1. Stationary rectangular loop. Consider a fixed single-turn
rectangular loop of area A with its plane perpendicular to a uniform
magnetic field. Find the voltage induced across the terminals of the
loop if the magnetic flux density is given by (a) B(t ) = B0te−αt and
(b) B(t ) = B0e−αt sin(ωt ). Get solution
2. Stationary circular loop. Consider a 20-turn circular loop of wire of 15 cm diameter with its plane perpendicular to a uniform magnetic field, as shown in Figure If the magnetic flux density B is given byFigure Emf induced in a coil.......find the rms value of the induced current through the 10Ω resistor at the following time instants: (a) t = 0, (b) t = 10 ms, (c) t = 100 ms, and (d) t = 1 s. Get solution
3. Two circular coils. Two circular coils of 5 cm radius each have the same axis of symmetry and are 1 m apart from each other. Coil 1 has 10 turns and coil 2 has 100 turns. (a) If an alternating current of 100 A amplitude and 1 kHz frequency is passed through coil 1, find the amplitude of the induced voltage across the open terminals of coil 2. (b) Repeat part (a) for a frequency of 10 kHz. Get solution
4. Two concentric coils. Consider the two concentric coils shown in Figure The magnetic flux density produced at the center of the two coils due to the current I flowing in the larger coil is given byFigure Two concentric coils......(a) If the larger coil has 30 turns, find its radius. (b) If the smaller coil has 75 turns and its radius is 1 cm, find the total flux linking the smaller coil. (c) If the current I in the larger coil is given by I (t ) = 10 cos(120πt ) A, find the induced current through the resistor R assuming R = 10Ω. Get solution
5. Triangular loop and long wire. An equilateral triangular loop is situated near a long current-carrying wire, as shown in Figure The wire is part of a power line carrying 60-Hz sinusoidal current. An ac ammeter inserted in the loop reads a current of amplitude 1 mA. Assume the total resistive impedance of the loop to be 0.01Ω. Find the amplitude of the current I in the long wire.Figure Triangular loop and long wire.... Get solution
6. Faraday’s law. Consider a circular loop (radius a=5 cm) of wire lying in the x-y plane with its center at the origin, in the presence of a z -directed magnetic field...where r is the polar coordinate and B0=10 mT (milliwebers/m2). The loop wire is made up of copper (σ =5.8×107 S/m) and has a cross-sectional area of 1 mm2. It is known that the wire would melt if the total current I flowing through it exceeds 20 A (I >20 A). Stating all assumptions, calculate the maximum frequency f for which this wire can be used in the presence of this magnetic field. Get solution
7. Toroidal coil around a long, straight wire. A long, straight wire carrying an alternating current of I (t ) = 100 cos(377t ) A coincides with the principal axis of symmetry of a 200- turn coil wrapped uniformly around a rectangular, toroidal-shaped iron core of inner and outer radii a = 6 cm and b = 8 cm, thickness t = 3 cm, and relative permeability μr = 1000, as shown in Figure Calculate the induced voltage Vind between the terminals of the coil. (This type of coil, called a current transformer, is used to measure the current in a conductor wire that passes through it.)Figure Toroidal coil around long wire. Problem.7.7... Get solution
8. Current transformer. A current transformer is used to measure the current in a high-voltage transmission line, as shown in Figure The circular toroidal core has a mean diameter of 6 cm, circular cross section of 1 cm diameter, and relative permeability of μr = 200. The winding consists of N = 300 turns. If the 60-Hz current of amplitude 1000 A flows through the high-voltage line, find the rms value of the open circuit voltage induced across the terminals of the toroid.Figure A current transformer. Problem.7.8... Get solution
9. Sliding bar in a constant magnetic field. A metal bar able to move on fixed rails is initially at rest on two stationary conducting rails that are l = 50 cm apart and connected to each other via a V0 = 10 V voltage source in series with a R = 10Ω resistor, as shown in Figure A constant magnetic field of B0 = ẑ1.5 T is turned on at t = 0. Assume the rails and the metal bar to be perfectly conducting and neglect the self-inductance of the loop formed by the rails and the bar. The initial resting position of the bar is at y0 = 75 cm. (a) With no friction and assuming that the mass of the bar is 0.5 kg, calculate the magnitudes and directions of its initial acceleration and its final velocity. Explain your reasoning. (b) Describe what happens if the voltage source is turned off (i.e., V0 = 0) at t = t0, when the bar has reached a velocity v0. (c) What happens if the voltage source remains at V0 but the magnetic field is turned off (i.e., B = 0) at t = t0?Figure Sliding bar in a constant magnetic field. Problem.... Get solution
10. Oscillating bar in a time-varying magnetic field. A metal wire is in contact with two long, parallel, stationary conducting rails connected with a resistance R at one end, as shown in Figure The wire oscillates symmetrically around x = x0 with a velocity v(t ) = xv0 cos(ω0t ) in the presence of a time-varying magnetic field that is perpendicular to the plane of the rails. The magnetic field is given by B(t ) = ˆzB0 sin(ω0t ). Find the induced current through the resistance R.Figure Oscillating bar in a time-varying magnetic field. Problem.... Get solution
11. Moving square loop. A uniform magnetic field of B0 = 1 T is confined to a square-shaped area 10 cm to a side as shown in Figure with zero magnetic field everywhere else. A square loop of side length 5 cm moves through the loop with a velocity of v = 10 cm-s−1. Find the electromotive force induced in the square loop and plot its value as a function of distance x, for 0 ≤ x ≤ 15 cm. Note that the plane of the loop is orthogonal to the magnetic field.Figure Moving square loop.Problem.... Get solution
12. Rectangular loop and long wire. A rectangular loop of wire lies in the plane of a long, straight wire carrying a steady current I, as shown in Figure If the loop is moving radially away from the long wire with a velocity v = rv0 without changing its shape, find the voltage induced across the open terminals of the loop with its polarity indicated.Figure Rectangular loop and long wire. Problem.... Get solution
13. Rotating wire in a constant magnetic field. A semicircular-shaped wire of radius a is rotated in a constant magnetic field B0 at a constant angular frequency ω, as shown in Figure If the wire forms a closed loop with a resistance R, find the induced current in R. Neglect the resistance of the wires.Figure Rotating wire in constant magnetic field. Problem.... Get solution
14. Rotating rectangular loop in a constant magnetic field. A single turn rectangular coil of dimensions 5 cm × 10 cm is rotated at 1500 rpm in a uniform magnetic field of 50 mT normal to the axis of rotation, as shown in Figure Find the induced voltage across the open terminals of the loop.Figure Elementary alternating current generator.... Get solution
15. Rotating rectangular loop in a nonuniform magnetic field. Consider the rectangular coil of 10 turns with dimensions 20 cm × 10 cm, rotating at an angular velocity of 2000 rpm about its axis, as shown in Figure in the presence of a magnetic field. If the B field vector is in the cylindrical radial direction, which only varies with the azimuthal angle φ as B = r3 cos φ mT, find the voltage Vab induced across the open terminals of the coil.Figure Rotating rectangular loop. Problem.... Get solution
16. Extracting power from a power line. Consider the possibility of putting your electromagnetics knowledge to practical use. You live out in the country and suddenly realize that a large power line passes by your farm. The power-line easement extends to 20 m on either side, so that the border of your property is at a ground distance of 20 m from the wire. You have at your disposal 200 m of number 6 gauge (4.1-mm diameter) copper wire, which can be deployed in the form of a 1-turn rectangular loop of side lengths a and b, as shown in Figure (a) Assuming the power line to carry a sinusoidal (60-Hz) current of I = 4000 A, find the maximum amount of power (in watts) that you can extract from the power line. State all assumptions. (b) Could you extract more power by using your 200-m wire in the form of a multiturn loop? Note that this is a hypothetical problem, which, like many engineering problems, may have legal ramifications. The power company may claim that it owns the spillover energy; on the other hand, you can claim that the spillover energy is an unauthorized intrusion into your private property and that the least you can do is to take advantage of it.Figure Extracting power from a power line. Problem.... Get solution
17. Induction. Two infinitely long wires carrying currents I1 and I2 cross (without electrical contact) at the origin, as shown in Figure A small rectangular loop is placed next to the wires. (a) If I1 = cos(ωt ) and I2 = sin(ωt ), determine the polarity and magnitude of the induced voltage Vind(t ). Sketch Vind(t ) together with I1(t ) and I2(t ). (b) If I1 and I2 are both constant so that I1 = I2 = I , and if we move the loop away from the wires at a constant velocity v, which direction should it be moved in order to produce the largest |Vind|? Find this value of |Vind|.Figure Induction. Problem.... Get solution
18. Induction. Consider the square loop located between two infinitely long parallel wires carrying currents I1 and I2, where the sides of the loop are equidistant from each wire, as shown in Figure (a) Starting at t = 0, the current I1 decreases exponentially over time according to I1 = I0e−t , while the current I2 stays constant at I2 = I0. Find an expression for the induced voltage Vind(t ) and sketch it as a function of time, paying particular attention to its polarity. (b) Repeat part (a) for I1 = I0 sin(ωt ). Your sketch of Vind(t ) should cover the range 0 t2π/ω.Figure Induction. Problem.... Get solution
19. Wave propagation. An electromagnetic pulse of length l traveling in the +x direction with a constant velocity vp has an electric field in the z direction, which is shown at t = 0 as a function of x. (See Figure.) An observer capable of measuring Ez is located at position x = d (where d > l ). (a) Sketch Ez as a function of x at t = (d − l)/vp , (d − l/2)/vp , (d + l/2)/vp, and (d + 3l)/vp . (b) Find Ez measured at t = (d − l/3)/vp , (d − 2l/3)/vp , and (d + l)/vp .Figure Wave propagation. Problem.... Get solution
20. Wave propagation. An observer located at y = y1 measures the magnetic field of an electromagnetic pulse propagating with a constant velocity vp in the +y direction as...Sketch Hz versus y at the following times: (a) t = 2t1 and (b) t = 4t1. Indicate the position of the observer in each sketch. Get solution
21. Interference of two waves. Two transverse electromagnetic pulses A and B propagate in the +z and −z directions, respectively, each with a velocity vp. At t = 0, the fronts of the two pulses are at a distance d from one another. The electric field waveform for pulse A is known, whereas the temporal shape of pulse B is unknown. If an observer located midway between the two waves measures a total electric field as shown in Figure sketch the electric field waveform of pulse B. Assume that the electric fields of both pulses are in the x direction.Figure Interference of two waves. Problem.... Get solution
22. Wave propagation. The electric field of an electromagnetic wave propagating in air has the shape of a Gaussian pulse, given by...in V-m−1, where vp = 3 × 108 m-s−1. An observer located at z = 1 km has a receiver that is capable of measuring a minimum electric field of 0.1 μV-m−1. (a) How long does the observer continue to detect a signal? (b) At what time does the field measured by the observer reach a maximum? What is the maximum E measured? (c) Sketch Ey (z , t ) as a function of z at t = 5 μs with the position of the observer indicated. Does the observer detect a measurable signal at this time? Get solution
23. 7.23 Phasors. Write the following real-time expressions in phasor form: (a) E(z , t ) = ŷcos(ωt − z ) V-m−1. (b) H(x, t ) = 0.1[ŷcos(ωt − 0.3x) + ẑ0.5 sin(ωt + 0.3x)] mA-m−1. (c) B(y, z , t ) = x40 sin(3 × 108t + 0.8y − 0.6z + π/4) μT. (d) E(x, y, t ) = ẑE0 sin(ax) cos(ωt + by). Get solution
24. 7.24 Phasors. Express the following phasors as real-time quantities: (a) E(y) = ẑ5e−j 40πy V-m−1. (b) B(z ) = x0.1e−j 2πz − ŷ0.3je−j 2πz μT. (c) E(x) = ẑ0.1(e−j 18x − 0.5ej 18x ) V-m−1. (d) H(x, z ) = ŷe−j 48πx ej 64πz mA-m−1. (e) J(y) = x40e−0.1y(1+j )+jπ/3 μA-m−2. Get solution
25. Displacement current in a capacitor. Consider a parallel-plate capacitor with metal plates of 1-cm2 area, each separated by mica (єr = 6) 1 mm thick. If an alternating voltage of V (t ) = 10 cos(2πft ) V is applied across the capacitor plates, find the displacement current Jd through the capacitor at (a) f = 10 kHz and (b) f = 1 MHz. Get solution
26. Propagation through lake water. The electric field of an electromagnetic wave propagating in a deep lake (for lake water, assume σ = 4 × 10−3 S-m−1, єr = 81, and μr = 1) transmitted by an electromagnetic probe submerged in the lake is approximately given by...Find the peak values of the vectors of conduction-current density Jc and displacementcurrent density Jd at (a) z = 0, (b) z = 10 m, and (c) z = 100 m. Get solution
27. Sea water. For sea water, σ = 4 S-m−1, єr = 81, and μr = 1. (a) Find the ratio of the magnitudes of the conduction-current density and the displacement-current density at 10 kHz, 1 MHz, 100 MHz, and 10 GHz. (b) Find the frequency at which the ratio is 1. Get solution
28. Dry soil. For dry soil, assume σ ≃ 10−4 S-m−1, єr ≃ 3, and μr = 1. Find the frequency at which the ratio of the magnitudes of the conduction-current and displacement-current densities is unity. Get solution
29. Maxwell’s equations. Consider an electromagnetic wave propagating in a source-free nonconducting medium represented by E and having components given by...where p1 and p2 are any two arbitrary functions and vp = 1/√(μє). Show that E satisfiesall of Maxwell’s equations, and find the components of the corresponding H. Get solution
30. Maxwell’s equations. The magnetic field in a source free (no charges or currents), lossless (i.e., σ =0), and simple non-magnetic dielectric medium (i.e., permittivity є and permeability μ0) is given as follows:...Determine all components of the electric field E(x, y, t ). Do the magnetic and electric fields constitute a propagating electromagnetic wave Get solution
31. AM radio waves. The electric-field and magnetic-field components of an AM radio signal propagating in air are given by...Find the values of β and η such that these expressions satisfy all of Maxwell’s equations. Get solution
32. Maxwell’s equations. The magnetic field phasor of an electromagnetic wave in air is given by...(a) Find the angular frequency ω of the wave such that H. satisfies all of Maxwell’s equations. (b) Find the corresponding time-harmonic electric field E. (c) Find the electric flux density D and the displacement-current density Jd. Get solution
33. Superposition of two waves. The sum of the electric fields of two time-harmonic (sinusoidal) electromagnetic waves propagating in opposite directions in air is given as...(a) Find the constant β. (b) Find the corresponding H Get solution
34. Electromagnetic wave in free space. An electromagnetic wave propagating in free space has an electric field given by...where a is a constant. Find the value of a and the corresponding expression for the magnetic field H. Get solution
35. Coaxial lines. The electric field of a transverse electromagnetic wave guided within a lossless coaxial transmission line along the z axis is expressed in cylindrical coordinates as...where β = ω √μє; μ and є are the permeability and permittivity of the dielectric material, respectively, and a and b are the inner and the outer radii of the coaxial line. (a) Find the corresponding H. (b) Write the time-domain expressions for E and H. (c) Sketch both Er and Hφ as functions of r over the range a ≤ r ≤ b at position z = 0 and time t = 0. (d) Sketch both Er and Hφ as functions of z at position r = a and time t = 0. Get solution
2. Stationary circular loop. Consider a 20-turn circular loop of wire of 15 cm diameter with its plane perpendicular to a uniform magnetic field, as shown in Figure If the magnetic flux density B is given byFigure Emf induced in a coil.......find the rms value of the induced current through the 10Ω resistor at the following time instants: (a) t = 0, (b) t = 10 ms, (c) t = 100 ms, and (d) t = 1 s. Get solution
3. Two circular coils. Two circular coils of 5 cm radius each have the same axis of symmetry and are 1 m apart from each other. Coil 1 has 10 turns and coil 2 has 100 turns. (a) If an alternating current of 100 A amplitude and 1 kHz frequency is passed through coil 1, find the amplitude of the induced voltage across the open terminals of coil 2. (b) Repeat part (a) for a frequency of 10 kHz. Get solution
4. Two concentric coils. Consider the two concentric coils shown in Figure The magnetic flux density produced at the center of the two coils due to the current I flowing in the larger coil is given byFigure Two concentric coils......(a) If the larger coil has 30 turns, find its radius. (b) If the smaller coil has 75 turns and its radius is 1 cm, find the total flux linking the smaller coil. (c) If the current I in the larger coil is given by I (t ) = 10 cos(120πt ) A, find the induced current through the resistor R assuming R = 10Ω. Get solution
5. Triangular loop and long wire. An equilateral triangular loop is situated near a long current-carrying wire, as shown in Figure The wire is part of a power line carrying 60-Hz sinusoidal current. An ac ammeter inserted in the loop reads a current of amplitude 1 mA. Assume the total resistive impedance of the loop to be 0.01Ω. Find the amplitude of the current I in the long wire.Figure Triangular loop and long wire.... Get solution
6. Faraday’s law. Consider a circular loop (radius a=5 cm) of wire lying in the x-y plane with its center at the origin, in the presence of a z -directed magnetic field...where r is the polar coordinate and B0=10 mT (milliwebers/m2). The loop wire is made up of copper (σ =5.8×107 S/m) and has a cross-sectional area of 1 mm2. It is known that the wire would melt if the total current I flowing through it exceeds 20 A (I >20 A). Stating all assumptions, calculate the maximum frequency f for which this wire can be used in the presence of this magnetic field. Get solution
7. Toroidal coil around a long, straight wire. A long, straight wire carrying an alternating current of I (t ) = 100 cos(377t ) A coincides with the principal axis of symmetry of a 200- turn coil wrapped uniformly around a rectangular, toroidal-shaped iron core of inner and outer radii a = 6 cm and b = 8 cm, thickness t = 3 cm, and relative permeability μr = 1000, as shown in Figure Calculate the induced voltage Vind between the terminals of the coil. (This type of coil, called a current transformer, is used to measure the current in a conductor wire that passes through it.)Figure Toroidal coil around long wire. Problem.7.7... Get solution
8. Current transformer. A current transformer is used to measure the current in a high-voltage transmission line, as shown in Figure The circular toroidal core has a mean diameter of 6 cm, circular cross section of 1 cm diameter, and relative permeability of μr = 200. The winding consists of N = 300 turns. If the 60-Hz current of amplitude 1000 A flows through the high-voltage line, find the rms value of the open circuit voltage induced across the terminals of the toroid.Figure A current transformer. Problem.7.8... Get solution
9. Sliding bar in a constant magnetic field. A metal bar able to move on fixed rails is initially at rest on two stationary conducting rails that are l = 50 cm apart and connected to each other via a V0 = 10 V voltage source in series with a R = 10Ω resistor, as shown in Figure A constant magnetic field of B0 = ẑ1.5 T is turned on at t = 0. Assume the rails and the metal bar to be perfectly conducting and neglect the self-inductance of the loop formed by the rails and the bar. The initial resting position of the bar is at y0 = 75 cm. (a) With no friction and assuming that the mass of the bar is 0.5 kg, calculate the magnitudes and directions of its initial acceleration and its final velocity. Explain your reasoning. (b) Describe what happens if the voltage source is turned off (i.e., V0 = 0) at t = t0, when the bar has reached a velocity v0. (c) What happens if the voltage source remains at V0 but the magnetic field is turned off (i.e., B = 0) at t = t0?Figure Sliding bar in a constant magnetic field. Problem.... Get solution
10. Oscillating bar in a time-varying magnetic field. A metal wire is in contact with two long, parallel, stationary conducting rails connected with a resistance R at one end, as shown in Figure The wire oscillates symmetrically around x = x0 with a velocity v(t ) = xv0 cos(ω0t ) in the presence of a time-varying magnetic field that is perpendicular to the plane of the rails. The magnetic field is given by B(t ) = ˆzB0 sin(ω0t ). Find the induced current through the resistance R.Figure Oscillating bar in a time-varying magnetic field. Problem.... Get solution
11. Moving square loop. A uniform magnetic field of B0 = 1 T is confined to a square-shaped area 10 cm to a side as shown in Figure with zero magnetic field everywhere else. A square loop of side length 5 cm moves through the loop with a velocity of v = 10 cm-s−1. Find the electromotive force induced in the square loop and plot its value as a function of distance x, for 0 ≤ x ≤ 15 cm. Note that the plane of the loop is orthogonal to the magnetic field.Figure Moving square loop.Problem.... Get solution
12. Rectangular loop and long wire. A rectangular loop of wire lies in the plane of a long, straight wire carrying a steady current I, as shown in Figure If the loop is moving radially away from the long wire with a velocity v = rv0 without changing its shape, find the voltage induced across the open terminals of the loop with its polarity indicated.Figure Rectangular loop and long wire. Problem.... Get solution
13. Rotating wire in a constant magnetic field. A semicircular-shaped wire of radius a is rotated in a constant magnetic field B0 at a constant angular frequency ω, as shown in Figure If the wire forms a closed loop with a resistance R, find the induced current in R. Neglect the resistance of the wires.Figure Rotating wire in constant magnetic field. Problem.... Get solution
14. Rotating rectangular loop in a constant magnetic field. A single turn rectangular coil of dimensions 5 cm × 10 cm is rotated at 1500 rpm in a uniform magnetic field of 50 mT normal to the axis of rotation, as shown in Figure Find the induced voltage across the open terminals of the loop.Figure Elementary alternating current generator.... Get solution
15. Rotating rectangular loop in a nonuniform magnetic field. Consider the rectangular coil of 10 turns with dimensions 20 cm × 10 cm, rotating at an angular velocity of 2000 rpm about its axis, as shown in Figure in the presence of a magnetic field. If the B field vector is in the cylindrical radial direction, which only varies with the azimuthal angle φ as B = r3 cos φ mT, find the voltage Vab induced across the open terminals of the coil.Figure Rotating rectangular loop. Problem.... Get solution
16. Extracting power from a power line. Consider the possibility of putting your electromagnetics knowledge to practical use. You live out in the country and suddenly realize that a large power line passes by your farm. The power-line easement extends to 20 m on either side, so that the border of your property is at a ground distance of 20 m from the wire. You have at your disposal 200 m of number 6 gauge (4.1-mm diameter) copper wire, which can be deployed in the form of a 1-turn rectangular loop of side lengths a and b, as shown in Figure (a) Assuming the power line to carry a sinusoidal (60-Hz) current of I = 4000 A, find the maximum amount of power (in watts) that you can extract from the power line. State all assumptions. (b) Could you extract more power by using your 200-m wire in the form of a multiturn loop? Note that this is a hypothetical problem, which, like many engineering problems, may have legal ramifications. The power company may claim that it owns the spillover energy; on the other hand, you can claim that the spillover energy is an unauthorized intrusion into your private property and that the least you can do is to take advantage of it.Figure Extracting power from a power line. Problem.... Get solution
17. Induction. Two infinitely long wires carrying currents I1 and I2 cross (without electrical contact) at the origin, as shown in Figure A small rectangular loop is placed next to the wires. (a) If I1 = cos(ωt ) and I2 = sin(ωt ), determine the polarity and magnitude of the induced voltage Vind(t ). Sketch Vind(t ) together with I1(t ) and I2(t ). (b) If I1 and I2 are both constant so that I1 = I2 = I , and if we move the loop away from the wires at a constant velocity v, which direction should it be moved in order to produce the largest |Vind|? Find this value of |Vind|.Figure Induction. Problem.... Get solution
18. Induction. Consider the square loop located between two infinitely long parallel wires carrying currents I1 and I2, where the sides of the loop are equidistant from each wire, as shown in Figure (a) Starting at t = 0, the current I1 decreases exponentially over time according to I1 = I0e−t , while the current I2 stays constant at I2 = I0. Find an expression for the induced voltage Vind(t ) and sketch it as a function of time, paying particular attention to its polarity. (b) Repeat part (a) for I1 = I0 sin(ωt ). Your sketch of Vind(t ) should cover the range 0 t2π/ω.Figure Induction. Problem.... Get solution
19. Wave propagation. An electromagnetic pulse of length l traveling in the +x direction with a constant velocity vp has an electric field in the z direction, which is shown at t = 0 as a function of x. (See Figure.) An observer capable of measuring Ez is located at position x = d (where d > l ). (a) Sketch Ez as a function of x at t = (d − l)/vp , (d − l/2)/vp , (d + l/2)/vp, and (d + 3l)/vp . (b) Find Ez measured at t = (d − l/3)/vp , (d − 2l/3)/vp , and (d + l)/vp .Figure Wave propagation. Problem.... Get solution
20. Wave propagation. An observer located at y = y1 measures the magnetic field of an electromagnetic pulse propagating with a constant velocity vp in the +y direction as...Sketch Hz versus y at the following times: (a) t = 2t1 and (b) t = 4t1. Indicate the position of the observer in each sketch. Get solution
21. Interference of two waves. Two transverse electromagnetic pulses A and B propagate in the +z and −z directions, respectively, each with a velocity vp. At t = 0, the fronts of the two pulses are at a distance d from one another. The electric field waveform for pulse A is known, whereas the temporal shape of pulse B is unknown. If an observer located midway between the two waves measures a total electric field as shown in Figure sketch the electric field waveform of pulse B. Assume that the electric fields of both pulses are in the x direction.Figure Interference of two waves. Problem.... Get solution
22. Wave propagation. The electric field of an electromagnetic wave propagating in air has the shape of a Gaussian pulse, given by...in V-m−1, where vp = 3 × 108 m-s−1. An observer located at z = 1 km has a receiver that is capable of measuring a minimum electric field of 0.1 μV-m−1. (a) How long does the observer continue to detect a signal? (b) At what time does the field measured by the observer reach a maximum? What is the maximum E measured? (c) Sketch Ey (z , t ) as a function of z at t = 5 μs with the position of the observer indicated. Does the observer detect a measurable signal at this time? Get solution
23. 7.23 Phasors. Write the following real-time expressions in phasor form: (a) E(z , t ) = ŷcos(ωt − z ) V-m−1. (b) H(x, t ) = 0.1[ŷcos(ωt − 0.3x) + ẑ0.5 sin(ωt + 0.3x)] mA-m−1. (c) B(y, z , t ) = x40 sin(3 × 108t + 0.8y − 0.6z + π/4) μT. (d) E(x, y, t ) = ẑE0 sin(ax) cos(ωt + by). Get solution
24. 7.24 Phasors. Express the following phasors as real-time quantities: (a) E(y) = ẑ5e−j 40πy V-m−1. (b) B(z ) = x0.1e−j 2πz − ŷ0.3je−j 2πz μT. (c) E(x) = ẑ0.1(e−j 18x − 0.5ej 18x ) V-m−1. (d) H(x, z ) = ŷe−j 48πx ej 64πz mA-m−1. (e) J(y) = x40e−0.1y(1+j )+jπ/3 μA-m−2. Get solution
25. Displacement current in a capacitor. Consider a parallel-plate capacitor with metal plates of 1-cm2 area, each separated by mica (єr = 6) 1 mm thick. If an alternating voltage of V (t ) = 10 cos(2πft ) V is applied across the capacitor plates, find the displacement current Jd through the capacitor at (a) f = 10 kHz and (b) f = 1 MHz. Get solution
26. Propagation through lake water. The electric field of an electromagnetic wave propagating in a deep lake (for lake water, assume σ = 4 × 10−3 S-m−1, єr = 81, and μr = 1) transmitted by an electromagnetic probe submerged in the lake is approximately given by...Find the peak values of the vectors of conduction-current density Jc and displacementcurrent density Jd at (a) z = 0, (b) z = 10 m, and (c) z = 100 m. Get solution
27. Sea water. For sea water, σ = 4 S-m−1, єr = 81, and μr = 1. (a) Find the ratio of the magnitudes of the conduction-current density and the displacement-current density at 10 kHz, 1 MHz, 100 MHz, and 10 GHz. (b) Find the frequency at which the ratio is 1. Get solution
28. Dry soil. For dry soil, assume σ ≃ 10−4 S-m−1, єr ≃ 3, and μr = 1. Find the frequency at which the ratio of the magnitudes of the conduction-current and displacement-current densities is unity. Get solution
29. Maxwell’s equations. Consider an electromagnetic wave propagating in a source-free nonconducting medium represented by E and having components given by...where p1 and p2 are any two arbitrary functions and vp = 1/√(μє). Show that E satisfiesall of Maxwell’s equations, and find the components of the corresponding H. Get solution
30. Maxwell’s equations. The magnetic field in a source free (no charges or currents), lossless (i.e., σ =0), and simple non-magnetic dielectric medium (i.e., permittivity є and permeability μ0) is given as follows:...Determine all components of the electric field E(x, y, t ). Do the magnetic and electric fields constitute a propagating electromagnetic wave Get solution
31. AM radio waves. The electric-field and magnetic-field components of an AM radio signal propagating in air are given by...Find the values of β and η such that these expressions satisfy all of Maxwell’s equations. Get solution
32. Maxwell’s equations. The magnetic field phasor of an electromagnetic wave in air is given by...(a) Find the angular frequency ω of the wave such that H. satisfies all of Maxwell’s equations. (b) Find the corresponding time-harmonic electric field E. (c) Find the electric flux density D and the displacement-current density Jd. Get solution
33. Superposition of two waves. The sum of the electric fields of two time-harmonic (sinusoidal) electromagnetic waves propagating in opposite directions in air is given as...(a) Find the constant β. (b) Find the corresponding H Get solution
34. Electromagnetic wave in free space. An electromagnetic wave propagating in free space has an electric field given by...where a is a constant. Find the value of a and the corresponding expression for the magnetic field H. Get solution
35. Coaxial lines. The electric field of a transverse electromagnetic wave guided within a lossless coaxial transmission line along the z axis is expressed in cylindrical coordinates as...where β = ω √μє; μ and є are the permeability and permittivity of the dielectric material, respectively, and a and b are the inner and the outer radii of the coaxial line. (a) Find the corresponding H. (b) Write the time-domain expressions for E and H. (c) Sketch both Er and Hφ as functions of r over the range a ≤ r ≤ b at position z = 0 and time t = 0. (d) Sketch both Er and Hφ as functions of z at position r = a and time t = 0. Get solution
