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electrical engineering

Questions and Answers of Electrical Engineering

Given B = x (z - 3y) + y(2x -3z) - z(x + y), find a unit vector parallel to B at point P(1, 0, -1)
When sketching or demonstrating the spatial variation of a vector field, we often use arrows, as in Fig. 3-25 (P3.17), wherein the length of the arrow is made to be proportional to the strength of
Use arrows to sketch each of the following vector fields: (a) E1 = xx - yy, (b) E2 = - Ф (c) E3 = y 1/x (d) E4 = r cos.
Problem 3.19 Convert the coordinates of the following points from Cartesian to cylindrical and spherical coordinates:(a) P1 (1, 2, 0,(b) P2 (0, 0, 2),(c) P3 (1, 1, 3,(d) P4 (–2, 2 – 2)
Convert the coordinates of the following points from cylindrical to Cartesian coordinates: (a) P1 (2, 4, 2), (b) P2 (3, 0, 2), (c) P3 (4,
Convert the coordinates of the following points from spherical to cylindrical coordinates: (a) P1 (5, 0, 0), (b) P2 (5, 0,  (c) P3 (3, 0).
Problem 3.22 Use the appropriate expression for the differential surface area ds to determine the area of each of the following surfaces: (a) r = 3; 0 < <
Find the volumes described by (a) 2 < r < 5; 2 < < ; 0 < z < 2, (b) 0 < R < 5; 0 < < 3; 0 < < 2. Also sketch the outline of each volume.
A section of a sphere is described by 0 < R < 2, 0 < < 90, and 30o < < 90 Find: (a) The surface area of the spherical section, (b) The enclosed volume. Also sketch the outline of the
A vector field is given in cylindrical coordinates by E = rr cos r sinzz2 Point P (2, 3) is located on
At a given point in space, vectors A and B are given in spherical coordinates by(a) The scalar component, or projection, of B in the direction of A,(b) The vector component of B in the direction of
Given vectorsFind(a) θ AB at (2, p/2, 0),(b) A unit vector perpendicular to both A and B at (2, p/3, 1)
Find the distance between the following pairs of points: (a) P1 (1, 2, 3) and P2 (- 2, - 3, -2) in Cartesian coordinates, (b) P3 (1,
Determine the distance between the following pairs of points. (a) P1 (1, 1, 2) and P2 (0, 2, 3), (b) P3 (2, 3, 1) and P4 (4,
Transform the following vectors into cylindrical coordinates and then evaluate them at the indicated points: (a) A = x(x + y) at P (1, 2, 3), (b) B = x(y – x) + y (x - y) at P2 (1, 0, 2), (c) C
Transform the following vectors into spherical coordinates and then evaluate them at the indicated points: (a) A = xy2 + yxz + z4 at P1 (1 – 1, 2), (b) B = y(x2 + y2 + z2) – z (x2 + y2) at P2
Find the gradient of the following scalar functions: (a) T = 3 / (x2 + z2), (b) V = xy2 z4|, (c) U =z cos1 + r2), (d) W = e-R sin, (e) S = 4x2e-z
The gradient of a scalar function T is given byIf T =10 at z =0, find T 􀀀z
Follow a procedure similar to that leading to Eq. (3.82) to derive the expression given by Eq. (3.83) for in spherical coordinates.
For the scalar function V = xy2 - z2, determine its directional derivative along the direction of vector A = (x – yz) and then evaluate it at P (1, - 1, 4).
For the scalar function T = ½ e r/5 cos, determine its directional derivative along the radial direction ˆr and then evaluate it at P (2, 4, 3).
For the scalar function U – 1/R sin2, determine its directional derivative along the range direction R and then evaluate it at P (5, 4, 2).
Repeat Problem 3.45 for the contour shown in Fig. P3.46 (b)
For the vector field E = xxz _ yyz2 _ zxy, verify the divergence theorem by computing: (a) The total outward flux flowing through the surface of a cube centered at the origin and with sides equal to
For the vector field E = r10e-r _ z3z, verify the divergence theorem for the cylindrical region enclosed by r =2, z =0, and z = 4.
For the vector field E = r10e-r _ z3z, verify the divergence theorem for the cylindrical region enclosed by r =2, z =0, and z = 4.
For the vector field D = R3R2, evaluate both sides of the divergence theorem for the region enclosed between the spherical shells defined by R = 1 and R = 2.
For the vector field E = xxy = y (x2 + 2y2) calculate (a) FC E.d1 around the triangular contour shown in Fig P3.43 (a), and (b) fS (X E) ds over the area of the triangle
Repeat Problem 3.43 for the contour shown in Fig. P3.43 (b)
Verify Stokes’s theorem for the vector field B = (rrcos sinby evaluating: (a) fC B.d1 over the semicircular contour shown in Fig. P3.46 (a),
Repeat Problem 3.45 for the contour shown in Fig. P3.46 (b)
Determine if each of the following vector fields is solenoid, conservative, or both:
Verify Stokes’s Theorem for the vector field A = Rcos sin by evaluating it on the hemisphere of unit radius.
Find the Laplacian of the following scalar functions: (a) V = 4xy2z3, (b) V = xy + yz + zx, (c) V = 3/ (x2, y2) (d) V = 5e-r cos, (e) V =10e-R sin
Find a vector G whose magnitude is 4 and whose direction is perpendicular to both vectors E and F, where E = x + y2 - z2 and F = y3 _ z6.
A given line is described by the equation:y = x – 1.Vector A starts at point P1 (0, 2) and ends at point P2 on the line such that A is orthogonal to the line. Fine an expression for A.
Vector field E is given byDetermine the component of E tangential to the spherical surface R = 2 at point P (2, 30o, 60o)
Transform the vectorInto cylindrical coordinates and then evaluate it at P (2, 2, 2).
Evaluate the line integral of E = xx_ yy along the segment P1 to P2 of the circular path shown in the figure.
Verify Stokes€™s theorem for the vector field B = (r cosË†sinby evaluating: (a) fC B. dl over the path
A voltage generator with vg (t) = 5 cos (2π x 109 t) V and internal impedance Zg – 50 Ω is connected to a 50-Ω is connected to a 50-Ω lossless air-spaced transmission line.
A 6-m section of 150-W lossless line is driven by a source with vg (t) = 5 cos (8π x 107t- 30o) (v) And Zg = 150Ω. If the line, which has a relative permittivity εr = 2.25, is
Find the total charge contained in a cylindrical volume defined by r < 2 m and 0 < z < 3 m if v = 20rz (mC/m3).
Find the total charge contained in a cone defined by R < 2 m and 0 < < 4, given that v = 10R2 cos2  (mC/m3).
If the line charge density is given by l = 24y2 (mC/m), find the total charge distributed on the y-axis from y = _ 5 to y =5.
The input impedance of a 31-cm-long lossless transmission line of unknown characteristic impedance was measured at 1 MHz. With the line terminated in a short circuit, the measurement yielded an input
If J = y4xz (A/m2), find the current I flowing through a square with corners at (0, 0, 0), (2, 0, 0), (2, 0, 2), and (0, 0, 2)
A 100-MHz FM broadcast station uses a 300-transmission line between the transmitter and a tower-mounted half-wave dipole antenna. The antenna impedance is 73 . You are asked
A 50-MHz generator with Zg = 50 is connected to a load ZL= 50 j25-. The time-average power transferred from the generator into the load is maximum when Zg = Z_L_ where Z _L is the complex
A square with sides 2 m each has a charge of 40 μC at each of its four corners. Determine the electric field at a point 5 m above the center of the square.
Three point charges, each with q = 3 nC, are located at the corners of a triangle in the x–y plane, with one corner at the origin, another at (2 cm, 0, 0), and the third at (0, 2 cm, 0). Find the
Charge q1 = 6 μC is located at (1 cm, 1 cm, 0) and charge q2 is located at (0, 0, 4 cm). What should q2 be so that E at (0, 2 cm, 0) has no y-component?
A line of charge with uniform density l = 8 (μC/m) exists in air along the z-axis between z = 0 and z = 5 cm. Find E at (0, 10 cm, 0).
Electric charge is distributed along an arc located in the x–y plane and defined by r = 2 cm and 0 < < 4. If l _ 5 (μC/m), find E at (0, 0, z) and then
A line of charge with uniform density extends between z = -L/2 and z = L/2 along the z-axis. Apply Coulomb’s law to obtain an expression for the electric field at any point P (r, 0) on the x–y
Repeat Example 4-5 for the circular disk of charge of radius a, but in the present case assume the surface charge density to vary with r as where s0 is a constant.
Multiple charges at different locations are said to be in equilibrium if the force acting on any one of them is identical in magnitude and direction to the force acting on any of the others. Suppose
Three infinite lines of charge, all parallel to the z-axis, are located at the three corners of the kite-shaped arrangement shown in Fig. 4-29 (P4.17). If the two right triangles are symmetrical and
Three infinite lines of charge, pl1 = 3 (nC/m), pl2 = _ 3 (nC/m), and pl3 = 3 (nC/m), are all parallel to the z-axis. If they pass through the respective points.
A horizontal strip lying in the x€“y plane is of width d in the y-direction and infinitely long in the x-direction. If the strip is in air and has a uniform charge distribution s,
Given the electric flux density determine(a) pv by applying Eq. (4.26),(b) The total charge Q enclosed in a cube 2 m on a side, located in the first octant with three of its sides coincident with the
Repeat Problem 4.20 for D = xxy3z3 (C/m2).
Charge Q1 is uniformly distributed over a thin spherical shell of radius a, and charge Q2 is uniformly distributed over a second spherical shell of radius b, with b < a. Apply Gauss’s law to find E
The electric flux density inside a dielectric sphere of radius a centered at the origin is given by D = R0R (C/m2), where 0 is a constant. Find the total charge inside the sphere.
In a certain region of space, the charge density is given in cylindrical coordinates by the function:Apply Gauss€™s law to find D.
An infinitely long cylindrical shell extending between r – 1 m and r – 3 m contains a uniform charge density v0. Apply Gauss’s law to find D in all regions.
If the charge density increases linearly with distance from the origin such that v = 0 at the origin and v = 40 C/m3 at R = 2 m, find the corresponding variation of D.
A square in the x–y plane in free space has a point charge of + Q at corner (a/2, a/2) and the same at corner (a/2, a/2) and a point charge of – Q at each of the other two corners. (a) Find the
The circular disk of radius a shown in Fig. 4-7 (P4.28) has uniform charge density s across its surface. (a) Obtain an expression for the electric potential V at a point P (0, 0, z) on the
A circular ring of charge of radius a lies in the x–y plane and is centered at the origin. If the ring is in air and carries a uniform density l , (a) Show that the electrical potential
A cylindrical bar of silicon has a radius of 4mmand a length of 8 cm. If a voltage of 5 V is applied between the ends of the bar and μe = 0.13 (m2/V.s), μh = 0 05 (m2/V _ s), Ne = 1.5 x
Find the electric potential V at a location a distance b from the origin in the x€“y plane due to a line charge with charge density l and of length l. The line charge is coincident
For the electric dipole shown in Fig. 4-13, d =1 cm and |E| = 4 (mV/m) at R = 1 m and 0o. Find E at R = 2 m and 90 o.
For each of the following distributions of the electric potential V, sketch the corresponding distribution of E (in all cases, the vertical axis is in volts and the horizontal axis is in meters):
Given the electric field find the electric potential of point A with respect to point B where A is at + 2 m and B at €“ 4 m, both on the z-axis.
An infinitely long line of charge with uniform density l = 9 (nC/m) lies in the x–y plane parallel to the y-axis at x _ 2 m. Find the potential VAB at point A (3m, 0, 4m) in Cartesian coordinates
The x–y plane contains a uniform sheet of charge with s1 = 0.2 (nC/m2 _ and a second sheet with s2 = - 0.2 (nC/m2) occupies the plane z = 6 m. Find VAB, VBC, and VAC for A (0, 0, 6m), B (0, 0, 0)
Show that the electric potential difference V12 between two points in air at radial distances r1 and r2 from an infinite line of charge with density l along the z-axis is V12
Repeat Problem 4.37 for a bar of germanium with μe = 0.4 (m2/V.s), μh =0.2 (m2/V.s), and Ne = Nh = 2.4 x 1019 electrons or holes/m3.
A 100-m-long conductor of uniform cross section has a voltage drop of 4 V between its ends. If the density of the current flowing through it is 1.4 x 106 (A/m2), identify the material of the
A coaxial resistor of length l consists of two concentric cylinders. The inner cylinder has radius a and is made of a material with conductivity 1, and the outer cylinder, extending between r
Apply the result of Problem 4.40 to find the resistance of a 20-cmlong hollow cylinder (Fig. P4.41) made of carbon with 3 x 104 (S/m).
A 2 x 10–3 -mm-thick square sheet of aluminum has 5 cm x 5 cm faces. Find:(a) The resistance between opposite edges on a square face, and(b) The resistance between the two square faces. (See
With reference to Fig. 4-19, find E1 if E2 = x3 – y2 + z2 (V/m), 1 = 20, 2 = 180, and the boundary has a surface charge density s = 3.54 x10-11 (C/m2). What
An infinitely long conducting cylinder of radius a has a surface charge density s. The cylinder is surrounded by a dielectric medium with r = 4 and contains no free charges. If the
A 2-cm conducting sphere is embedded in a charge-free dielectric medium with 2r = 9. If E2 = R 3cos3sin (V/m) in the
If E = R150 (V/m) at the surface of a 5-cm conducting sphere centered at the origin, what is the total charge Q on the sphere’s surface?
Figure 4-34(a) (P4.47) shows three planar dielectric slabs of equal thickness but with different dielectric constants. If E0 in air makes an angle of 45 _ with respect to the z-axis, find the angle
Determine the force of attraction in a parallel-plate capacitor with A = 5 cm2, d=_ 2 cm, and r = 4 if the voltage across it is 50 V.
Dielectric breakdown occurs in a material whenever the magnitude of the field E exceeds the dielectric strength anywhere in that material. In the coaxial capacitor of Example 4-12, (a) At what value
An electron with charge Qe = 1.6 x 10–19 C and mass me = 9.1 x 10–31 kg is injected at a point adjacent to the negatively charged plate in the region between the plates of an air-filled
In a dielectric medium with r4, the electric field is given byCalculate the electrostatic energy stored in the region €“ 1 m
Figure 4-34a (P4.52 (a)) depicts a capacitor consisting of two parallel, conducting plates separated by a distance d. The space between the plates contains two adjacent dielectrics,One with
Use the result of Problem 4.52 to determine the capacitance for each of the following configurations:(a) Conducting plates are on top and bottom faces of rectangular structure in Fig. 4-35(a) (P4.53
The capacitor shown in Fig. 4-36 (P4.54) consists of two parallel dielectric layers. We wish to use energy considerations to show that the equivalent capacitance of the overall capacitor, C, is equal
Use the expressions given in Problem 4.54 to determine the capacitance for the configurations in Fig. 4.35(a) (P4.55) when the conducting plates are placed on the right and left faces of the structure
With reference to Fig. 4-37 (P4.56), charge Q is located at a distance d above a grounded half-plane located in the x€“y plane and at a distance d from another grounded half-plane in the
Conducting wires above a conducting plane carry currents I1 and I2 in the directions shown in Fig. 4-38 (P4.57). Keeping in mind that the direction of a current is defined in terms of the movement of
Use the image method to find the capacitance per unit length of an infinitely long conducting cylinder of radius a situated at a distance d from a parallel conducting plane, as shown in Fig. 4-39
A circular beam of charge of radius a consists of electrons moving with a constant speed u along the + z direction. The beam’s axis is coincident with the z-axis and the electron charge density is

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