Ko

v (observed)

1-0001387 1-000132

Authority

Mean v

Hydrogen

1-000132

1

2

1-000135

Air

1-000294

1-0002927 1-000293

1 2

1-000293

Carbon Monoxide

1-000346

1-0003350

1

1-000335

Carbon Dioxide

1-000482

1-000449 1-000451

3

2

1-000450

Nitrous Oxide ...

1-000541

1-0005151 1-000503

1 3

1-000509

Ethylene

1-000692

1-000720 1-000678

1 2

1-000699

Authorities :— 1. Mascart. 2. G. W. Walker {Phil. Trans. A. 201, p. 435). 3. Preston {Theory of Light, p. 137).

• From material collected by Pidduck, A Treatise on Electricity (Camb. Univ. Press, 1916), p. 451.

t Infra-red radiation. £ Sodium light.

534

The Electromagnetic Theory of Light [oh. xviii

Waves of Light in non-conducting Media.

Solution of Differential Equation for Plane Waves. 591. The equation of wave-propagation

^n, = a2VV dt2 *

has, as a particular solution,

y = Agi* (lx+my+nz-at) (547)

provided I2 -f m2 + v? — 1. This value of % is a complex quantity of which the real and imaginary parts separately must be solutions of the original equation. Thus we have the two solutions

% = A cos k ilx + my + nz— at) (548),

% = A sin k (Ix -f my + nz — at).

Either of these solutions represents the propagation of a plane wave. The direction-cosines of the direction of propagation are I, m, n, and the velocity of propagation is a. Usually it will be found simplest to take the value of x given by equation (547) as the solution of the equation and reject imaginary terms after the anal}Tsis is completed. This procedure will be followed throughout the present chapter; it will of course give the same result as would be obtained by taking equation (548) as the solution of the differential equation.

Propagation of a Plane Wave.

592. Let us now consider in detail the propagation of a plane wave of light, the direction of propagation being taken, for simplicity, to be the axis of x. The values of X, Y, Z, a, @, 7 must all be solutions of the differential equation, each being of the form

X = A eiK (x~at) (549).

The six values of X, Y, Z, a, ft, <y are not independent, being connected by the six equations of § 577, namely

KdX

G dt

KdY

C dt

Kd2[

C dt

d<y dy

da ds'

dft dx

dz

dx

da dy

•(A),

fi da Cdi''

fidft Cdt "

fi dy G~di

dZ dy dX

97

' dz

d_Zy

dz dx

d_Y dx

d_X

•(B).

591-592 a] Crystalline Media 535

From the form of solution (equation (549)), it is clear that all the differ- ential operators may be replaced by multipliers. We may put

d . 3 _ . 3 3 A

dt~~tKa' dx%K> fy~"3* "

The equations now become

X = 0 \ a=0

Ka v_ _ \ V±q- C 7l (A'), Cp- "\ (B').

^Z= fi

/May = -7

G

Since Kfia? = C2, it is clear that the second and third equations in (A') are identical with the third and second equations respectively in (B').

Since X = 0, a = 0, it appears that both the electric and magnetic forces are, at every instant, at right angles to the axis of x, i.e. to the direction of propagation. From the last two equations of system (A') we obtain

/3Y+yZ=0,

shewing that the electric force and the magnetic force are also at right angles to one another.

On comparing the results obtained from the electromagnetic theory of light, with those obtained from physical optics, it is found that the wave of light which we have been examining is a plane-polarised ray whose plane of polarisation is the plane containing the magnetic force and the direction of propagation. Thus the magnetic force is in the plane of polarisation, while the electric force is at right angles to this plane.

Crystalline Dielectric Media.

592 a. Let us consider the propagation of light, on the electromagnetic theory, in a crystalline medium in which the ratio of the polarisation to the electric force is different in different directions.

By equation (92), the electric energy W per unit volume in such a medium is given by

W = i (/fnZ2 + 2K12XY+ . . .).

If we transform axes, taking as new axes of reference the principal axes of

the quadric

Knx2 + 2K12xy + ... = 1,

then the energy per unit volume assumes the form

W = i (K,X* + K2Y> + KZZ%

536

The Electromagnetic Theory of Light [ch. xviii

The components of polarisation are now given by (cf. equations (89))

4tt/ = K,X, 47r# = K2 Y, 4,-irh = KZZ, so that the general equations (529) and (528) of § 574 assume the forms

G dt dy dz

KzdY_ da _ dy

G dt ~ dz dx?

K,dZ

G dt

d_l doc

da dy}

•(A"),

/j, da dZ

' dz

G dt dy

fid/3 dX G dt~ dz'

dZ dx

\

fx dy dY Gdi"dx

dX dy i

(B").

These may be compared with equations (A) and (B) of § 592.

If we differentiate the system of equations (A") with respect to the time,

and substitute the values of -j-, -j-, -j from system (B"), we obtain

K^dtX =V2Z _ d_ fdX +dY + dZ

etc.

G2 dt2 ' ' ~ dx\dx ' dy ' dz, On assuming a solution in which X, Y, Z are each proportional to

g?K (Iz+my+nz- Vt)

these equations become

KlfM

G*

V2X = X-l (IX + mY+ nZ) = 0, etc.

On eliminating X, Y and Z from these three equations, we obtain

P

V^-l

m*

n*

C2

7

C2

V

C2

• 1=0.

C2

If we put ^— = Vj2, etc., and simplify, this becomes

P

m'

w

= 0.

v2-v* v2-v2" V2-vs<

This equation gives the velocity of propagation V in terms of the direction- cosines I, in, n of the normal to the wave-front. The equation is identical with that found by Fresnel to represent the results of experiment. It can be shewn that the corresponding wave-surface is the well-known Fresnel wave- surface, and all the geometrical phenomena of the propagation of light in a crystalline medium follow directly. For the development of this part of the theory, the reader is referred to books on physical optics.

Assuming that a, /3, y as well as X, Y, Z are proportional to the exponen- tial e^te+my+nz-vt) t the equations of system (A") become

and two similar equations.

G

X = my — n(3,

592 a, 592 b] Mechanical Action 537

If we multiply these three equations by I, m, n respectively and add, we obtain

IKXX + mK2Y + nK3Z = 0,

shewing that the electric polarisation is in the wave-front.

The system (B") of equations reduce to

V

[x p a = mZ — nY,

and two similar equations, and on again multiplying by I, m, n and adding, we obtain

la + m/3 + ny = 0,

which shews that the magnetic force also is in the wave-front.

We shall not discuss crystalline media in detail in the present book since their special peculiarities are the same on the electromagnetic as on any other theory of light. The discussion of these peculiarities is a branch of the science of optics rather than of electromagnetism.

Mechanical Action.

Energy in Light-waves.

592 b. For a wave of light propagated along the axis of Ox, and having the electric force parallel to Oy, we have (cf. § 592) the solution

X = Z = 0; Y=Y0cosic(x-at),

a = /8 = 0 ; y = y0 cos k(x — at),

and this satisfies all the electromagnetic equations, provided the ratio of <y0 to Y0 is given by

jo_Ka_G_ IK

Y0 ~ G ~ fia ~ V fM '

The energy per unit volume at the point x is

~ (K P + ^72) = i (K F02 + /z7o2) cos2 k(x- at).

07T 07T

Since /xy02 = KY02, it appears that the electric energy is equal to the

magnetic at every point of the wave. The average value of cos2 k (x — at),

averaged with respect either to x or to t, is £, so that the average energy per

unit volume

= KYf = wl

8tt 8tt '

As Maxwell has pointed out*, these formulae enable us to determine the magnitude of the electric and magnetic forces involved in the propagation of

• Maxwell, Electricity and Magnetism (Third Edition), § 793.

Thus the total pressure per unit area

Sir 87T

cos2 k(x — at).

This is exactly the expression just found for the energy per unit volume Thus we see that over every wave-front there ought, on the electromagnetic theory, to be a pressure of amount per unit area equal to the energy of the wave per unit volume at that point. The existence of this pressure has been demonstrated experimentally by Lebedew * and by Nichols and Hull f, and their results agree quantitatively with those predicted by Maxwell's Theory.

Refraction and Reflection.

Conditions at a Boundary between two different media.

593. Let us next consider what happens when a wave meets a boundary between two different dielectric media 1, 2. Let the suffix 1 refer to quanti- ties evaluated in the first medium, and the suffix 2 to quantities evaluated in the second medium. For simplicity let us suppose the boundary to coincide with the plane of yz.

• Annalen der Physik, 6, p. 433. t Physical Review, 13, p. 307.

538 The Electromagnetic Theory of Light [ch. xvin

light. According to the determination of Langley, the mean energy of sun- light, after allowing for partial absorption by the earth's atmosphere, is 4-3 x 10~5 ergs per unit volume. This gives, as the maximum value of the electric intensity,

T0 = *33 C.G.S. electrostatic units = 9'9 volts per centimetre,

and, as the maximum value of the magnetic force,

7o = "033 c.G.S. electromagnetic units,

which is about one-sixth of the horizontal component of the earth's field in England.

The Pressure of Radiation.

592 c. In virtue of the existence of the electric intensity Y, there is in any

KY2

medium (§ 165) a pressure — — per unit area at right angles to the lines of

electric force. There is therefore a pressure of this amount per unit area over each wave-front. Similarly the magnetic field results (§471) in a pres-

o

sure oi amount £-*■ per unit area. brr

592 5-594]

Refraction and Reflection

539

At the boundary, the conditions to be satisfied are (§§ 137, 467) :

(1) the tangential components of electric force must be continuous,

(2) the normal components of electric polarisation must be continuous,

(3) the tangential components of magnetic force must be continuous,

(4) the normal components of magnetic induction must be continuous.

Analytically, these conditions are expressed by the equations

K.X^K.X,, Yx=Y.2y Z, = Z, (550),

/i1a1 = /u2a2, /3i = /32, 7i = 72 (551).

It will be at once seen that these six equations are not independent : if the last two of equations (550) are satisfied, then the first of equations (551) is necessarily satisfied also, as a consequence of the relation

pda = dZ_dY C dt dy dz

being satisfied in each medium, while similarly, if the last two of equations (551) are satisfied, then the first of equations (550) is necessarily satisfied. Thus there are only four independent conditions to be satisfied at the boundary, and each of these must be satisfied for all values of y, z and t. It is most convenient to suppose the four boundary conditions to be the continuity of Y, Z, j3, 7.

Refraction of a Wave polarised in plane of incidence.

594. Let us now imagine a wave of light to be propagated through medium (1), and to meet the boun- dary, this wave being supposed polar- ised in the plane of incidence. Let the boundary, as before, be the plane of yz, and let the plane of incidence be supposed to be the plane of xy. Since the wave is supposed to be polar- ised in the plane of incidence, the magnetic force must be in the plane of xy, and the electric force must be parallel to the axis of z. Hence for this wave, we may take

X=F=0, Z = Z' ei<x (x cos 8l+y sin 9l ~ Fl*'

a _ a' gt'ici (x cos 61+ y sin 0i- V\t) Q _ Q' gi«i (a:cos0i+y sin 9i- V\t)

7 = 0,

X

0

62/

/

\

(1) /,

\

/

\

540 The Electromagnetic Theory of Light [ch. xviii

and it is found that the six equations (A), (B) of p. 534 are satisfied if

of £' Z'

(552).

sin #i — cos v1

The angle Bx is seen to be the " angle of incidence " of the wave, namely, the angle between its direction of propagation and the normal (Ox) to the boundary.

Let us suppose that in the second medium there is a refracted wave, given by

X=Y=0,

2, = Z" g'^tecosdj + y sin02— V2t) tt _ a" git2 (x cos 02 + y sin 0, — V2t) Q _ Q" QiKi (x cos 63+y sin 02 - Vd)

7 = 0,

where, in order that the equations of propagation may be satisfied, we must have

°" . J8" - *" fM8)

sin#2 -cos ^2 //j..

Wit

It will be found on substitution in the boundary equations (550) and (551) that the presence of an incident and refracted wave is not sufficient to enable these equations to be satisfied. The equations can, however, all be satisfied if we suppose that in the first medium, in addition to the incident wave, there is a reflected wave given by

X=F=0,

Z = Z'" e:K^xeose3+ysine3-vit)

a __ ft'" q%k% (x cos 03+ y sin 03 - VJ) Q _ Q"i g«s(x cos 8s+y sine,- Vit)

7 = 0,

where, in order that the equations of propagation may be satisfied, we must have

/// QUI 17/1/

■4^—S-g.- 4— (5M).

sin 63 — cos vz ixx

The boundary conditions must be satisfied for all values of y and t. Since

y and t enter only through exponentials in the different waves, this requires

that we have

«! sin #x = /£„ sin 62 = k3 sin 6Z (555),

k1V1 = k2V2 = k3V1 (556).

594, 595] Refraction and Reflection 541

From (556) we must have k1 = k3, and hence from (555), sin 6X = sin 03. Since 6X and 03 must not be identical, we must have 91 = it — 03. Thus

The angle of incidence is equal to the angle of reflection.

We further have, from equations (555) and (556),

sin^ Vx

^errrv (5o7X

where v is the index of refraction on passing from medium 1 to medium 2, so that the sine of the angle of incidence is equal to v times the sine of the angle of refraction.

Thus the geometrical laws of reflection and refraction can be deduced at once from the electromagnetic theory. These laws can, however, be deduced from practically any undidatory theory of light. A more severe test of a theory is its ability to predict rightly the relative intensities of the incident, reflected and refracted waves, and this we now proceed to examine.

595. The only boundary conditions to be satisfied are the continuity, at the boundary, of Z and /3 (cf. § 593). Thus we must have

Z' + Z'" = Z" (558),

/3' + £"' = /3" (559).

On substituting from equations (552), (553) and (554), the last relation becomes

^/^cos01(Z'-Z"')=^/^cos02Z" (560),

so that all the boundary conditions are satisfied if

Z' Z" Z"'

T^^Y^T^u (561>'

K2 Mi cos2 do

where u°=^k^i (S62)-

For all media in which light can be propagated, we may take fi = l, so that

.2 cos 62 _ sin Bx cos 62 tan dx

" cos 6X sin #2 cos ti1 tan d2 ^ °''

Thus the ratio of the amplitude of the reflected to the incident ray is

Z'" _ 1 — u _ tan 02 — tan Qx _ sin(#2— #i)

'W " 1 + u ~ tan 02 + tan dt = sin (<92 + 0~) ( ^

This prediction of the theory is in good agreement with experiment.

Z" This being so, the predicted ratio of -=> is necessarily in agreement with ex-

periment, since both in theory and experiment the energy of the incident wave must be equal to the sum of the energies of the reflected and refracted waves.

542 The Electromagnetic Theory of Light [ch. xviii

Total reflection.

596. We  have  seen  (equation  (557))  that  the  angle  62  is  given  by 
     

sin 02 = -sin du v

where v is the index of refraction for light passing from medium 1 to

medium 2. If v is less than unity, the value of - sin dY may be either

greater or less than unity according as 81> or < sin-1 v. In the former case sin #2 is greater than unity, so that the value of 62 is imaginary.

This circumstance does not affect the value of the foregoing analysis in a case in which 61 > sin-1 v, but the geometrical interpretation no longer holds.

Let us denote - sin 6X by p, and /p2 —1 by q. Then in the analysis we

may replace sin 62 by p, and cos 62 by iq, both p and q being real quantities. The exponential which occurs in the refracted wave is now

qik2 (x cos 08+y sin 63 - F^t ) = giK2(.iqx+py -Vit) = q-k&x Qix^PU- V2t)m

Thus the refracted wave is propagated parallel to the axis of y, i.e. normal to the boundary, and its magnitude decreases proportionally to the factor e~K*qx. At a small distance from the boundary the refracted wave becomes imperceptible.

Algebraically, the values of Z', Z" and Z'" are still given by equations (561), but we now have

/Kofi^ cos #2 _ . I KvfjL! q V yiiaif x COS #! V yttaA'j cos 6X '

so that u is an imaginary quantity, say u = iv, and, from equations (561),

Z"' -u 1-iv

u

Since v is real, we have

Z' ' 1 + u 1+iv'

1 — iv

1 +iv

Z'" =Z'

=■ 1, so that we may take

/l — iv\ where % = arg ( J = — 2 tan-1?;.

In the reflected wave, we now have

2, — Z'" g**i (-« cos 0,+y sin <9,-F,*)

= Z e**1 (~xc°s9i+y sine,-F1t-2tau-1»)

596-598] Refraction and Reflection 543

Comparing with the incident wave, in which

Z= Z' eiKi (x cos 6i+y sin °i ~ v^

we see that reflection is now accompanied by a change of phase — 2k tan-1 v, but the amplitude of the wave remains unaltered, as obviously it must from the principle of energy.

Refraction of a Wave polarised 'perpendicular to plane of incidence.

597. The analysis which has been already given can easily be modified so as to apply to the case in which the polarisation of the incident wave is perpendicular to the plane of incidence. All that is necessary is to inter- change corresponding electric and magnetic quantities : we then have an incident wave in which the magnetic force is perpendicular to the plane of incidence, and this is what is required.

Clearly all the geometrical laws which have already been obtained will remain true without modification, and the analysis of § 596 (total reflection) will also hold without modification.

Formula (563), giving the amplitude of the reflected ray, will, however, require alteration. We have, as in equation (564), for the ratio of the amplitudes of the incident and reflected rays,

— -= r~i — (o65),

7 1 4- u \ /»

but the value of u, instead of being given by equation (563), must now be supposed to be given by

2 _ ix2 K^ cos2 02 K2 fix cos2 #j '

this equation being obtained by the interchange of electric and magnetic terms in equation (562). Taking /x2 = /u,1= 1, we obtain

cos 02 sin 02 cos 02 sin 202

tt=vl

2 cos 0! sin 01 cos Qx sin 2QX whence, from equation (565),

ry'" tan (d2 - ex)

.(566),

7 tan(0a + 0a)

giving the ratio of the amplitudes of the incident and reflected waves. This result also agrees well with experiment.

598. We notice that if 01 + 62 = 90°, then 7"' = 0. Thus there is a certain angle of incidence such that no light is reflected. Beyond this angle y" is negative, so that the reflected light will shew an abrupt change of phase of 180°. This angle of incidence is known as the polarising angle, because if a beam of non-polarised light is incident at this angle, the reflected beam will

544 The Electromagnetic Theory of Light [ch. xviii

consist entirely of light polarised in the plane of incidence, and will accordingly be plane-polarised light.