Engineering electromagnetics pdf

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The dipole axis is vertical, with the positive charge on the top. The streamlines for the electric field are obtained by applying the methods of Section 2. Six equipotential surfaces are labeled with relative values of V. Equation 38 is independent of any coordinate system. Since it is equal to the product of the charge and the separation, neither the dipole moment nor the potential will change as Q increases and d decreases, provided the product remains constant.

The limiting case of a point dipole is achieved when we let d approach zero and Q approach infinity such that the product p is finite. Turning our attention to the resultant fields, it is interesting to note that the potential field is now proportional to the inverse square of the distance, and the electric field intensity is proportional to the inverse cube of the distance from the dipole.

Each field falls off faster than the corresponding field for the point charge, but this is no more than we should expect because the opposite charges appear to be closer together at greater distances and to act more like a single point charge of zero coulombs. Symmetrical arrangements of larger numbers of point charges produce fields proportional to the inverse of higher and higher powers of r.

These charge distributions are called multipoles, and they are used in infinite series to approximate more unwieldy charge configurations. Imagine that the external source carries the charge up to a point near the fixed charge and then holds it there. Energy must be conserved, and the energy expended in bringing this charge into position now represents potential energy, for if the external source released its hold on the charge, it would accelerate away from the fixed charge, acquiring kinetic energy of its own and the capability of doing work.

In order to find the potential energy present in a system of charges, we must find the work done by an external source in positioning the charges. Bringing a charge Q1 from infinity to any position requires no work, for there is no field present. We represent this potential as V, where the first subscript indicates the location and the second subscript the source.

That is, V is the potential at the location of Q2 due to Q1. How much energy is required to bring two half-charges into coincidence to make a unit charge?

Similar expressions may be easily written in terms of line or surface charge density. Usually we prefer to use Eq. This may always be done by considering point charges, line charge density, or surface charge density to be continuous distributions of volume charge density over very small regions. We will illustrate such a procedure with an example shortly. We begin by making the expression a little bit longer. This volume, first appearing in Eq. We may therefore consider the volume as infinite in extent if we wish.

We found in Section 3. The integrand in Eq. Potential energy can never be pinned down precisely in terms of physical location. Someone lifts a pencil, and the pencil acquires potential energy. Is the energy stored in the molecules of the pencil, in the gravitational field between the pencil and the earth, or in some obscure place?

Is the energy in a capacitor stored in the charges themselves, in the field, or where? No one can offer any proof for his or her own private opinion, and the matter of deciding may be left to the philosophers. Electromagnetic field theory makes it easy to believe that the energy of an electric field or a charge distribution is stored in the field itself, for if we take Eq.

The interpretation afforded by Eq. Attwood, S. Electric and Magnetic Fields. There are a large number of well-drawn field maps of various charge distributions, including the dipole field. Vector analysis is not used. Gradient is described on pp. The directional derivative and the gradient are presented on pp. The transverse electric mode field, TE11, defined for 0 4. What is the potential at the inner sphere surface in this case? A perfectly conducting cylindrical shell, whose axis is the z axis, surrounds the line charge.

The cylinder of radius b is at ground potential. The electron therefore moves approximately along a streamline. Where does it leave the plate and in what direction is it moving at the time? Under what conditions does the answer agree with Eq. Find the total stored energy by applying a Eq. Again find the total stored energy. In the first part of the chapter, we consider conducting materials by describing the parameters that relate current to an applied electric field.

We then develop methods of evaluating resistances of conductors in a few simple geometric forms. Conditions that must be met at a conducting boundary are obtained next, and this knowledge leads to a discussion of the method of images. The properties of semiconductors are described to conclude the discussion of conducting media. In the second part of the chapter, we consider insulating materials, or dielectrics.

Such materials differ from conductors in that ideally, there is no free charge that can be transported within them to produce conduction current. Instead, all charge is confined to molecular or lattice sites by coulomb forces. An applied electric field has the effect of displacing the charges slightly, leading to the formation of ensembles of electric dipoles.

The extent to which this occurs is measured by the relative permittivity, or dielectric constant. Polarization of the medium may modify the electric field, whose magnitude and direction may differ from the values it would have in a different medium or in free space. Boundary conditions for the fields at interfaces between dielectrics are developed to evaluate these differences.

It should be noted that most materials will possess both dielectric and conductive properties; that is, a material considered a dielectric may be slightly conductive, and a material that is mostly conductive may be slightly polarizable. These departures from the ideal cases lead to some interesting behavior, particularly as to the effects on electromagnetic wave propagation, as we will see later. The unit of current is the ampere A , defined as a rate of movement of charge passing a given reference point or crossing a given reference plane of one coulomb per second.

Current density is a vector flux density1 represented by J. To simplify the explanation, assume that the charge element is oriented with its edges parallel to the coordinate axes and that it has only an x component of velocity. Current in an exceedingly fine wire, or a filamentary current, is occasionally defined as a vector, but we usually prefer to be consistent and give the direction to the filament, or path, and not to the current.

Note that the convection current density is related linearly to charge density as well as to velocity. The mass rate of flow of cars cars per square foot per second in the Holland Tunnel could be increased either by raising the density of cars per cubic foot or by going to higher speeds, if the drivers were capable of doing so.

The principle of conservation of charge states simply that charges can be neither created nor destroyed, although equal 2 The lowercase v is used both for volume and velocity. The continuity equation follows from this principle when we consider any region bounded by a closed surface.

In circuit theory we usually associate the current flow into one terminal of a capacitor with the time rate of increase of charge on that plate. The current of 4 , however, is an outward-flowing current. To see why this happens, we need to look at the volume charge density and the velocity. In summary, we have a current density that is inversely proportional to r, a charge density that is inversely proportional to r2, and a velocity and total current that are proportional to r.

The total energy is the sum of the kinetic and potential energies, and because energy must be given to an electron to pull it away from the nucleus, the energy of every electron in the atom is a negative quantity. Even though this picture has some limitations, it is convenient to associate these energy values with orbits surrounding the nucleus, the more negative energies corresponding to orbits of smaller radius. According to the quantum theory, only certain discrete energy levels, or energy states, are permissible in a given atom, and an electron must therefore absorb or emit discrete amounts of energy, or quanta, in passing from one level to another.

A normal atom at absolute zero temperature has an electron occupying every one of the lower energy shells, starting outward from the nucleus and continuing until the supply of electrons is exhausted.

At a temperature of absolute zero, the normal solid also has every level occupied, starting with the lowest and proceeding in order until all the electrons are located. The electrons with the highest least negative energy levels, the valence electrons, are located in the valence band.

If there are permissible higher-energy levels in the valence band, or if the valence band merges smoothly into a conduction band, then additional kinetic energy may be given to the valence electrons by an external field, resulting in an electron flow. The solid is called a metallic conductor. The filled valence band and the unfilled conduction band for a conductor at absolute zero temperature are suggested by the sketch in Figure 5.

If, however, the electron with the greatest energy occupies the top level in the valence band and a gap exists between the valence band and the conduction band, then the electron cannot accept additional energy in small amounts, and the material is an insulator. This band structure is indicated in Figure 5.

Note that if a relatively large amount of energy can be transferred to the electron, it may be sufficiently excited to jump the gap into the next band where conduction can occur easily.

Here the insulator breaks down. Small amounts of energy in the form of heat, light, or an electric field may raise the energy of the electrons at the top of the filled band and provide the basis for conduction. These materials are insulators which display many of the properties of conductors and are called semiconductors.

Let us first consider the conductor. Here the valence electrons, or conduction, or free, electrons, move under the influence of an electric field. In free space, the electron would accelerate and continuously increase its velocity and energy ; in the crystalline material, the progress of the electron is impeded by continual collisions with the thermally excited crystalline lattice structure, and a constant average velocity is soon attained.

This velocity vd is termed the drift velocity, and it is linearly related to the electric field intensity by the mobility of the electron in the given material.

Note that the electron velocity is in a direction opposite to the direction of E. Equation 6 also shows that mobility is measured in the units of square meters per volt-second; typical values3 are 0.

For these good conductors, a drift velocity of a few centimeters per second is sufficient to produce a noticeable temperature rise and can cause the wire to melt if the heat cannot be quickly removed by thermal conduction or radiation.

Substituting 6 into Eq. One siemens 1 S is the basic unit of conductance in the SI system and is defined as one ampere per volt. First it is informative to note the conductivity of several metallic conductors; typical values in siemens per meter are 3. Data for other conductors may be found in Appendix C. On seeing data such as these, it is natural to assume that we are being presented with constant values; this is essentially true.

The conductivity is a function of temperature, however. The resistivity, which is the reciprocal of the conductivity, varies almost linearly with temperature in the region of room temperature, and for aluminum, copper, and silver it increases about 0. Copper and silver are not superconductors, although aluminum is for temperatures below 1.

Initially, assume that J and E are uniform, as they are in the cylindrical region shown in Figure 5. Karl became a British subject and was knighted, becoming Sir William Siemens. If the fields are not uniform, the resistance may still be defined as the ratio of V to I, where V is the potential difference between two specified equipotential surfaces in the material and I is the total current crossing the more positive surface into the material.

The diameter of the wire is 0. Using a conductivity of 5. With this current, the potential difference between the two ends of the wire is V, the electric field intensity is 0. A copper conductor has a diameter of 0. Assume that it carries a total dc current of 50 A. Suppose, for the sake of argument, that there suddenly appear a number of electrons in the interior of a conductor.

The electric fields set up by these electrons are not counteracted by any positive charges, and the electrons therefore begin to accelerate away from each other. This continues until the electrons reach the surface of the conductor or until a number of electrons equal to the number injected have reached the surface.

Here, the outward progress of the electrons is stopped, for the material surrounding the conductor is an insulator not possessing a convenient conduction band.

No charge may remain within the conductor. If it did, the resulting electric field would force the charges to the surface.

Hence the final result within a conductor is zero charge density, and a surface charge density resides on the exterior surface. This is one of the two characteristics of a good conductor. Physically, we see that if an electric field were present, the conduction electrons would move and produce a current, thus leading to a nonstatic condition.

Summarizing for electrostatics, no charge and no electric field may exist at any point within a conducting material. Charge may, however, appear on the surface as a surface charge density, and our next investigation concerns the fields external to the conductor. We wish to relate these external fields to the charge on the surface of the conductor.

The problem is a simple one, and we may first talk our way to the solution with a little mathematics. If the external electric field intensity is decomposed into two components, one tangential and one normal to the conductor surface, the tangential component is seen to be zero. If it were not zero, a tangential force would be applied to the elements of the surface charge, resulting in their motion and nonstatic conditions. Because static conditions are assumed, the tangential electric field intensity and electric flux density are zero.

The electric flux leaving a small increment of surface must be equal to the charge residing on that incremental surface. The flux cannot penetrate into the conductor, for the total field there is zero.

It must then leave the surface normally. If we use some of our previously derived results in making a more careful analysis and incidentally introducing a general method which we must use later , we should set up a boundary between a conductor and free space Figure 5. Both fields are zero in the conductor. The tangential field may be determined by applying Section 4. The condition on the normal field is found most readily by considering DN rather than EN and choosing a small cylinder as the gaussian surface.

Taking the cross product or the dot product of either field quantity with n gives the tangential or the normal component of the field, respectively. An immediate and important consequence of a zero tangential electric field intensity is the fact that a conductor surface is an equipotential surface.

To summarize the principles which apply to conductors in electrostatic fields, we may state that 1. The static electric field intensity inside a conductor is zero. The static electric field intensity at the surface of a conductor is everywhere directed normal to that surface. The conductor surface is an equipotential surface.

Using these three principles, there are a number of quantities that may be calculated at a conductor boundary, given a knowledge of the potential field. Because the conductor is an equipotential surface, the potential at the entire surface must be V. Let us assume arbitrarily that the solid conductor lies above and to the right of the equipotential surface at point P, whereas free space is down and to the left. Such a plane may be represented by a vanishingly thin conducting plane that is infinite in extent.

Thus, if we replace the dipole configuration shown in Figure 5. Below the conducting plane, all fields are zero, as we have not provided any charges in that region. Of course, we might also substitute a single negative charge below a conducting plane for the dipole arrangement and obtain equivalence for the fields in the lower half of each region. If we approach this equivalence from the opposite point of view, we begin with a single charge above a perfectly conducting plane and then see that we may maintain the same fields above the plane by removing the plane and locating a negative charge at a symmetrical location below the plane.

This charge is called the image of the original charge, and it is the negative of that value. If we can do this once, linearity allows us to do it again and again, and thus any charge configuration above an infinite ground plane may be replaced by an arrangement composed of the given charge configuration, its image, and no conducting plane. This is suggested by the two illustrations of Figure 5. In many cases, the potential field of the new system is much easier to find since it does not contain the conducting plane with its unknown surface charge distribution.

The field at P may now be obtained by superposition of the known fields of the line charges. The electrons are those from the top of the filled valence band that have received sufficient energy usually thermal to cross the relatively small forbidden band into the conduction band.

The forbidden-band energy gap in typical semiconductors is of the order of one electronvolt. The vacancies left by these electrons represent unfilled energy states in the valence band which may also move from atom to atom in the crystal.

Both carriers move in an electric field, and they move in opposite directions; hence each contributes a component of the total current which is in the same direction as that provided by the other. These values are given in square meters per volt-second and range from 10 to times as large as those for aluminum, copper, silver, and other metallic conductors.

The electron and hole concentrations depend strongly on temperature. At K the electron and hole volume charge densities are both 0. These values lead to conductivities of 0. As temperature increases, the mobilities decrease, but the charge densities increase very rapidly. As a result, the conductivity of silicon increases by a factor of 10 as the temperature increases from to about K and decreases by a factor of 10 as the temperature drops from to about K.

Note that the conductivity of the intrinsic semiconductor increases with temperature, whereas that of a metallic conductor decreases with temperature; this is one of the characteristic differences between the metallic conductors and the intrinsic semiconductors. The number of charge carriers and the conductivity may both be increased dramatically by adding very small amounts of impurities.

Donor materials provide additional electrons and form n-type semiconductors, whereas acceptors furnish extra holes and form p-type materials. The process is known as doping, and a donor concentration in silicon as low as one part in causes an increase in conductivity by a factor of The range of value of the conductivity is extreme as we go from the best insulating materials to semiconductors and the finest conductors.

These values cover the remarkably large range of some 25 orders of magnitude. Using the values given in this section for the electron and hole mobilities in silicon at K, and assuming hole and electron charge densities are 0. These are not free charges, and they cannot contribute to the conduction process. Rather, they are bound in place by atomic and molecular forces and can only shift positions slightly in response to external fields. They are called bound charges, in contrast to the free charges that determine conductivity.

The bound charges can be treated as any other sources of the electrostatic field. Therefore, we would not need to introduce the dielectric constant as a new parameter or to deal with permittivities different from the permittivity of free space; however, the alternative would be to consider every charge within a piece of dielectric material.

This storage takes place by means of a shift in the relative positions of the internal, bound positive and negative charges against the normal molecular and atomic forces. The source of the energy is the external field, the motion of the shifting charges resulting perhaps in a transient current through a battery that is producing the field.

The actual mechanism of the charge displacement differs in the various dielectric materials. Normally the dipoles are oriented in a random way throughout the interior of the material, and the action of the external field is to align these molecules, to some extent, in the same direction.

A sufficiently strong field may even produce an additional displacement between the positive and negative charges. A nonpolar molecule does not have this dipole arrangement until after a field is applied.

The negative and positive charges shift in opposite directions against their mutual attraction and produce a dipole that is aligned with the electric field. Either type of dipole may be described by its dipole moment p, as developed in Section 4. We note again that the units of p are coulomb-meters. However, a random orientation may cause ptotal to be essentially zero. We will treat P as a typical continuous field, even though it is obvious that it is essentially undefined at points within an atom or molecule.

To be specific, assume that we have a dielectric containing nonpolar molecules. Figure 5. This relationship will, of course, be a function of the type of material, and we will essentially limit our discussion to those isotropic materials for which E and P are linearly related. In an isotropic material, the vectors E and P are always parallel, regardless of the orientation of the field. Although most engineering dielectrics are linear for moderate-to-large field strengths and are also isotropic, single crystals may be anisotropic.

The periodic nature of crystalline materials causes dipole moments to be formed most easily along the crystal axes, and not necessarily in the direction of the applied field. In ferroelectric materials, the relationship between P and E not only is nonlinear, but also shows hysteresis effects; that is, the polarization produced by a given electric field intensity depends on the past history of the sample.

Important examples of this type of dielectric are barium titanate, often used in ceramic capacitors, and Rochelle salt. Using this relationship in Eq. The dielectric constants are given for some representative materials in Appendix C. Anisotropic dielectric materials cannot be described in terms of a simple susceptibility or permittivity parameter. Certain choices of axis directions lead to simpler matrices. We will concentrate our attention on linear isotropic materials and reserve the general case for a more advanced text.

However, when anisotropic or nonlinear materials must be considered, the relative permittivity, in the simple scalar form that we have discussed, is no longer applicable. We seek values for D, E, and P everywhere. The dielectric constant of the Teflon is 2.

Now, any of the last four or five equations will enable us to relate the several fields inside the material to each other. The difficulty lies in crossing over the boundary from the known fields external to the dielectric to the unknown ones within it. To do this we need a boundary condition, and this is the subject of the next section.

We will complete this example then. We will limit our discussion to isotropic materials. A slab of dielectric material has a relative dielectric constant of 3. This is another example of a boundary condition, such as the condition at the surface of a conductor whereby the tangential fields are zero and the normal electric flux density is equal to the surface charge density on the conductor.

Now we take the first step in solving a two-dielectric problem, or a dielectric-conductor problem, by determining the behavior of the fields at the dielectric interface.

The continuity of DN is shown by the gaussian surface on the right, and the continuity of Etan is shown by the line integral about the closed path at the left.

It cannot be a bound surface charge density, because we are taking the polarization of the dielectric into effect by using a dielectric constant different from unity; that is, instead of considering bound charges in free space, we are using an increased permittivity. Also, it is extremely unlikely that any free charge is on the interface, for no free charge is available in the perfect dielectrics we are considering.

This charge must then have been placed there deliberately, thus unbalancing the total charge in and on this dielectric body.

This construction was used previously in Eqs. These conditions may be used to show the change in the vectors D and E at the surface. The magnitude of D in region 2 may be found from Eq. The boundary conditions must be used to help us determine the fields on both sides of the boundary from the other information that is given. Continue Problem D5. Fano, R. Chu, and R. Electromagnetic Fields, Energy, and Forces.

Cambridge, Mass. Polarization in dielectrics is discussed in Chapter 5. This junior-level text presupposes a full-term physics course in electricity and magnetism, and it is therefore slightly more advanced in level. The introduction beginning on p. Dekker, A. Electrical Engineering Materials. This admirable little book covers dielectrics, conductors, semiconductors, and magnetic materials.

Electromagnetics is too important in too many fields for knowledge to be gathered on the fly. A deep understanding gained through structured presentation of concepts and practical problem solving is the best way to approach this important subject. Fundamentals of Engineering Electromagnetics provides such an understanding, distilling the most important.

Fundamentals of Electric Power Engineering. This book serves as a tool for any engineer who wants tolearn about circuits, electrical machines and drives, powerelectronics, and power systems basics From time to time, engineers find they need to brush up oncertain fundamentals within electrical engineering. This clear andconcise book is the ideal learning tool for them. Graphic Violence ; Graphic Sexual Content ; texts. How is Chegg Study better than a printed Engineering Electromagnetics student solution manual from the bookstore?

Hayt] with solution manual Thanks a lot for providing this book. Library of Congress Cataloging-in-Publication Data. Hayt, William Hart, —. Our solutions are written by Chegg experts so you can be assured of the highest quality! Engineering Electromagnetics William Hayt. Also if visitors will get caught uploading multiple copyrighted files, their IP will be permanently banned from using our service.

It will be useful for 8th edition also..