Chapter 6

Vector Calculus

References: Skilling, Fundamentals of Electric Waves 2; Lorrain & Corson, Intro. to Electromag-netic Fields and Waves, 1; Hecht, Optics, 3.2, Appendix 1In 1864, James Clerk Maxwell published a paper on the dynamics of electromagnetic elds, in

which he collected four previously described equations which relate electric and magnetic forces,modied one, and combined them to demonstrate the true nature of light waves. The four equationsare now collected into a group that bears his name.To interpret the four Maxwell equations, we must rst understand some concepts of dierential

vector calculus, which seems intimidating but is really just an extension of normal dierentiationapplied to scalar and vector elds. For our purposes, a scalar eld is a description of scalar values inspace (one or more spatial dimensions). One example of a scalar eld is the temperature distributionin the air throughout the atmosphere. Obviously, a single number is assigned to each point in thespace. On the other hand, a vector eld denes the values of a vector quantity throughout a volume.For example, the vector eld of wind velocity in the atmosphere assigns a three-dimensional vectorto each point in space. In my notation, scalar quantities are denoted by normal-face type andvectors (usually) by overscored bold face, e.g., f [x, y, z] and g[x,y,z] describe scalar and vector elds,respectively. Unit vectors (vectors with unit magnitude, also called unit length) are indicated bybold-faced characters topped by a caret,In vector calculus, spatial derivatives are performed on vector AND scalar elds to derive other

vector or scalar elds. The rst-order dierential operator (called del) has three components:

=

x,

y,

z

where are unit vectors in the x, y, and z directions respectively. The del-operator may be appliedto a scalar eld to create a 3-D vector eld (the gradient operation), or to a vector eld to createa scalar eld (the divergence), or to a vector eld to create a 3-D vector eld (the curl). The rsttwo operations may be generalized to operate on or generate 2-D vectors, whereas the curl is denedonly for 3-D vector elds.

6.1 GRADIENT

Derives a Vector Field f from a Scalar Field fApplying to a scalar eld f [x, y, z] with three dimensions (such as the temperature of air at all

points in the atmosphere) generates a eld of 3-D vectors which describes the spatial rate-of-changeof the scalar eld, i.e., the gradient of the temperature at each point in the atmosphere is a vectorthat describes the direction and magnitude of the change in air temperature. In the 2-D case wherethe scalar eld describes the altitude of landform topography, the gradient vector is the size anddirection of the maximum slope of the landform.

f [x, y, z] f

x,f

y,f

z

= x

f

x+y

f

y+z

f

z= a vector

43

44 CHAPTER 6. VECTOR CALCULUS

As implied by its name, the gradient vector at [x, y, z] points uphill in the direction of maximumrate-of-change of the eld; the magnitude of the gradient |f | is the slope of the scalar eld.x:

Scalar eld represented as contour map and as 3-D display.

Gradient of the scalar eld is a vector eld. At each coordinate [x, y], the vector points uphilland its length is equal to the slope.

6.2 DIVERGENCE

Derives a Scalar Field g from a Vector Field g

g [x, y, z]

x,

y,

z

[gx, gy, gz]

=gxx

+gyy

+gzz

= a scalar

6.3. CURL 45

The divergence at each point in a vector eld is a number that describes the total spatial rate-of-change, such as the total outgoing vector ux per unit volume. Consider a vector eld g and aninnitesmal surface element which is described by the normal vector da directed outward.from thevolume. The ux z of the vector eld through the surface element is the scalar (dot) product ofthe eld with the normal to the surface:

gda = dz= z =

Zsurface area

gda

The divergence of the eld describes total ux through the surface area A enclosing the volume.Unless the volume encloses a net source or sink of the vector eld (a point from which thevector eld diverges or converges), then the divergence must be zero:

g =Zsurface area

gda= 0 if no source or sink of vector eld in volume

.

A vector eld with nonzero divergence has a disparity between input and output ux

6.3 CURL

Derives a 3-D VectorField from a 3-D Vector Field gThe curl of a vector eld describes a spatial nonuniformity of the 3-D vector eld g[x,y,z].

g [x, y, z] = x y z

xy

z

gx gy gz

= x

gzy gy

z

+ y

gxz gz

x

+ z

gyx gx

y

= a vector

To visualize curl, imagine a vector eld that describes motion of a uid (e.g., water or wind). Ifa paddle-wheel placed in the uid does not revolve, the eld has no curl. If the wheel does revolve,the curl is nonzero. The direction of the curl vector is that of the axis of the paddlewheel when

46 CHAPTER 6. VECTOR CALCULUS

the rotation is maximized and its magnitude is that rotation rate. The algebraic sign of the curl isdetermined by the direction of rotation (clockwise = positive curl). The paddle will rotate only ifthe vector eld is spatially nonuniform. Note that some points in the eld can have zero curl whileothers have nonvanishing curl. Both vector elds shown in the examples of divergence have zerocurl, since a paddle wheel placed at any point in either eld will not rotate.

Two vector elds with nonzero curl. The paddlewheel rotates in both cases.

6.4 Maxwells Equations

By 1864, much was known about electric and magnetic eects on materials. Faraday had discoveredthat a time-varying magnetic eld (such as from a moving magnet) can generate an electric eld,and Ampere demonstrated the corresponding eect that a time-varying electric eld (as from amoving electric charge) produces a magnetic eld. Both electric and magnetic elds were known tobe vectors that could vary in time and space: E[x,y,z,t] and B[x,y,z,t]. The electric eld E has unitsof Volts/meter, while the magnetic eld B is measured in Tesla or Gauss:

1 Tesla = 104 Gauss = 1Weber

meter= 1

Newton

AmpmeterTwo other vector elds are required when describing propagation of electromagnetism through

matter: the electric displacement D and the magnetic eld intensity H. D denes the total electriceld in a material, which is the sum of the external eld E and any local eld P generated withinthe matter due to polarization from the incident E eld. H is a similar construct for magnetic elds.E and D, and B and H are related by the so-called constitutive equations:

D = E

B = H

where is the electric permittivity of the material and is its magnetic permeability. These re-spective describe how well the electric and magnetic elds penetrate matter. Since we will considerpropagation of light only in vacuum, we can ignore the elds D and H in matter. In vacuum, andare denoted 0 and 0 and both are set to unity in CGS units. In MKS units, the quantities are:

0 = 4 107 newton/amp20 = 8.85 1012 farad/meter.

As is true for the refractive index n, the permittivity and permeability in matter are larger than invacuum, > 0, and > 0. In fact (though we wont discuss it in detail), and in fact determine

BTHHighlight

BTHHighlight

6.4. MAXWELLS EQUATIONS 47

v and n via:

v =1

n =c

v=

00

Maxwell collected the four dierential equations relating the electric vector eld E and themagnetic vector eld B listed below and solved them to derive the character of electromagneticwaves.

E = Gauss Law for Electric Fields, Coulombs Law

B = 0, Gauss Law for Magnetic FieldsB

t= E, Faradays Law of Magnetic Induction

+E

t= B

J, Amperes Law

The scalar is the electric charge density [coulombs/m3] and the vector eld J is the electric currentdensity [amps/m3]. These source terms only are nonzero in media, such as copper wire. If weconsider the propagation of light only in a vacuum, neither electric charges nor conductors arepresent and both source terms vanish.The denition of curl may be used to rewrite the four vector equations of Maxwell as eight scalar

equations:

Exx

+Eyy

+Ezz

= 0 GaussLaw for E (1)

Bxx

+Byy

+Bzz

= 0 Gauss Law for B (2)

Bxt

=Ezy

Eyz

Faradays Law (3)

Byt

=Exz

Ezx

(4)

Bzt

=Eyx

Exy

(5)

+Ext

=Bzy

Byz

Amperes Law (6)

+Eyt

=Bxz

Bzx

(7)

+Ezt

=Byx

Bxy

(8)

In the general case, the electric eld and magnetic elds can have the form:

E [x, y, z, t] = xEx [x, y, z, t] + yEy [x, y, z, t] + zEz [x, y, z, t]

B[x, y, z, t] = xBx[x, y, z, t] + yBy[x, y, z, t] + zBz[x, y, z, t]

We will now solve these equations for a single specic case: an innite plane electric eld wavepropagating in vacuum toward z = +. The locus of points of constant phase (often called awavefront) of a plane wave is (obviously) a plane. The electric eld E has no variation along x or yat a particular value of z, but can vary with z; this variation will be shown to be sinusoidal. Thisconstraint aects the derivatives of the components of the electric eld:

Exx

=Exy

=Eyx

=Eyy

=Ezx

=Ezy

=0 (9)

48 CHAPTER 6. VECTOR CALCULUS

and the 4-D vector eld E[x,y,z,t] can be written as:

E [x, y, z, t] = E [z, t] = Ex [z, t] +Ey [z, t] +Ez [z, t] (10)

From (9) and Gauss law for electric elds (1), we nd that:

Exx

+Eyy

+Ezz

= 0 = Ezz

= 0 (11)

Since the derivative of Ez with respect to z vanishes, then The z-component of the electric eld Ezis an arbitrary constant, which we select to be 0:

Ez [x, y, z] = 0 (12)

Therefore, the electric eld is now expressable as:

E [x, y, z] = Ex [z, t] +Ey [z, t] (13)

i.e., the only existing electric eld is transverse to z! This alone is a signicant result. We cansimplify eq. (13) by rotating the coordinate system about the z axis such that E is aligned with thex-axis so that

Ey[z, t] = 0 by assumption (14)

The expression for the electric eld is:

E [x, y, z] = Ex [z, t] (15)

Given the expression for E[x,y,z,t], we can substitute these results into Faradays Law (eqs. 3,4,5)to nd the magnetic eld:

Bxt

=Ezy

Eyz

= 0 Eyz

= Bxt

=Eyz

= 0 = Bx [t] is constant (3)

Byt

=Exz

Ezx

=Exz

= Byt

= Exz

(4)

Bzt

=Eyx

Exy

= 0 = Bz [t] is constant (5)

We can arbitrarily set the constant term Bz = 0, so the only remaining equation is:

Byt

=Exz

(4)

which says that the time derivative of the magnetic eld By is equal to to the negative of the spacederivative of Ex. We can now nd a relation between By and Ex by standard solution techniquesof dierential equations. Assume that: E is a vector eld that varies sinusoidally with z :

E [x, y, z, t] = x Ex [z, t] = x E0 cos [kz t]= Ex

z= kE0 sin [kz t]

= By [z, t] =Z

Exz

dt = (kE0)Z

sin [kz t] dt

By [z, t] = +kE0 cos [kz t]

= E0k

cos [kz t]

=E0k

cos [kz t]=

E0vcos [kz t]

= B [z, t] = yE0vcos [kz t]

6.5. PHASE VELOCITY OF ELECTROMAGNETIC WAVES 49

where v is the phase velocity of the electromagnetic wave

By =E0vcos [kz t] = Ex

v= Ex = vBy

Note that the only existing component of B is By, which is perpendicular to Ex. Also note thatthe sinusoidal variations of E and B have the same arguments, which means that they oscillate inphase.

6.5 PHASEVELOCITYOF ELECTROMAGNETICWAVES

Given the form for the plane electromagnetic wave in a vacuum, we can now use the three Ampererelations to nd something else useful:

+Ext

=Bzy

Byz

(6)

+Eyt

=Bxz

Bzx

(7)

+Ezt

=Byx

Bxy

(8)

Because E = xE0, only (6) does not vanish:

t(E0 cos [kz t]) =

z

E0vcos [kz t]

= E0 ( sin [kz t]) = E0v (k sin [kz t])

= E0 = E0kv= = k

v=1

v2= v2 =

1

v =q

1

In vacuum, 0, 0,v c, and c =q

100

. The permittivity and permeability of free

space (vacuum) can be measured in laboratory experiments, thus allowing a calculation of the phasevelocity of electromagnetic waves. Using the values for 0 and 0 given above, we nd:

00 =

8.85 1012 coul

v-m

1.26 106 J

amp2 m

= 1.11 1017 coulv-m

Jcoul2m

s2

= 1.11 1017 1

v-m J-s

2

coul-m

= 1.11 1017 s2

m2

= c =s

1

00= 2.99 108m

s, which agrees well with experiment .

50 CHAPTER 6. VECTOR CALCULUS

6.6 CONSEQUENCES OF MAXWELLS EQUATIONS

1. The wave requires both electric and magnetic elds to propagate, and they copropagate, i.e.,they pull each other along.

2. he electric and magnetic elds of an electromagnetic wave are mutually perpendicular.

3. The Copropagation of E and B: The wave travels in a direction mutually perpendicular toboth E and B, and in fact the propagation direction is dened by the direction:

S = EB the Poynting vector

4. In vacuum, E and B are in-phase, which means that the phases of the sinusoidal variation ofE and B are identical (in matter, the phases of the elds often are not so).

5. Both E and B travel at c, the phase velocity of the wave.

6. Energy is carried by both the electric and magnetic elds, and the magnitude of the energyE E20 .

7. There is no limitation on the possible frequencies of the waves, i.e., [0 ], which impliesthe allowed wavelengths are in the interval [ 0]

Propagation of electric and magnetic elds in vacuum.

6.7 Optical Frequencies Detector Response

The general equation for a traveling electromagnetic wave is:

y [z, t] = A0 cos [kz t] = A0 cosh2 z t

i= A0Re

nei(kzt)

oWe see electromagnetic radiation with detectors, i.e., devices which respond in some way to

incident electromagnetic radiation. The human eye is sensitive only to visible light, i.e., light withwavelengths in the range [400nm 700nm]. This is not the case for all life, however. Thepit viper can see radiation emitted by humans at a wavelength of about 10 m; it needs specialreceptors on the sides of its head to do this.As shown in the plot of the electromagnetic spectrum, the frequencies of visible wavelengths are

quite high: ' 1015 Hz. The temporal period of an optical wave is therefore T = 1 ' 1015 S.Human visual receptors cannot respond fast enough to detect the periodic oscillation of the waveamplitude; we see an invariant brightness. Note that this limitation exists for all optical detectors;they all respond to the average brightness. The same is true for hearing; your ear cannot detectthe variation of sound pressure due to the oscillation at frequencies above a few Hz. Because waterwaves have a much lower frequency, the amplitude and phase of the wave can be measured.

6.7. OPTICAL FREQUENCIES DETECTOR RESPONSE 51

The average amplitude of a sinusoidal wave is:

hy [z, t]i = 1T d

Z Td0

y [z, t] dt

=1

T d

Z Td0

A0 cos [kz t]dt

= A0Td

sin [kz t] |t=Tdt=0

Since y [z, t] is sinusoidal, the average value of the wave will tend to zero unless Td is smaller than thewaves temporal period. However, the intensity (squared-magnitude) of the wave does not averageto zero:

E y2 [z, t] = 1Td

Z Td0

y2 [z, t]dt

=1

Td

Z Td0

A20 cos2 [kz t] dt

=A20Td

Z Td0

cos2 [kz t] dt

=A20TdTd2=A202

= E y2 [z, t] = A202

if Td >> 1

because the average value of cos2 [x] = 12 .

Optical detectors are sensitive to time-averaged intensity, not amplitude.

52 CHAPTER 6. VECTOR CALCULUS