The capacitance will be Qnet What resistance is measured between the two perfect conductors? Two coaxial conducting cones have their vertices at the origin and the z axis as their axis. Cone A has the point A 1, 0, 2 on its surface, while cone B has the point B 0, 3, 2 on its surface. The solution is found from Eq. Using thirteen terms,. The series converges rapidly enough so that terms after the sixth one produce no change in the third digit.
The four sides of a square trough are held at potentials of 0, 20, , and 60 V; the highest and lowest potentials are on opposite sides. Find the potential at the center of the trough: Here we can make good use of symmetry. The solution for a single potential on the right side, for example, with all other sides at 0V is given by Eq. In Fig. Functions of this form are called circular harmonics. Referring to Chapter 6, 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. 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. 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: 0.
The four distances and potentials are now substituted into the given equation:. Using the method described in Problem 7. Use a computer to obtain values for a 0. Work to the nearest 0. Values along the vertical line of symmetry are included, and the original grid values are underlined. The Biot-Savart method was used here for the sake of illustration. I will work this one from scratch, using the Biot-Savart law. It is also possible to work this problem somewhat more easily by using Eq.
Each carries a current I in the az direction. Determine the side length b in terms of a , such that H at the origin is the same magnitude as that of the circular loop of part a. Applying Eq. A disk of radius a lies in the xy plane, with the z axis through its center. Find H at any point on the z axis. Since the limits are symmetric, the integral of the z component over y is zero. Find H in spherical coordinates a inside and b outside the sphere. The sketch below shows one of the slabs of thickness D oriented with the current coming out of the page.
For example, if the sketch below shows the upper slab in Fig. Thus H will be in the positive x direction above the slab midpoint, and will be in the negative x direction below the midpoint. We are now in a position to solve the problem. This point lies within the lower slab above its midpoint. Referring to Fig. Since 0. There sec. The only way to enclose current is to set up the loop which we choose to be rectangular such that it is oriented with two parallel opposing segments lying in the z direction; one of these lies inside the cylinder, the other outside.
The loop is now cut by the current sheet, and if we assume a length of the loop in z of d, then the enclosed current will be given by Kd A. Thus H would not change with z. There would also be no change if the loop was simply moved along the z direction. We would expect Hz outside to decrease as the Biot-Savart law would imply but the same amount of current is always enclosed no matter how far away the outer segment is.
Inner and outer currents have the same magnitude. We can now proceed with what is requested: a PA 1. We obtain 2. A balanced coaxial cable contains three coaxial conductors of negligible resistance.
Assume a solid inner conductor of radius a, an intermediate conductor of inner radius bi , outer radius bo , and an outer conductor having inner and outer radii ci and co , respectively. The intermediate conductor carries current I in the positive az direction and is at potential V0. A solid conductor of circular cross-section with a radius of 5 mm has a conductivity that varies with radius.
The value of H at each point is given. Each curl component is found by integrating H over a square path that is normal to the component in question. The x component of the curl is thus:. To do this, we use the result of Problem 8. This leaves only the path segment that coindides with the axis, and that lying parallel to the axis, but outside. Their centers are at the origin.
Integrals over x, to complete the loop, do not exist since there is no x component of H. The path direction is chosen to be clockwise looking down on the xy plane. A long straight non-magnetic conductor of 0. A solid nonmagnetic conductor of circular cross-section has a radius of 2mm. All surfaces must carry equal currents. Thus, using the result of Section 8. The simplest form in this case is that involving the inverse hyperbolic sine.
Compute the vector magnetic potential within the outer conductor for the coaxial line whose vector magnetic potential is shown in Fig. By expanding Eq. Use Eq. The initial velocity in x is constant, and so no force is applied in that direction. Make use of Eq. Solve these equations perhaps with the help of an example given in Section 7.
A circular orbit can be established if the magnetic force on the particle is balanced by the centripital force associated with the circular path. In either case, the centripital force must counteract the magnetic force. A rectangular loop of wire in free space joins points A 1, 0, 1 to B 3, 0, 1 to C 3, 0, 4 to D 1, 0, 4 to A. Note that by symmetry, the forces on sides AB and CD will be equal and opposite, and so will cancel.
Find the total force on the rectangular loop shown in Fig. A planar transmission line consists of two conducting planes of width b separated d m in air, carrying equal and opposite currents of I A. Take the current in the top plate in the positive z direction, and so the bottom plate current is directed along negative z.
The rectangular loop of Prob. Assume that an electron is describing a circular orbit of radius a about a positively-charged nucleus. Calculate the vector torque on the square loop shown in Fig. So we must use the given origin. Then M 0. At radii between the currents the path integral will enclose only the inner current so, 3. Find a H everywhere: This result will depend on the current and not the materials, and is: I 1. The core shown in Fig.
A coil of turns carrying 12 mA is placed around the central leg. We now have mmf In Problem 9. Using this value of B and the magnetization curve for silicon. Using Fig. A toroidal core has a circular cross section of 4 cm2 area. The mean radius of the toroid is 6 cm.
There is a 4mm air gap at each of the two joints, and the core is wrapped by a turn coil carrying a dc current I1. The reluctance of each gap is now 0. From Fig. Then, in the linear material, 1. This is still larger than the given value of.
The result of 0. A toroid is constructed of a magnetic material having a cross-sectional area of 2. There is also a short air gap 0. This is d 0. A toroidal core has a square cross section, 2. The currents return on a spherical conducting surface of 0. Second method: Use the energy computation of Problem 9. The core material has a relative permeability of A coaxial cable has conductor dimensions of 1 and 5 mm.
Find the inductance per meter length: The interfaces between media all occur along radial lines, normal to the direction of B and H in the coax line. B is therefore continuous and constant at constant radius around a circular loop centered on the z axis. The rings are coplanar and concentric. We use the result of Problem 8. Now for the right hand side. The location of the sliding bar in Fig.
The rails in Fig. Then D 1. Now B 2. Find the total displacement current through the dielectric and compare it with the source current as determined from the capacitance Sec.
The parallel plate transmission line shown in Fig. Thus 1. A transmitter and receiver are connected using a cascaded pair of transmission lines. At the operating frequency, Line 1 has a measured loss of 0. The link is composed of 40m of Line 1, joined to 25m of Line 2.
At the joint, a splice loss of 2 dB is measured. If the transmitted power is mW, what is the received power? The total loss in the link in dB is 40 0. Suppose a receiver is rated as having a sensitivity of -5 dBm — indicating the minimum power that it must receive in order to adequately interpret the transmitted data. Consider a transmitter having an output of mW connected to this receiver through a length of transmission line whose loss is 0.
What is the maximum length of line that can be used? For this impedance to equal 50 ohms, the imaginary parts must cancel. If so what are they? At the input end of the line, a DC voltage source, V0 , is connected.
In a circuit in which a sinusoidal voltage source drives its internal impedance in series with a load impedance, it is known that maximum power transfer to the load occurs when the source and load impedances form a complex conjugate pair.
The condition of maximum power transfer will be met if the input impedance to the line is the conjugate of the internal impedance. What average power is delivered to each load resistor?
First, we need the input impedance. The parallel resistors give a net load impedance of 20 ohms. For the transmission line represented in Fig. A ohm transmission line is 0. The line is operating in air with a wavelength of 0. Determine the average power absorbed by each resistor in Fig. The next step is to determine the input impedance of the 2. The power dissipated by the ohm resistor is now 1 V 2 1 A lossless transmission line is 50 cm in length and operating at a frequency of MHz.
Determine s on the transmission line of Fig. To achieve this, the imaginary part of the total impedance of part c must be reduced to zero so we need an inductor. Using normalized impedances, Eq. A line drawn from the origin through this point intersects the outer chart boundary at the position 0. With a wavelength of 1. On the WTL scale, we add 0. A straight line is now drawn from the origin though the 0.
A compass is then used to measure the distance between the origin and zin. This is close to the value of the VSWR, as we found earlier. Problem This is close to the computed inverse of yL , which is 1. Now, the position of zL is read on the outer edge of the chart as 0. The point is now transformed through the line length distance of 1. Drawing a line between this mark on the WTG scale and the chart center, and scribing the compass arc length on this line, yields the normalized input impedance.
On the WTG scale, we read the zL location as 0. The distance is then 0. What is s on the remainder of the line? This will be just s for the line as it was before. This would return us to the original point, requiring a complete circle around the chart one- half wavelength distance. With the aid of the Smith chart, plot a curve of Zin vs.
Then, using a compass, draw a circle beginning at zL and progressing clockwise to the positive real axis. The intersections of the lines and the circle give a total of 11 zin values. The table below summarizes the results. A fairly good comparison is obtained. We mark this on the positive real axis of the chart see next page. The load position is now 0. A line is drawn from the origin through this point on the chart, as shown. We then scribe this same distance along the line drawn through the.
A line is drawn from the origin through this location on the chart. Drawing a line from the chart center through this point yields its location at 0. Alternately, use the s scale at the bottom of the chart, setting the compass point at the center, and scribing the distance on the scale to the left. This distance in wavelengths is just the load position on the WTL scale, since the starting point for this scale is the negative real axis. So the distance is 0. Transforming the load through this distance toward the generator involves revolution once around the chart 0.
A line is drawn between this point and the chart center. This is plotted on the Smith chart below. We then set on the compass the distance between yL and the origin. The same distance is then scribed along the positive real axis, and the value of s is read as 2. We note a reading on that scale of about 0. To this we add 0. A line drawn from the 0. This is at the zero position on the WTL scale.
The load is at the approximate 0. The wavelength on a certain lossless line is 10cm. We read the input location as slightly more than 0. The line length of 12cm corresponds to 1. Thus, to transform to the load, we go counter-clockwise twice around the chart, plus 0. A line is drawn to the origin from that position, and the compass with its previous setting is scribed through the line.
A standing wave ratio of 2. Probe measurements locate a voltage minimum on the line whose location is marked by a small scratch on the line. When the load is replaced by a short circuit, the minima are 25 cm apart, and one minimum is located at a point 7 cm toward the source from the scratch. This is a possible location for the scratch, which would otherwise occur at multiples of a half-wavelength farther away from that point, toward the generator.
As a check, I will do the problem analytically. At the point X, indicated by the arrow in Fig. With the short circuit removed, a voltage minimum is found 5cm to the left of X, and a voltage maximum is located that is 3 times voltage of the minimum.
This point is then transformed, using the compass, to the negative real axis, which corresponds to the location of a voltage minimum. On the chart, we now move this distance from the Vmin location toward the load, using the WTL scale.
A line is drawn from the origin through the 0. With a short circuit replacing the load, a minimum is found at a point on the line marked by a small spot of puce paint. The 1m distance is therefore 3. Therefore, with the actual load installed, the Vmin position as stated would be 3. This being the case, the normalized load impedance will lie on the positive real axis of the Smith chart, and will be equal to the standing wave ratio.
This point is to be transformed to a location at which the real part of the normalized admittance is unity. The stub is connected at either of these two points. The stub input admittance must cancel the imaginary part of the line admittance at that point. This point is marked on the outer circle and occurs at 0. The length of the stub is found by computing the distance between its input, found above, and the short-circuit position stub load end , marked as Psc.
The length of the main line between its load and the stub attachment point is found on the chart by measuring the distance between yL and yin2 , in moving clockwise toward generator.
This occurs at 0. The attachment point is found by transforming yL to yin1 , where the former point is located at 0. The lossless line shown in Fig. For the line to be matched, it is required that the sum of the normalized input admittances of the shorted stub and the main line at the point where the stub is connected be unity. So the input susceptances of the two lines must cancel. This line is one-quarter wavelength long, so the normalized load impedance is equal to the normalized input admittance.
To cancel the input normalized susceptance of We therefore write 2. The two-wire lines shown in Fig. In this case, we have a series combination of the loaded line section and the shorted stub, so we use impedances and the Smith chart as an impedance diagram. The requirement for matching is that the total normalized impedance at the junction consisting of the sum of the input impedances to the stub and main loaded section is unity.
In the transmission line of Fig. First, the load voltage is found by adding voltages along the right side of the voltage diagram at the indicated times. The load voltage as a function of time is found by accumulating voltage values as they are read moving up along the right hand boundary of the chart.
In the charged line of Fig. This problem accompanies Example Plots of the voltage and current at the resistor are then found by accumulating values from the left sides of the two charts, producing the plots as shown. A simple frozen wave generator is shown in Fig.
Determine and plot the load voltage as a function of time: Closing the switches sets up a total of four voltage waves as shown in the diagram below. Note that in Problem Given, a MHz uniform plane wave in a medium known to be a good dielectric. Also, the specified distance in part f should be 10m, not 1km. We use the good dielectric approximations, Eqs. Perfectly-conducting cylinders with radii of 8 mm and 20 mm are coaxial. The external and internal regions are non-conducting. The inner and outer dimensions of a copper coaxial transmission line are 2 and 7 mm, respec- tively.
The dielectric is lossless and the operating frequency is MHz. A hollow tubular conductor is constructed from a type of brass having a conductivity of 1. The inner and outer radii are 9 mm and 10 mm respectively. Calculate the resistance per meter length at a frequency of a dc: In this case the current density is uniform over the entire tube cross-section.
Most microwave ovens operate at 2. A good conductor is planar in form and carries a uniform plane wave that has a wavelength of 0. The outer conductor thickness is 0. Use information from Secs. The force on a differential length. Thus the force per unit length acting on. The differential force produced by this. The force per unit area is. Find E at the origin if the following charge distributions are present in free space: point charge, 12 nC.
An electric dipole discussed in detail in Sec. Using rectangular coordinates, determine. Find the equation of the streamline passing through. For fields that do not vary with z in cylindrical coordinates, the equations of the streamlines are. Find the equation of the line passing. An empty metal paint can is placed on a marble table, the lid is removed, and both parts are.
Determine the total flux. State whether the divergence of the following vector fields is positive, negative, or zero: a the. A spherical surface of radius 3 mm is centered at P 4, 1, 5 in free space. Evaluate the outward flux of F 1 through the. Our flux integral. Given an initial point P 2, 1, 1 and a final point Q 4, 3, 1 , find. Find the potential at any point, using.
Three identical point charges of 4 pC each are located at the corners of an equilateral triangle. Evaluate each of the following quantities. Find the total. Assuming free space conditions, find:. Use the electric field intensity of the dipole Sec. Find the total stored. Four 0. Assume that a uniform electron beam of circular cross-section with radius of 0. Using data tabulated in Appendix C, calculate the required diameter for a 2-m long nichrome. The region. A hollow cylindrical tube with a rectangular cross-section has external dimensions of 0.
It is known that. At P , find. At a certain temperature, the electron and hole mobilities in intrinsic germanium are given as. If the electron and hole concentrations are both 2. Electron and hole concentrations increase with temperature. For pure silicon, suitable expressions. Find the dielectric constant of a material in which the electric flux density is four times the.
The planar interface. An air-filled parallel-plate capacitor with plate separation d and plate area A is connected to. D will, as usual, be x-directed, originating at the.
A parallel-plate capacitor is made using two circular plates of radius a, with the bottom plate. For the conductor configuration of Problem 6.
Construct a curvilinear square map of the potential field between two parallel circular cylinders,. A solid conducting cylinder of 4-cm radius is centered within a rectangular conducting cylinder.
For the coaxial capacitor of Problem 6. A two-wire transmission line consists of two parallel perfectly-conducting cylinders, each having. Itisknown that both. At this stage, it is helpful to recall that the x coordinate in rectangular coordinates. The electric field is now. Only V 2 is,. Consider the parallel-plate capacitor of Problem 7. Consider the. Repeat Problem 7. Two coaxial conducting cones have their vertices at the origin and the z axis as their axis.
The four sides of a square trough are held at potentials of 0, 20, , and 60 V; the highest. In Fig. The integral on the right hand side picks the nth term out of the series, enabling the coefficients,. Referring to Chapter 6, Fig. Use the iteration method to estimate the potentials at points x and y in the triangular trough.
Use iteration methods to estimate the potential at point x in the trough shown in Fig. Using the grid indicated in Fig. Consider the configuration of conductors and potentials shown in Fig. Using the method. Use a computer to obtain values for a 0. Work to the nearest 0. Find H in cartesian components at P 2, 3, 4 if there is a current filament on the z axis carrying.
A filamentary conductor is formed into an equilateral triangle with sides of length l carrying. Now, x 0 lies at the center of the equilateral triangle, and from the geometry of. The final answer is therefore.
A disk of radius a lies in the xy plane, with the z axis through its center. Surface charge of. The differential field at point. For the finite-length current element on the z axis, as shown in Fig. A hollow spherical conducting shell of radius a has filamentary connections made at the top. Three uniform cylindrical. A hollow cylindrical shell of radius a is centered on the z axis and carries a uniform surface. A toroid having a cross section of rectangular shape is defined by the following surfaces: the.
The construction is similar to that of the toroid of round cross section as done on p. Assume that there is a region with cylindrical symmetry in which the conductivity is given by. A balanced coaxial cable contains three coaxial conductors of negligible resistance. Assume a. The inner and. A current filament on the z axis carries a current of 7 mA in the a z direction, and current. A solid conductor of circular cross-section with a radius of 5 mm has a conductivity that varies.
Find the total current in the. A solid nonmagnetic conductor of circular cross-section has. Use an expansion in rectangular coordinates to show that the curl of the gradient of any scalar.
A filamentary conductor on the z axis carries a current of 16A in the a z direction, a conducting. Assume a direct current I amps flowing in the a z direction in a filament extending between. Show that the line integral of the vector potential A about any closed path is equal to the.
Compute the vector magnetic potential within the outer conductor for the coaxial line whose. By expanding Eq. Integrating a second time yields the. Make use of Eq. Solve these equations. We can construct the differential equations for the forces in x and in y as. Show that a charged particle in a uniform magnetic field describes a circular orbit with an orbital. A rectangular loop of wire in free space joins points A 1, 0, 1 to B 3, 0, 1 to C 3, 0, 4 to D 1, 0, 4.
Note that. Wewish to find the force acting to split the outer. Since the outer cylinder is a two-dimensional current sheet, its field exists only just outside the. If this cylinder possessed a finite thickness, then we would need. A planar transmission line consists of two conducting planes of width b separated d minair,. Let z 1 indicate the z coordinate along I 1 , and z 2 indicate the z coordinate along I 2.
This second method is really just the first over again, since we recognize the inside integral of. A current of 6A flows from M 2, 0, 5 to N 5, 0, 5 in a straight solid conductor in free space. Compute the. The rectangular loop of Prob. With this field, forces will be acting only on the wire segments that are parallel to the y axis. An infinite filament on the z axis carries 5.
0コメント