Engineering Electromagnetics 7th Edition William H. Hayt Solution Manual. of engineering electromagnetics 6th edition william h. hayt, john a. tvnovellas.info, Past. Electromagnetics Hayt 6th Edition Solution Manual [PDF] [EPUB] Engineering electromagnetics. [solution manual] (william h. hayt jr. john a. [solution manual] (william h. hayt jr. john a. buck - 6th edition) - Fri, 05 Apr GMT. (PDF) Engineering Electromagnetics Hayt _ Buck 8th edition.

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pdf. Engineering electromagnetics [solution manual] (william h. hayt jr. john a. buck - 6th edition) The vector from the origin to the point A is given as (6, −2, −4 ), and the unit vector directed from the origin toward point B is (2, −2, 1)/3. If points . Title: Solutions of engineering electromagnetics 6th edition william h hayt, john a buck pdf, Author: Erwin Aguilar, Name: Solutions of engineering. Solutions of engineering electromagnetics 6th edition william h. hayt, john a. tvnovellas.info, Past Exams for Electromagnetic Engineering. University.

Skip to main content. Log In Sign Up. Engineering electromagnetics [solution manual] william h. Hasibullah Mekaiel. If points A and B are ten units apart, find the coordinates of point B.

The perspective has been broadened by an expanded emphasis toward optics concepts and applications, which are presented along with the more traditional lower-frequency discussions. This again seemed to be a logical step, as the importance of optics and optical communications has increased significantly since the earlier editions were published. The theme of the text has not changed since the first edition of An inductive approach is used that is consistent with the historical development.

In it, the experimental laws are presented as individual concepts that are later unified in Maxwell's equations. Apart from the first chapter on vector analysis, the mathematical tools are introduced in the text on an as-needed basis.

Throughout every edition, as well as this one, the primary goal has been to enable students to learn independently.

Numerous examples, drill problems usually having multiple parts , and end-of-chapter problems are provided to facilitate this. Answers to the drill problems are given below each problem. This expression simplifies to the following quadratic: The field will take the general form: The total field at P will be: Determine E at P 0, y, 0: This field will be: Now, since the charge is at the origin, we expect to obtain only a radial component of EM.

This will be: Calculate the total charge present: A uniform volume charge density of 0. With the limits thus changed, the integral for the charge becomes: What is the average volume charge density throughout this large region?

Each cube will contain the equivalent of one little sphere. Neglecting the little sphere volume, the average density becomes 3. Find the charge within the region: Uniform line charges of 0. This field will in general be: With the infinite line, we know that the field will have only a radial component in cylindrical coordinates or x and y components in cartesian. Therefore, at point P: Since all line charges are infinitely-long, we can write: What force per unit length does each line charge exert on the other?

The charges are parallel to the z axis and are separated by 0. We use the superposition integral: The integral becomes: This is evident just from the symmetry of the problem. Performing the z integration first on the x component, we obtain using tables: First, we recognize from symmetry that only a z component of E will be present. The superposition integral for the z component of E will be: Surface charge density is positioned in free space as follows: For this reason, the net field magnitude will be the same everywhere, whereas the field direction will depend on which side of a given sheet one is positioned.

This will be the magnitude at the other two points as well. Find E at the origin if the following charge distributions are present in free space: The sum of the fields at the origin from each charge in order is: A uniform surface charge density of 0. Find E at the origin: Since each pair consists of equal and opposite charges, the effect at the origin is to double the field produce by one of each type.

Taking the sum of the fields at the origin from the surface and line charges, respectively, we find: An empty metal paint can is placed on a marble table, the lid is removed, and both parts are discharged honorably by touching them to ground.

An insulating nylon thread is glued to the center of the lid, and a penny, a nickel, and a dime are glued to the thread so that they are not touching each other. The assembly is lowered into the can so that the coins hang clear of all walls, and the lid is secured.

The outside of the can is again touched momentarily to ground. The device is carefully disassembled with insulating gloves and tools. All coins were insulated during the entire procedure, so they will retain their original charges: Again, since the coins are insulated, they retain their original charges.

A point charge of 12 nC is located at the origin. Note that this point lies in the center of a symmetric arrangement of line charges, whose fields will all cancel at that point. This sphere encloses the point charge, so its flux of 12 nC is included. The flux from the line charges will equal the total line charge that lies within the sphere. We integrate over the surface to find: We just integrate the charge density on that surface to find the flux that leaves it.

This point lies at radius 5 cm, and is thus inside the outermost charge layer. This layer, being of uniform density, will not contribute to D at P.

We know that in cylindrical coordinates, the layers at 1 and 2 cm will produce the flux density: Of the 6 surfaces to consider, only 2 will contribute to the net outward flux. These fluxes will thus cancel. The net outward flux becomes: Find the total flux leaving a sphere of radius 2 m if it is centered at a A 3, 1, 0: In this case the result will be the same if we move the sphere to the origin and keep the charges where they were.

To find the charge we integrate: The gaussian surface is a spherical shell of radius 1 mm. The enclosed charge is the result of part a. This will be the same computation as in part b, except the gaussian surface now lies at 20 mm. The sheet charge can be thought of as an assembly of infinitely-long parallel strips that lie parallel to the y axis in the yz plane, and where each is of thickness dz. The field from each strip is that of an infinite line charge, and so we can construct the field at P from a single strip as: Note also that the expression is valid for all x positive or negative values.

Find D everywhere: Since the charge varies only with radius, and is in the form of a cylinder, symmetry tells us that the flux density will be radially-directed and will be constant over a cylindrical surface of a fixed radius.

The set up is the same, except the upper limit of the above integral is 1 instead of r. These are plotted on the next page.

Noting that the charges are spherically-symmetric, we ascertain that D will be radially-directed and will vary only with radius. The Gaussian cylinder now lies outside the charge, so 2. Volume charge density is located as follows: In this application, Eq. We note that there is no z component of D, so there will be no outward flux contributions from the top and bottom surfaces.

This is. Evaluate both sides of Eq. Evaluate the partial derivatives at the center of the volume. A spherical surface of radius 3 mm is centered at P 4, 1, 5 in free space. Use the.

A cube of volume a 3 has its faces parallel to the cartesian coordinate surfaces. The equation reads: Using the divergence formula for cylindrical coordinates see problem 3.

Show that div D is zero everywhere except at the origin. Using the formula for divergence in spherical coordinates see problem 3. Find div D everywhere: We note that D has only a radial component, and so flux would leave only through the cylinder sides. To find this, we integrate over the volume: This is in the region where the second field expression is valid.

The total surface charge should be equal and opposite to the total volume charge. The plot is zero at larger radii. Evaluate the surface integral side for the corresponding closed surface: Note that since the x component of D does not vary with x, the outward fluxes from the front and back surfaces will cancel each other.

The same is true for the left and right surfaces, since Dy does not vary with y. This leaves only the top and bottom surfaces, where the fluxes are: Taking the surface integral side first, the six sides over which the flux must be evaluated are only four, since there is no z component of D. We use the divergence theorem and first evaluate the surface integral side. This is where the divergence theorem really saves you time! Find the incremental work done in moving a 4-C charge a distance of 1 mm in the direction specified by: The computation is similar to that of part a, but we change the direction: A little thought is in order here: We could just as well position the two points at the same z location and the problem would not change.

Halfway along this line is a point of symmetry in the field make a sketch to see this. This means that when starting from either point, the initial force will be the same. This is also found by going through the same procedure as in part a, but with the direction roles of P and Q reversed. We set up the computation as follows, and find the the result does not depend on the path.

Repeat Problem 4. Now things are different in that the path does matter: We obtain: A point charge Q1 is located at the origin in free space. Find the work done in carrying a charge Q2 from: Since the charge density is uniform and is spherically-symmetric, the angular coordinates do not matter.

Again, the angles do not matter because of the spherical symmetry. We need to find a potential function for the combined charges. Combining the integration con- stants, we obtain: We need to find a potential function for the combined charges which is zero at M.

Combining the integration constants, we obtain: Three point charges, 0. The sketch will show that V maximizes to a value of 8. Three identical point charges of 4 pC each are located at the corners of an equilateral triangle 0. How much work must be done to move one charge to a point equidistant from the other two and on the line joining them?

Since the charges are positioned symmetrically about the z axis, the potential at z will be double that from one charge.

This becomes: To find this we need to differentiate the part c result and find its zero: If the potential at the origin is V, find V at P 4, 1, 3: The net potential function for the two charges would in general be: We keep in mind the definition of absolute potential as the work done in moving a unit positive charge from infinity to location r.

At radii outside all three spheres, the potential will be the same as that of a point charge at the origin, whose charge is the sum of the three sphere charges: Using the results of part a, we substitute these distances in meters into the appropriate formulas to obtain: This last condition enables us to write the potential at P as a superposition of point charge potentials. The result is the integral: We use the superposition integral form: Line charge along the y axis: Line charge along y axis: Substitute P directly to obtain: This will be just Assuming free space conditions, find: Since the field is constant at constant radius, we obtain the product: How much charge lies within the cylinder?

Two point charges, 1 nC at 0, 0, 0. Use q. From the above field expression, the radial component magnitude is twice that of the theta component.

We use the general expression for the potential in the far field: This will be the surface of a cone, centered at the origin, along which E, in the ar direction, will exist. Therefore, the given surface cannot be an equipotential the problem was ill-conceived. Only a surface of constant r could be an equipotential in this field. Four 0. Again find the total stored energy: This will be the energy found in part a plus the amount of work done in moving the fifth charge into position from infinity.

The latter is just the potential at the square center arising from the original four charges, times the new charge value, or 4. Note first that current through the top and bottom surfaces will not exist, since J has no z component. We find the net outward current through the surface of the cube by integrating the divergence of J over the cube volume.

Find the total current I crossing the surface: Make the assumption that the electrons are emitted continuously as a beam with a 0. This is found using the equation of continuity: In part c, the net outward flux was found to be zero, and in part b, the divergence of J was found to be zero as will be its volume integral.

Therefore, the divergence theorem is satisfied. Assuming that there is no transformation of mass to energy or vice-versa, it is possible to write a continuity equation for mass. If we assume that the cube is an incremental volume element, determine an approximate value for the time rate of change of density at its center. We may write the continuity equation for mass as follows, also invoking the divergence theorem: The continuity equation for mass equates the divergence of the mass rate of flow mass per second per square meter to the negative of the density mass per cubic meter.

After setting up a cartesian coordinate system inside a star, Captain Kirk and his intrepid crew make measurements over the faces of a cube centered at the origin with edges 40 km long and parallel to the coordinate axes. We make the estimate using the definition of divergence, but without taking the limit as the volume shrinks to zero: The continuity equation for mass reads: Therefore, the rate of change of density at the origin will be just the negative.

Using data tabulated in Appendix C, calculate the required diameter for a 2-m long nichrome wire that will dissipate an average power of W when V rms at 60 Hz is applied to it: Thus 3. Let the total current carried by this hybrid conductor be 80 A dc. We begin with the fact that electric field must be the same in the aluminum and steel regions.

This comes from the requirement that E tangent to the boundary between two media must be continuous, and from the fact that when integrating E over the wire length, the applied voltage value must be obtained, regardless of the medium within which this integral is evaluated. The total current passing radially outward through the medium between the cylinders is 3 A dc.

Assume the cylinders are both of length l. We first find J as a function of radius by dividing the current by the area of a sphere of radius r: So it works. A hollow cylindrical tube with a rectangular cross-section has external dimensions of 0. A current of A dc is flowing down the tube.

Converting all measurements to meters, the tube resistance over a 1 m length will be: The resistance of the filling will be: Find the magnitude of the electric field intensity in a conductor if: Find the equation of the conductor surface: Set the given potential function equal to 20, to find: Since E is at the perfectly-conducting surface, it will be normal to the surface, so we may write: The surface will be an equipotential, where the value of the potential is VP: I will work parts b and c in reverse order Find E at P: Such a unit normal can be construced from the result of part c: There are two reasons for this: Therefore the surface charge density at P is 0.

A more general method involves deriving the potential from the given field: The general procedure is to adjust the functions, f , such that the result for V is the same in all three integrations. Setting the given potential function equal to 0 and 60 and simplifying results in: First, find the electric field there: For each line charge, this will be: Vectors from all four charges to point A are: First, E at the origin is done as per the setup in part a, except the vectors are directed from the charges to the origin: We use the far-field potential for a z-directed dipole: The potential at any point is now: We just set the potential exression of part a equal to V to obtain: The concentration of both holes and electrons is 2.

Determine both the conductivity and the resistivity of this silicon sample: Electron and hole concentrations increase with temperature. Atomic hydrogen contains 5. In a certain region where the relative permittivity is 2. In Fig. The origin lies in region 1. We need to find the components of E1 that are normal and tangent to the boundary, and then apply the appropriate boundary conditions. Since the magnitude is negative, the normal component points into region 1 from the surface.

So in region 1: Again calculate E, D, Q, and the energy: With the source disconnected, the charge is constant, and thus so is D: Capacitors tend to be more expensive as their capacitance and maximum voltage, Vmax , increase.

The voltage Vmax is limited by the field strength at which the dielectric breaks down, EBD. Which of these dielectrics will give the largest CVmax product for equal plate areas: Trying this with the given materials yields the winner, which is barium titanate. A dielectric circular cylinder used between the plates of a capacitor has a thickness of 0. R 0S 8. D will, as usual, be x-directed, originating at the top plate and terminating on the bottom plate.

The key here is that D will be constant over the distance between plates. This can be understood by considering the x-varying dielectric as constructed of many thin layers, each having constant permittivity. The permittivity changes from layer to layer to approximate the given function of x. The approximation becomes exact as the layer thicknesses approach zero. The width of the region containing R1 in Fig.

Having parallel capacitors, the capacitances will add, so R1 0 2. Let w1 be the width of region 1. The above conditions enable us to write: Calculate the capacitance per square meter of surface area if: The region 0.

Two coaxial conducting cylinders of radius 2 cm and 4 cm have a length of 1m. Perfect dielectrics occupy the interior region: Then 6. These fields are plotted below. For a spherical capacitor, we know that: Again, find C: With reference to Fig. A 2 cm diameter conductor is suspended in air with its axis 5 cm from a conducting plane.

Let the potential of the cylinder be V and that of the plane be 0 V. Find the surface charge density on the: The cylinder will image across the plane, producing an equivalent two-cylinder problem, with the second one at location 5 cm below the plane.

These dimensions are suitable for the drawing. The sketch is shown below. The capacitance is thus. These dimensions are suitable for the actual sketch if symmetry is considered. As a check, compute the capacitance per meter both from your sketch and from the exact formula.

Symmetry allows us to plot the field lines and equipotentials over just the first quadrant, as is done in the sketch below shown to one-half scale. Construct a curvilinear square map of the potential field between two parallel circular cylinders, one of 4-cm radius inside one of 8-cm radius.

The two axes are displaced by 2. As a check on the accuracy, compute the capacitance per meter from the sketch and from the exact expression: The drawing is shown below. Use of the drawing produces: A solid conducting cylinder of 4-cm radius is centered within a rectangular conducting cylinder with a cm by cm cross-section. The result below could still be improved a little, but is nevertheless sufficient for a reasonable capacitance estimate.

Note that the five-sided region in the upper right corner has been partially subdivided dashed line in anticipation of how it would look when the next-level subdivision is done doubling the number of field lines and equipotentials.

In this case NQ is the number of squares around the full perimeter of the circular conductor, or four times the number of squares shown in the drawing. NV is the number of squares between the circle and the rectangle, or 5. The inner conductor of the transmission line shown in Fig. The axes are displaced as shown. Some improvement is possible, depending on how much time one wishes to spend. Let the inner conductor of the transmission line shown in Fig.

Construct a grid, 0. Work to the nearest volt: The drawing is shown below, and we identify the requested voltage as 38 V. Use the iteration method to estimate the potentials at points x and y in the triangular trough of Fig.

Work only to the nearest volt: The result is shown below. The mirror image of the values shown occur at the points on the other side of the line of symmetry dashed line. Use iteration methods to estimate the potential at point x in the trough shown in Fig. Working to the nearest volt is sufficient. The result is shown below, where we identify the voltage at x to be 40 V. Note that the potentials in the gaps are 50 V. Using the grid indicated in Fig.

The voltages at the grid points are shown below, where VA is found to be 19 V. Half the figure is drawn since mirror images of all values occur across the line of symmetry dashed line. Conductors having boundaries that are curved or skewed usually do not permit every grid point to coincide with the actual boundary.

Figure 6. The other two distances are found by writing equations for the circles: The four distances and potentials are now substituted into the given equation: Consider the configuration of conductors and potentials shown in Fig. Using the method described in Problem 10, write an expression for Vx not V0: Again, work to the nearest volt: The result is shown below, with values for the original grid points underlined: Use the computer to obtain values for a 0.

Work to the nearest 0. Values for the left half of the configuration are shown in the table below. Values along the vertical line of symmetry are included, and the original grid values are underlined.

Perfectly-conducting concentric spheres have radii of 2 and 6 cm. Find E and J everywhere: From symmetry, E and J will be radially-directed, and we note the fact that the current, I , must be constant at any cross-section; i. Since we know the voltage between spheres 1V , we can find the value of I through: This we find through. The cross-section of the transmission line shown in Fig.

The result is independent of a, provided the proportions are maintained. The four radii are 1, 1. Connections made to the two rings show a resistance of ohms between them. Using the two radii 1.

The square washer shown in Fig.

The inside and outside surfaces are perfectly-conducting. A few curvilinear squares are suggested: Having found this, we can construct the total resistance by using the fundamental square as a building block. These numbers are found from the curvilinear square plot shown. A two-wire transmission line consists of two parallel perfectly-conducting cylinders, each having a radius of 0. A V battery is connected between the wires. A coaxial transmission line is modelled by the use of a rubber sheet having horizontal dimensions that are times those of the actual line.

The model is 8 cm in height at the inner conductor and zero at the outer. If the potential of the inner conductor is V: We use the part a result, since the gravitational function must be the same as that for the electric potential. We replace V0 by the maximum height, and multiply all dimensions by to obtain: The streamline is now specified by the equations: No, since the charge density is not zero.

It is known that both Ex and V are zero at the origin. Our two equations are: We integrate the charge density found in part a over the specified volume: First, we find E: The problem did not provide information necessary to determine this.

Uniqueness specifies that there is only one potential that will satisfy all the given boundary conditions. So they apply to different situations.

No for V1 V2. Only V2 is, since it is given as satisfying all the boundary conditions that V1 does.

The others, not satisfying the boundary conditions, are not the same as V1. Find and sketch: We begin with the general solution of the one-dimensional Laplace equation in rectangular coordinates: The planes are parallel, and so we expect variation in potential in the direction normal to them. Using the two boundary condtions, our general potential function can be written: Working in volts and meters, we write the general one-dimensional solution to the Laplace equation in cylindrical coordinates, assuming radial variation: B is then found through either equation; e.

The region between the cylinders is filled with a homogeneous perfect dielectric. If the inner cylinder is at V and the outer at 0V, find: From Eq. A narrow insulating strip separates them along the z axis. The boundary conditions are then substituted: Subtract the latter equation from the former to obtain: The two conducting planes illustrated in Fig.

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