UNIT 1. INTRODUCTION Objective: The aim of this chapter is to gain knowledge on Basics of electromagnetic fields Scalar and vector quantities, vector calculus Various co-ordinate systems namely Cartesian, cylindrical and spherical systems. Divergence and Stoke s theorems INTRODUCTION TO ELECTROMAGNETIC FIELDS Electromagnetic fields are a combination of invisible electric and magnetic fields of force. They are generated by natural phenomena like the Earth s magnetic field but also by human activities, mainly through the use of electricity. Mobile phones, power lines and computer screens are examples of equipment that generates electromagnetic fields. Most man-made electromagnetic fields reverse their direction at regular intervals of time, ranging from high radio frequencies (mobile phones) through intermediate frequencies (computer screens) to extremely low frequencies (power lines). Figure. Frequency ranges for different types and sources of electromagnetic fields 1
Typical sources of electromagnetic fields Frequency range Frequencies Some examples of exposure sources Static 0 Hz video display units; MRI (medical imaging) and other diagnostic or scientific instrumentation; industrial electrolysis; welding devices ELF [Extremely 0-300 Hz Low Frequencies] power lines; domestic distribution lines; domestic appliances; electric engines in cars, trains and tramways; welding devices video display units; anti-theft devices in shops; IF [Intermediate 300 Hz - 100 hands-free access control systems, card readers and Frequencies] khz metal detectors; MRI; welding devices mobile telephones; broadcasting and TV; microwave RF [Radio 100 khz - 300 ovens; radar and radio transceivers; portable radios; Frequencies] GHz MRI What are electromagnetic waves? Electricity can be static, like what holds a balloon to the wall or makes your hair stand on end. Magnetism can also be static like a refrigerator magnet. But when they change or move together, they make waves - electromagnetic waves. 2
Electromagnetic waves are formed when an electric field (shown as blue arrows) couples with a magnetic field (shown as red arrows). The magnetic and electric fields of an electromagnetic wave are perpendicular to each other and to the direction of the wave. Electromagnetic Waves have different wavelengths When you listen to the radio, watch TV, or cook dinner in a microwave oven, you are using electromagnetic waves. Radio waves, television waves, and microwaves are all types of electromagnetic waves. They differ from each other in wavelength. Wavelength is the distance between one wave crest to the next. 3
Waves in the electromagnetic spectrum vary in size from very long radio waves the size of buildings, to very short gamma-rays smaller than the size of the nucleus of an atom. Vector calculus deals with vectors and with their derivatives and integrals. Vector calculus plays a central role in the study of electromagnetic theory. The equations for the electric and magnetic fields (Maxwell's equations) link components of these fields in different directions; such equations can be expressed most concisely in vector notation. In fact, much of vector calculus was invented for the specific purpose of simplifying the equations of electromagnetic theory. DIFFERENT CO-ORDINATE SYSTEMS specified way. It is a system for specifying points using coordinates measured in some Some coordinate systems are the following: The Cartesian coordinate system (also called the "rectangular coordinate system"), which, for three-dimensional flat space, uses three numbers representing distances. Curvilinear coordinates are a generalization of coordinate systems generally; the system is based on the intersection of curves. 4
The polar coordinate systems: o Circular coordinate system (commonly referred to as the polar coordinate system) represents a point in the plane by an angle and a distance from the origin. o Cylindrical coordinate system represents a point in space by an angle, a distance from the origin and a height. o Spherical coordinate system represents a point in space with two angles and a distance from the origin. 1.CARTESIAN CO-ORDINATE SYSTEM OR RECTANGULAR CO-ORDINATE SYSTEM The most common coordinate system for representing positions in space is one based on three perpendicular spatial axes generally designated x, y, and z. History Cartesian means relating to the French mathematician and philosopher René Descartes (Latin: Cartesius), who, among other things, worked to merge algebra and Euclidean geometry. This work was influential in the development of analytic geometry, calculus, and cartography. Three-dimensional coordinate system The three dimensional Cartesian coordinate system provides the three physical dimensions of space length, width, and height. The three Cartesian axes defining the system are perpendicular to each other. The relevant coordinates are of the form (x,y,z). The x-, y-, and z-coordinates of a point can also be taken as the distances from the yz-plane, xz-plane, and xy-plane respectively. The xy-, yz-, and xz-planes divide the three-dimensional space into eight subdivisions known as octants, similar to the quadrants of 2D space. The standard orientation, where the xy-plane is horizontal and the z-axis points up is called right-handed or positive. 5
Right handed system: The name derives from the right-hand rule. If the index finger of the right hand is pointed forward, the middle finger bent inward at a right angle to it, and the thumb placed at a right angle to both, the three fingers indicate the relative directions of the x-, y-, and z-axes in a right-handed system. The thumb indicates the x-axis, the index finger the y-axis and the middle finger the z-axis. Fig. The right-handed Cartesian coordinate system indicating the coordinate planes. Any other point in a Cartesian coordinate system is obtained by incrementing the coordinates in their respective directions. For example, a point Q is located with reference to a point P(x,y,z) by incrementing x,y,z by an amount of dx,dy and dz. The coordinates: (x, y, z) are the coordinates of a Rectangular coordinate system. Differential length elements: dx, dy, dz Differential length vector dl = dxa x + dya y + dza z Differential Surface vector 6
ds x = ± dydza x ds y = ± dxdza y ds z = ± dxdya z Differential volume dv = ± dx dy dz. Applications: Cartesian coordinates are often used to represent two or three dimensions of space, but they can also be used to represent many other quantities (such as mass, time, force, etc.). In such cases the coordinate axes will typically be labelled with other letters (such as m, t, F, etc.) in place of x, y, and z. Each axis may also have different units of measurement associated with it (such as kilograms, seconds, pounds, etc.). 2. CYLINDRICAL COORDINATES If you replace the x-y plane in the rectangular coordinates (x,y,z) with the standard polar coordinate system, you have the cylindrical coordinate system. Its variables are (r, 0,z). The third coordinate of any point is the same in both systems. The conversion equations are the same as in polar coordinates: The coordinate surfaces for the rectangular coordinate system are the planes perpendicular to the coordinate axes, x = a, y = b, and z = c. The coordinate surfaces for the cylindrical coordinate system is: r = a (a cylinder), 0 = b (a plane perpendicular to the x-y plane and through the z-axis) and z = c (a plane perpendicular to the z-axis). Notice that the system is named after the r = a surface. A point is located at the intersection of three mutually perpendicular surfaces. r = a = constant 0 = b = constant 7
Cylindrical Co-ordinate System The cylindrical coordinate system specifies the position of a point P by the combination of its distance to the z-axis (radius r), the angle j between the x-axis and the line OQ, and the elevation along the z-axis. Note, that the angle is positive in counter-clockwise direction and negative in clockwise direction. The value of the angle is measured in radians or degrees (0..360 ). To specify a particular point in a cylindrical coordinate system, you indicate the coordinate values in the form [r,, z]. The r ( or ) coordinate is called radial coordinate, the coordinate is called azimuthal coordinate, and the z coordinate is also called the applicate. Any other point in a cylindrical coordinate system is obtained by incrementing the coordinates in their respective directions. For example, a point Q is located with reference to a point P(r,, z) by incrementing r,, z by an amount of dr, d and dz. The coordinates: (r,, z) are the coordinates of a cylindrical coordinate system. Differential length elements: dr, d, dz 8
Differential length vector dl = dra r + d a + dza z Differential Surface vector ds r = ± d dza r ds = ± drdza ds z = ± drd a z Differential volume dv = ± dr d dz. A point in the cylindrical coordinate system. 3. SPHERICAL COORDINATES The spherical coordinate system is defined by a sphere of radius, a cone with half angle and a plane making an angle with the reference plane (i.e) xz plane. Thus it has two polar coordinates. The plane is just as in cylindrical coordinates. 9
The coordinate surfaces are rho = a (a sphere), 0 = b (a plane perpendicular to the x-y plane through the z-axis), and phi = c (a cone whose apex is at the origin and whose axis is along the z-axis.) The system is named after the rho = a surface. rho = a phi = c Spherical Co-ordinate System To specify a particular point in a spherical coordinate system, you indicate the coordinate values in the form [r,, ]. 10
A point in the spherical co-ordinate system. Any other point in a spherical coordinate system is obtained by incrementing the coordinates in their respective directions. For example, a point Q is located with reference to a point P(r,, ) by incrementing r,, by an amount of dr,d and d. The coordinates: (r,, ) are the coordinates of a spherical coordinate system. Differential length elements: dr, r sin d, r d Differential length vector dl = dra r + r sin d a + r d a Differential Surface vector ds r = ± r 2 sin d d a r ds = ± r drd a ds = ± r sin drd a 11
Differential volume dv = ± r 2 sin dr d d. CONVERSION BETWEEN 3D CO-ORDINATE SYSTEMS Conversion from cartesian to cylindrical co-ordinates: Cartesian [x, y, z] Cylindrical [r,, z] r = (x 2 + y 2 ) = tan - 1 (y/x) z =z Conversion from cylindrical to cartesian co-ordinates: Cylindrical [r,, z] Cartesian [x, y, z] x = r cos y = r sin z = z Conversion from spherical to cartesian co-ordinates: Spherical [r,, ] Cartesian [x, y, z] x = r sin cos y = r sin sin z = r cos Conversion from cartesian to spherical co-ordinates: Cartesian [x, y, z] Spherical [r,, ] 12
Conversion from spherical to cylindrical co-ordinates: Spherical [r,, ] Cylindrical [r,, z] r = r sin = z = r cos Conversion from cylindrical to spherical co-ordinates: Cylindrical [r,, z] Spherical [r,, ] = arctan(r/z) = LINE INTEGRAL A line integral (sometimes called a path integral or curve integral) is an integral where the function to be integrated is evaluated along a curve. Various different line integrals are in use. In the case of a closed curve it is also called a contour integral. The function to be integrated may be a scalar field or a vector field. The value of the line integral is the sum of values of the field at all points on the curve, weighted by some scalar function on the curve (commonly arc length or, for a vector field, the scalar product of the vector field with a differential vector in the curve). This weighting distinguishes the line integral from simpler integrals defined on intervals. The line integral finds the work done on an object moving through an electric or gravitational field, for example. 13
Applications The line integral has many uses in physics. For example, the work done on a particle traveling on a curve C inside a force field represented as a vector field F is the line integral of F on C. SURFACE INTEGRAL A surface integral is a definite integral taken over a surface (which may be a curved set in space); it can be thought of as the double integral analog of the line integral. Given a surface, one may integrate over it scalar fields (that is, functions which return numbers as values), and vector fields (that is, functions which return vectors as values). Surface integrals have applications in physics, particularly with the classical theory of electromagnetism. The definition of surface integral relies on splitting the surface into small surface elements. It is a double integral. Theorems involving surface integrals Various useful results for surface integrals can be derived using differential geometry and vector calculus, such as the divergence theorem, and its generalization, Stokes' theorem. VOLUME INTEGRAL A volume integral refers to an integral over a 3-dimensional domain. Volume integral is a triple integral of the constant function 1, which gives the volume of the region D. It can also mean a triple integral within a region D in R 3 of a function f(x,y,z). 14
CURL In vector calculus, curl (or: rotor) is a vector operator that shows a vector field's "rotation"; that is, the direction of the axis of rotation and the magnitude of the rotation. It can also be described as the circulation density. Definition The curl of a vector field curl F about a point is defined as the circulation of F per unit surface as the contour and surface shrink to zero. The net circulation of a vector field F about some closed contour C is the line integral of F along C. The circulation about a point can be obtained by shrinking the contour C. The vector differential operator has the following form 2 2 2 r x y z We denote the curl of a vector field F this way r z 2 2 Notice here that both the differential operator and the field F are vector quantities. That means that the above is a cross product. Curl is therefore a vector. If we have a field F which has the form where i,j,k are unit vectors in the x,y and z directions, and where F x,f y and F z are functions with partial derivatives, then the curl of F is given by 15
The above is the determinant form of the formula for curl. The first line is made up of unit vectors, the second of scalar operators, and the third of scalar functions, so this is not a determinant in the strict mathematical sense. Consider a vector field with only one component. We then get the situation shown in the applet below. This situation can happen for example in flowing water. Then the velocity of the water at different positions is the vector field. Try changing the gradient of the vector field, and see what the effect on the paddlewheel is. If you have an arbitrary vector field, then you can imagine the curl something like this: The curl vector is perpendicular to the vector field. DIVERGENCE In vector calculus, the divergence is an operator that measures the magnitude of a vector field's source or sink at a given point; the divergence of a vector field is a (signed) scalar. For example, for a vector field that denotes the velocity of air expanding as it is heated, the divergence of the velocity field would have a positive value because the air expands. If the air cools and contracts, the divergence is negative. The divergence could be thought of as a measure of the change in density. 16
A vector field that has zero divergence everywhere is called solenoidal. Definition It is defined as the net outward flux of the vector field F per unit volume as the volume shrinks to zero and is a scalar quantity. The divergence of a continuously differentiable vector field F = F x i + F y j + F z k is defined to be the scalar-valued function: GRADIENT In vector calculus, the gradient of a scalar field is a vector field which points in the direction of the greatest rate of increase of the scalar field, and whose magnitude is the greatest rate of change. In the above two images, the scalar field is in black and white, black representing higher values, and its corresponding gradient is represented by blue arrows. The gradient vector provides both the magnitude and direction of the maximum space rate of change of the scalar field f. df = grad f. dl grad f = f df = f. dl ( dot product) df = f dl cos 17
df / dl = f cos (df / dl) max = f DIVERGENCE THEOREM In vector calculus, the divergence theorem, also known as Gauss's theorem (Carl Friedrich Gauss), Ostrogradsky's theorem (Mikhail Vasilievich Ostrogradsky), or Gauss-Ostrogradsky theorem is a result that relates the flow (that is, flux) of a vector field through a surface to the behavior of the vector field inside the surface. More precisely, the divergence theorem states that the outward flux of a vector field through a surface is equal to the triple integral of the divergence on the region inside the surface. Intuitively, it states that the sum of all sources minus the sum of all sinks gives the net flow out of a region. The divergence theorem is an important result for the mathematics of engineering, in particular in electrostatics and fluid dynamics. The theorem is a special case of the more general Stokes' theorem, which generalizes the fundamental theorem of calculus. STOKES' THEOREM (OR STOKES'S THEOREM) It is a statement about the integration of differential forms which generalizes several theorems from vector calculus.it relates the closed line integral of a vector field to the surface integral of the curl of that vector field. Summary: Scalar and vector quantities have been studied. Scalar product produces a scalar and a vector product gives a vector. The three common coordinate systems namely Cartesian, cylindrical and spherical systems and the transformation between them have been dealt with. The differential displacement, area and volume have been discussed. The divergence theorem relates a surface integral over a closed surface to a volume integral. Stokes theorem relates a line integral over a closed path to a surface integral. 18