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Overview
Starting with chapters on vector analysis and electrostatics, the text covers electrostatic boundary value problems, formal and microscopic theories of dielectric electrostatics and of magnetism and matter, electrostatic energy, steady currents, and induction. Additional topics include magnetic energy, circuits with nonsteady currents, Maxwell's equations, radiation, electromagnetic boundary value problems, and the special theory of relativity. Exercises appear at the end of each chapter and answers to odd-numbered problems are included in one of several helpful appendixes.
Product Details
| ISBN-13: | 9780486802992 |
|---|---|
| Publisher: | Dover Publications |
| Publication date: | 01/09/2015 |
| Sold by: | Barnes & Noble |
| Format: | eBook |
| Pages: | 640 |
| File size: | 51 MB |
| Note: | This product may take a few minutes to download. |
About the Author
Munir H. Nayfeh is Professor of Physics at the Univeristy of Illinois at Urbana-Champaign.
Morton K. Brussel is Professor Emeritus of Physics at the University of Illinois at Urbana-Champaign.
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Electricity & Magnetism
By Munir H. Nayfeh, Morton K. Brussel
Dover Publications, Inc.
Copyright © 1985 Munir H. Nayfeh and Morton K. BrusselAll rights reserved.
ISBN: 978-0-486-80299-2
Contents
ONE: VECTOR ANALYSIS,TWO: ELECTROSTATICS,
THREE: ELECTROSTATIC BOUNDARY VALUE PROBLEMS,
FOUR: FORMAL THEORY OF DIELECTRIC ELECTROSTATICS,
FIVE: THE MICROSCOPIC THEORY OF DIELECTRICS,
SIX: ELECTROSTATIC ENERGY,
SEVEN: STEADY CURRENTS,
EIGHT: MAGNETISM OF STEADY CURRENTS,
NINE: FORMAL THEORY OF MAGNETISM AND MATTER,
TEN: THE MICROSCOPIC THEORY OF MAGNETISM,
ELEVEN: INDUCTION,
TWELVE: MAGNETIC ENERGY,
THIRTEEN: CIRCUITS WITH NONSTEADY CURRENTS,
FOURTEEN: MAXWELL'S EQUATIONS,
FIFTEEN: RADIATION,
SIXTEEN: ELECTROMAGNETIC BOUNDARY VALUE PROBLEMS,
SEVENTEEN: THE SPECIAL THEORY OF RELATIVITY,
APPENDIX I: SYSTEM OF UNITS,
APPENDIX II: DIVERGENCE, CURL, GRADIENTS, AND LAPLACIAN,
APPENDIX III: SOME FUNDAMENTAL CONSTANTS OF PHYSICS,
APPENDIX IV: SOME SI DERIVED UNITS WITH SPECIAL NAMES,
ANSWERS TO ODD-NUMBERED PROBLEMS,
INDEX,
CHAPTER 1
VECTOR ANALYSIS
1.1 Properties of Vectors and Coordinate Systems
Ordinary numbers are called scalars. They may be real or complex numbers. In contrast to scalars, we have other quantities called vectors. These quantities combine with each other differently than scalars. In physics they are used to represent objects that have both magnitude and direction, the prototype of which is a displacement. Mathematically, vectors are simply quantities that behave and combine according to the following rules.
1. The sum of two vectors u and v is another vector: u + v = w. The sum is a commutative binary operation; that is, u + v = v + u.
2. Under summation, the associative law holds. For vectors u, v, and w,
(u + v) + w = u + (v + w) = u + v + w
3. Any vector can be "multiplied" by a scalar to yield another vector.
We shall represent vectors geometrically by directed line segments (i.e., arrows). The magnitude of the vector is proportional to the length of the line segment, and the direction is given by the orientation of the arrow—that is, the direction in which it points. The rules to be followed in performing this (vector) addition geometrically are these (see Fig. 1.1): On a diagram drawn to scale lay out the displacement vector u; then draw v with its tail at the head of u, and draw a line from the tail of u to the head of v to construct the vector sum w. This is a displacement equivalent in length and direction to the successive displacements u and v. This procedure can be generalized to obtain the sum of any number of successive displacements.
1.1.1 Base Vectors and Coordinate Systems
Choosing a coordinate system in space is essentially equivalent to choosing a set of base vectors. If we choose a cartesian coordinate system (Fig. 1.2), our base vectors are chosen to be along three fixed mutually perpendicular (orthogonal) fixed directions called the x, y, and z directions. If we represent a vector by an arrow, the perpendicular projections of the arrow upon the three coordinate axes are called the cartesian components of the vector in these directions. In terms of these components, the magnitude of a vector A is as follows:
Magnitude of A [equivalent to] |A| = (A2x + A2y + A2z)1/2
A unit vector [??] is that vector which when multiplied by the magnitude |A|, yields the vector A; that is, A [equivalent to] |A|[??]. It provides a means for indicating direction. Unit vectors along x, y, and z coordinate axes (cartesian) are denoted by [??], [??], [??], respectively. They provide a convenient and fundamental set of base vectors. In terms of cartesian unit vectors, any vector A is represented by
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.1)
where Ax, Ay and Az are the components of A along the [??], [??], and [??] directions, respectively.
We shall restrict our attention to cases where the base vectors form an orthogonal set. Moreover the magnitude of each base vector will be taken as unity (orthonormal). The most commonly used base vectors in ordinary three-dimensional space are the unit base vectors ([??], [??], [??]). These vectors are considered to be constant vectors. Neither their directions nor their magnitudes depend on where they are located with respect to some reference point in space. It is the constancy of this orthonormal set of base vectors that we wish to emphasize by the word cartesian.
The representation of vectors using the unit vectors is very useful in vector manipulations. For example, to add A to B we simply add the cartesian components:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
It is frequently convenient to use other sets of base vectors whose directions do happen to depend on their locations (curvilinear base vectors). For example, we shall define and often use a spherical coordinate system and a cylindrical coordinate system. For each of these systems we shall find an orthonormal set of associated base vectors that depend on where in space they are located.
Recall that the cartesian unit vector [??] may be defined as the unit vector that is perpendicular to any plane x = constant. Similarly, for the [??] and [??] unit vectors we respectively associate the planes y = constant and z = constant. Now, there are other surfaces that one can describe that correspond to some geometrical variable being constant. If we can find three surfaces, defined by (three) geometrical variables, that intersect each other perpendicularly at a point, then at this point we can define three associated mutually perpendicular vectors that are normal to these surfaces. In describing the spherical and cylindrical systems, we cite two instances where we find it useful to do so. (There are many others.)
In the cylindrical coordinate system (Fig. 1.3) we define a set of base vectors at a point by considering surfaces, two of which are planes and one of which is a cylinder. The surfaces are denoted by the following equations:
(a) z = constant
(b) ρ = constant = √ x2 + y2
(c) φ = constant = tan-1 (y/x)
In equation (a) the z coordinate specifies a set of parallel planes. It is defined by reference to a reference plane called the z = 0 plane. The unit vector z is then a constant vector pointing in the (positive) direction (which may be chosen arbitrarily), perpendicular to the z = constant planes. The z axis is chosen to be a line pointing in the z direction (for -∞ < z< + ∞).
In equation (b) the ρ coordinate is defined with reference to the z axis by a set of cylindrically circular surfaces that intersect the z = constant planes perpendicularly. The distance ρ from a particular surface to the z axis is the radius of the cylindrical surface. The unit vector [??] is perpendicular to the cylindrical surface, pointing away from the z axis. Its direction depends upon what geometrical point of the circle resulting from intersection of the planes z = constant and ρ = constant is considered. Thus, in Fig. 1.3, [??] is a function of the variable φ defined in equation (c) (for ρ > 0).
In equation (c) the only surfaces that can perpendicularly intersect the previously defined surfaces at all points of intersection are planes that contain the z axis. One such plane is called the φ = 0 plane and is chosen arbitrarily. The [??] unit vectors lie perpendicular to the φ = constant surfaces, and depend upon the angle φ of the plane with reference to the φ = 0 plane.
The intersection of the surfaces described by equations (a), (b), and (c) above locate points in space, just as the intersection of the cartesian coordinate planes do. However, the cylindrical unit vectors are well specified only when a point (not on the z axis) is specified. (The origin is specified by setting z = 0 and ρ = 0.) Once this has been done by assigning values of (ρ, φ, z) or (x, y, z) to the point, any vector may be expressed in terms of the cylindrical unit vectors ([??],[??], [??]) at that point.
One can easily show that these cylindrical unit vectors are related to the cartesian unit vectors by the following relations.
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.2)
Remember that [??] and [??] depend upon the coordinate φ. Thus, for any vector A and for a point at which the unit vectors are [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] since {[??], [??],[??]} form an orthonormal set. If A(r) is a vector point field, the natural triad of base vectors used to express A will be that defined by the location r. Note that the displacement vector r to a point (z, ρ, φ) is given by r = ρ[??] + z[??].
We shall not describe the spherical coordinate system (Fig. 1.4) in the detail used above for the cylindrical system, except to note that the constant surfaces chosen are as follows:
(a) r = constant √ x2 + y2 + z2, which describes a sphere of radius r with respect to the origin.
(b) θ = constant = cos -1(z/r), which describes a right circular cone with opening angle θ.
(c) φ = constant = tan-1(y/x), which describes a plane containing the axis of the cone in (b).
The unit vectors prescribed by these surfaces are denoted [??], [??], [??] respectively, and form an orthonormal set once the point (not at the origin and not at the z axis) located by the intersection of the three orthogonal surfaces is determined. These unit vectors are given in terms of the cartesian unit vectors by the following relations.
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.3)
If A is a member of a vector field, A (r), then at every point given by the displacement r one can express A (r) in terms of the base vectors associated with that point: A(r) = Ar[??] + Aθ[??] + Aψ[??] where Ar is the projection of A on [??] and so on. The displacement vector to a point (r, θ, φ) is given simply r = [??]r.
1.1.2 The Scalar Product (Dot Product)
An important concept in vector algebra is that of the scalar product of two vectors. It is denoted by A · B and also called dot product or inner product. It is defined according to the following rule: A · B = |A B| cos α where |A| and |B| are the magnitudes of A and B, and α is the angle between A and B. It can easily be seen that the scalar product, as defined, has the following properties: Two vectors whose scalar product is zero are said to be orthogonal; that is, if A · B = 0, A is said to be orthogonal to B. The unit cartesian vectors [??], [??], and [??] are said to constitute an orthonormal set of base vectors because they are orthogonal to each other and their magnitudes are normalized to unity.
1.1.3 The Vector Product (Cross Product)
We have seen that we can assign a scalar to any pair of vectors. The operation that does this is called the scalar product. We now wish to assign a vector quantity to any pair of vectors, A and B, and so we define what is known as a vector product (or cross product); it is denoted by A × B. The direction of the vector product is taken to be perpendicular to the plane determined by the pair of vectors. Its magnitude is given by the area of the parallelogram whose sides are formed by the vector pair. Therefore, if [??] is a unit vector perpendicular to the plane formed by the vector pair (A, B), then the vector product is defined according to the following rule.
A x B [equivalent to] |A B| sin α [??] (1.4)
So far the definition given remains ambiguous because the normal to the plane of A and B may point "up" or "down." To specify which way points, we use the right-hand screw convention. We say that if A is rotated to the direction of B, through the angle a (≤ 180°), then the same sence of rotation given to a right-handed screw determines n, which points along the direction of advance of the screw as it is rotated (Fig. 1.5).
In terms of cartesian unit vectors the vector product is expressed as
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.5)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.6)
Whenever we have a set of three orthonormal vectors, [??]1, [??]2, and [??]3, we say we have a right-handed system when [??]i × [??]j = [??]k where i, j, k are in the order (1, 2, 3), (2, 3, 1), or (3, 1, 2). These are cyclic permutations of the integers 1, 2, 3. Note that for a right-handed system given by the triplet {[??]1 [??]2, [??]3}, one has [??]1 · ([??]2 × [??]3) = 1. T he cartesian coordinate system we have used is right-handed if we identify x with 1, y with 2, and z with 3. A left-handed system is a mirror image of a right-handed system.
Two useful identities to be remembered are as follows:
1. Triple scalar product
A · (B x C) = (A x B) · C = (C x A) · B (1.7)
(It is the volume of a parallelepiped whose edges are A, B, and C.)
2. Triple vector product
A x (B x C) = B(A · C) - C(A · B) (1.8)
The latter is frequently known as the "back cab" rule. It will be noted in Eq. (1.7) that the dot (·) and the cross (×) may be freely interchanged so long as {A, B, C} remain in cyclic order.
1.2 Elements of Displacement, Area, and Volume; Solid Angle
1.2.1 Element of Displacement
Consider two points in space (x, y, x) and (x + Δx, y + Δy, z + Δz). The first point is displaced relative to the second by the displacement Δr; that is,
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.9)
expressed in a cartesian system. A differential element of displacement is consequently written
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.10)
We now wish to express Δr (or dr) in terms of cylindrical and spherical coordinates and their associated unit vectors. We again assume that Δr may be made arbitrarily small, in the limit calling it dr.
Consider Fig. 1.6, where two points are displaced by Δr. In the cylindrical coordinate system we have base vectors that are different at the two points 1 and 2. Thus
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.11)
where
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.12)
(Continues...)
Excerpted from Electricity & Magnetism by Munir H. Nayfeh, Morton K. Brussel. Copyright © 1985 Munir H. Nayfeh and Morton K. Brussel. Excerpted by permission of Dover Publications, Inc..
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