Field Theory

1.01Introduction to Electromagnetics

Stated in a simple fashion, electromagnetics is the study of the effects of electric charges at rest and in motion. From elementary physics we know that there are two kinds of charges: positive and negative. Both positive and negative charges are sources of an electric field. Moving charges produce a current, which gives rise to a magnetic field. Here we tentatively speak of electric field and magnetic field in a general way. A field is a spatial distribution of a quantity, which may or may not be a function of time. In other words, time varying electric and magnetic field, and vice versa. In other words, time varying electric and magnetic fields are coupled, resulting in an electromagnetic field. Under certain conditions, time-dependent electromagnetic fields produce waves that radiate from the source.

The concept of fields and waves is essential in the explanation of action at a distance. The possibilities of satellite communication and of receiving signals from space probes millions of miles away can be explained only by postulating the existence of electric and magnetic fields and electromagnetic waves. The subject of electromagnetics is concerned with the principles and applications of the laws of electromagnetism that govern electromagnetic phenomena.

Electromagnetics is of fundamental importance and indispensable in understanding the principle of cathode-ray oscilloscopes, radar, satellite communication, electromagnetic compatibility problems and so on. Circuit concepts represent a simplistic, a special case, of electromagnetic concepts. When the source frequency is very low so that the dimensions of a conducting network are much smaller than the wavelength, resulting in a quasi-static situation, which simplifies an electromagnetic problem to a circuit problem.

Two situations illustrate the drawback of circuit-theory concept and the need for electromagnetic-field concepts. A monopole antenna of the type we see on a walkie-talkie. On transmit; the source at the base feeds the antenna with a message-carrying current at an appropriate carrier frequency. From a circuit theory of view, the source feeds into an open circuit because the upper tip of the antenna is not connected to anything physically; hence no current would flow, and nothing would happen. This viewpoint cannot explain how communication takes place. When the length of the antenna is an appreciable part of the carrier wavelength, a non-uniform current will flow along the open-ended antenna. This current radiates a time-varying electromagnetic field in space, which propagates as an electromagnetic wave and induces currents in other antennas at a distance.

In another situation in which an electromagnetic wave is incident from the left on a large conducting wall containing a small hole. Electromagnetic fields will exit on the right side of the wall at points, such as points that are not necessarily directly behind the hole. Circuit theory is obviously inadequate for the determination of the field.

1.02 Vector AnalysisThe use of vector analysis in the study of electromagnetic field theory helps to give a clearer understanding of the physical laws that mathematics describes. Vectors allow expressing these physical relations in a more concise form as a whole, rather than in its scalar component parts.

Scalar. A quantity that is characterized only by magnitude and algebraic sign is called a scalar. Examples of physical quantities that are scalars are time, mass, temperature etc. They are represented by italic letters, such as A, B, C, a, b and c.

Vector. A quantity that has direction as well as magnitude is called is vector. Force, velocity, displacement, and acceleration are examples of vector quantities. They are represented by letters in boldface type, such as A, B, C, a, b and c. A vector can be represented geometrically by an arrow whose direction is appropriately chosen and whose length is proportional to the magnitude of the vector.

Field. If at each point of a region there is a corresponding value of some physical function, the region is called a field. Fields may be classified as either scalar or vector, depending upon the type of function involved.

If the value of the physical function at each point is a scalar quantity, then the field is a scalar field, the temperature of the atmosphere, the height of the surface of the earth above sea level are examples of scalar fields.

When the value of the function at each point is a vector quantity, the field is a vector field. The wind velocity of the atmosphere, the force of gravity on a mass in space and the force on a charge body placed in an electric field, are examples of vector fields.

Sum and Difference of Two Vectors. The sum of any two vectors A and B is illustrated in Fig. 1.1(a). It is apparent that it makes no difference whether B is added to A or A is added to B. Hence

(1-1)When the order of the operation may be reversed with no effect on the result, the operation is said to obey th commutative law.

Figure 1-1(b) illustrates the difference of any two vectors A and B

(a)

(b)

Figure 1.1It is to be remembered that the negative of a vector is a vector of the same magnitude, but with a reversed direction.1.3 The Rectangular Coordinate SystemIn the rectangular coordinate system there are three coordinate axes mutually at right angles to each other, indicated as x,y and z axes. It is customary to choose a right-handed coordinate system, in which a rotation of the x-axis into y-axis would cause a right-handed screw to progress in the direction of the z-axis. Figure 1.2 shows a right-handed rectangular co-ordinate system. Figure 1.2A point is located by giving its x, y, and z coordinates. These are the distances from the origin to the intersection of the perpendicular dropped from the point to the x, y, and z axes. An alternative way of interpreting a point is to consider the point as the intersection of three surfaces as shown in the figure 1.3(a), the planes x = constant, y = constant, and z = constant, the constants being the coordinate value of the point.

Figure 1.3b shows the points P and Q whose coordinates are (1, 2, 3) and (2,-2, 1), respectively. Point P is therefore located at the common point of intersection of the planes x = 1, y = 2, and z = 3, while point Q is located at the intersection of the planes x = 2, y = -2, z = 1. (a) (b)

Figure 1.3 (a) A point is defined as intersection of three orthogonal surfaces. (b)The location of points P(1,2,3) and Q(2,-2,1). 1.4 The Dot ProductThe dot product , or scalar product of two vectors is a scalar quantity whose magnitude is equal to the product of the magnitude of the two vectors and the cosine of the angle between them. This type of multiplication is indicated by a . (dot) placed between the two vectors to be multiplied.

Given two vectorsAandB, the dot product,orscalar product,isdefined as the product of the magnitude of A, the magnitude of B, and the cosine of the smaller angle between them.

The dot, or scalar, product is a scalar, and it obeys the commutative law, for the sign of the angle does not affect the cosine term. The expression AB is read A dot B.

Figure 1.4 : a) The scalar component of Bin the direction of the unit vector ais

Ba.(b) The vector component of Bin the direction of the unit vector ais (Ba)a.A more helpful result is obtained by considering two vectors whose rectangular components are given, such as A=Axax +Ayay+Azaz and B=Bxax +Byay+Bzaz. The dot product also obeys the distributive law, and, therefore, AByields the sum of nine scalar terms, each involving the dot product of two unit vectors. Because the angle between two different unit vectors of the

rectangular coordinate system is 90,we then have

The remaining three terms involve the dot product of a unit vector with itself, which is unity, giving finally

A vector dotted with itself yields the magnitude squared, or

and any unit vector dotted with itself is unity,

1.5 The Cross Product

Given two vectors A and B, we now define the cross product, or vector product, of A and B, written with a cross between the two vectors as AB and read A cross B. The cross product AB is a vector; the magnitude of AB is equal to the product of the magnitudes of A, B, and the sine of the smaller angle between A and B; the direction of AB is perpendicular to the plane containing A and B and is along one of the two possible perpendiculars which is in the direction of advance of a right-handed screw as A is turned into B.

As an equation we can write,

Figure 1.5: The direction of AB is in the

direction of advance of a right-handed screw

as A is turned into B.

Figure 1.6: Cross Product of A and B, AB

Consider two vectors whose rectangular component are given such as A = Axax+Ayay+Azaz and

B = Bxax+Byay+Bzaz . The cross product of the two vectors yields the sum of nine vector terms each containing the vector product of two unit vectors. Since the angle between any two unit vectors is 90o, we then have,

axay = az, ay az = ax, azax = ayaxax = ayay = azaz = 0

Finally we have ,

AB = (Ay Bz - Az By) ax + (Az Bx - AxBz) ay + (Ax By - Ay Bx) azIn the determinant form,

1.6 Cartesian Coordinates

A point P(x1, y1, z1) in cartesian coordinates is the intersection of three planes specified by x = x1, y= y1, z = z1 as shown in Fig. It is a right-handed system with base vectors ax, ay, and az satisfying the following relations:

The position vector to the point P(x1, y1, z1)

A vector A in cartesian Coordinates can be written as

Figure 1.7: The differential volume element in rectangular coordinates; dx, dy, and dz

are, in general, independent differentials.

1.8 Cylindrical Coordinate system

In cylindrical coordinating a point P(1, 1, z1) is the intersection of a circular cylinder surface = 1, a half-plane containing the z-axis and making an angle = 1 with xz-plane, and a plane parallel to the xy-plane at z = z1. As indicated in figure, angle is measured from the positive x-axis, and the base vector a is tangential to the cylindrical surface. The following right-hand relations apply:

Figure 1.8: (a) The three mutually perpendicular surfaces of the circular cylindrical co-ordinate system. (b) The three unit vectors of the circular cylindrical coordinate system.Cylindrical Coordinates are important for problems with long line charges or currents, and in places where cylindrical or circular boundaries exist. The two-dimensional polar coordinates are a special case at z=0.

A vector in cylindrical coordinates is written as

A = A a + A a + AzazTwo of the three coordinates, r and z are themselves lengths, however, is an angle requiring a metric coefficient to convert d to dl. The general expression for a differential length in cylindrical coordinates is,

1.9 Spherical CoordinatesA point P(R1, 1, 1) in spherical coordinates is specified as the interaction of the following three surfaces: a spherical surface at the origin with a radius R = R1; a right circular cone with its apex at the origin, its axis coinciding with the z-axis