# Problems

# Sections 6.1 to 6.6: Faraday’s Law and its Applications

1. *6.1 The switch in the bottom loop of Fig. P6.1 is closed at t=0 and then opened at a later time t 1. What is the direction of the current I in the top loop (clockwise or counterclockwise) at each of these two times?

   # Figure P6.1 Loops of Problem 6.1.

   Figure P6.1 Full Alternative Text

2. 6.2 The loop in Fig. P6.2 is in the x−y plane and B=z ˆ B 0 sin ω t with B 0 positive. What is the direction of I (ϕ^ or −ϕ^) at:

   1. t=0

   2. ω t=π/4

   3. ω t=π/2

   # Figure P6.2 Loop of Problem 6.2.

3. 6.3 A coil consists of 100 turns of wire wrapped around a square frame of sides 0.25 m. The coil is centered at the origin with each of its sides parallel to the x or y axis. Find the induced emf across the open-circuited ends of the coil if the magnetic field is given by

   1. *a. B=z ˆ 20 e−3 t (T)

   2. b.B=z ˆ 20 cos x cos 10 3 t (T)

   3. c.B=z ˆ 20 cos x sin 2 y cos 10 3 t (T)

4. 6.4 A stationary conducting loop with an internal resistance of 0.5 Ω is placed in a time-varying magnetic field. When the loop is closed, a current of 5 A flows through it. What will the current be if the loop is opened to create a small gap and a 2 Ω resistor is connected across its open ends?

5. *6.5 A circular-loop TV antenna with 0.02 m 2 area is in the presence of a uniform-amplitude 300 Mhz signal. When oriented for maximum response, the loop develops an emf with a peak value of 30 (mV). What is the peak magnitude of B of the incident wave?

6. 6.6 The square loop shown in Fig. P6.6 is coplanar with a long, straight wire carrying a current

   I(t)=5 cos(2 π×10 4 t) (A).

   1. Determine the emf induced across a small gap created in the loop.

   2. Determine the direction and magnitude of the current that would flow through a 4 Ω resistor connected across the gap. The loop has an internal resistance of 1 Ω.

   # Figure P6.6 Loop coplanar with long wire (Problem 6.6).

   Figure P6.6 Full Alternative Text

7. *6.7 The rectangular conducting loop shown in Fig. P6.7 rotates at 6,000 revolutions per minute in a uniform magnetic flux density given by

   B=y ˆ 50 (m T).

   # Figure P6.7 Rotating loop in a magnetic field (Problem 6.7).

   Figure P6.7 Full Alternative Text

   Determine the current induced in the loop if its internal resistance is 0.5 Ω.

8. 6.8 The transformer shown in Fig. P6.8 consists of a long wire coincident with the z axis carrying a current I=I 0 cos ω t, coupling magnetic energy to a toroidal coil situated in the x−y plane and centered at the origin. The toroidal core uses iron material with relative permeability μ r, around which 100 turns of a tightly wound coil serves to induce a voltage V e m f, as shown in the figure.

   1. Develop an expression for V e m f.

   2. Calculate V e m f for f=60 Hz, μ r=4000, a=5 cm, b=6 cm, c=2 cm, and I 0=50 A.

   # Figure P6.8 Problem 6.8.

   Figure P6.8 Full Alternative Text

9. 6.9 A rectangular conducting loop 5 cm×10 cm with a small air gap in one of its sides is spinning at 7200 revolutions per minute. If the field B is normal to the loop axis and its magnitude is 6×10−6 T, what is the peak voltage induced across the air gap?

10. *6.10 A 50-cm long metal rod rotates about the z axis at 90 revolutions per minute with end 1 fixed at the origin as shown in Fig. P6.10. Determine the induced emf V 12 if B=z ˆ 2×10−4 T.

    # Figure P6.10 Rotating rod of Problem 6.10.

11. 6.11 The loop shown in Fig. P6.11 moves away from a wire carrying a current I 1=10 A at a constant velocity u=y ˆ 7.5 (m/s). If R=10 Ω and the direction of I 2 is as defined in the figure, find I 2 as a function of y 0, which is the distance between the wire and the loop. Ignore the internal resistance of the loop.

    # Figure P6.11 Moving loop of Problem 6.11.

    Figure P6.11 Full Alternative Text

12. *6.12 The electromagnetic generator shown in Fig. 6-12 is connected to an electric bulb with a resistance of 150 Ω. If the loop area is 0.1 m 2 and it rotates at 3,600 revolutions per minute in a uniform magnetic flux density B 0=0.4 T, determine the amplitude of the current generated in the light bulb.

13. 6.13 The circular, conducting disk shown in Fig. P6.13 lies in the x−y plane and rotates with uniform angular velocity ω about the z axis. The disk is of radius a and is present in a uniform magnetic flux density B=z ˆ B 0. Obtain an expression for the emf induced at the rim relative to the center of the disk.

    # Figure P6.13 Rotating circular disk in a magnetic field (Problem 6.13).

# Section 6.7: Displacement Current

1. 6.14 The plates of a parallel-plate capacitor have areas of 10 c m 2 each and are separated by 2 cm. The capacitor is filled with a dielectric material with ε=4 ε 0, and the voltage across it is given by V(t)=30 cos 2 π×10 6 t (V). Find the displacement current.

2. *6.15 A coaxial capacitor of length l=6 cm uses an insulating dielectric material with ε r=9. The radii of the cylindrical conductors are 0.5 cm and 1 cm. If the voltage applied across the capacitor is

   V(t)=50 sin(120 π t) (V),

   what is the displacement current?

3. 6.16 The parallel-plate capacitor shown in Fig. P6.16 is filled with a lossy dielectric material of relative permittivity ε r and conductivity σ. The separation between the plates is d and each plate is of area A. The capacitor is connected to a time-varying voltage source V(t).

   1. Obtain an expression for I c, the conduction current flowing between the plates inside the capacitor, in terms of the given quantities.

   2. Obtain an expression for I d, the displacement current flowing inside the capacitor.

   3. Based on your expressions for parts (a) and (b), give an equivalent-circuit representation for the capacitor.

   4. Evaluate the values of the circuit elements for A=4 cm 2, d=0.5 cm, ε r=4, σ=2.5 (S/m), and

      V(t)=10 cos(3 π×10 3 t)(V).

   # Figure P6.16 Parallel-plate capacitor containing a lossy dielectric material (Problem 6.16).

   Figure P6.16 Full Alternative Text

4. *6.17 In wet soil characterized by σ=10−2 (S/m), μ r=1, and ε r=36, at what frequency is the conduction current density equal in magnitude to the displacement current density?

5. 6.18 An electromagnetic wave propagating in seawater has an electric field with a time variation given by E=z ˆ E 0 cos ω t. If the permittivity of water is 81 ε 0 and its conductivity is 4 (S/m), find the ratio of the magnitudes of the conduction current density to displacement current density at each of the following frequencies:

   1. a.1 kHz

   2. *b. 1 MHz

   3. c.1 GHz

   4. d.100 GHz

# Sections 6.9 and 6.10: Continuity Equation and Charge Dissipation

1. 6.19 At t=0, charge density ρ v 0 was introduced into the interior of a material with a relative permittivity ε r=9. If at t=1 μ s the charge density has dissipated down to 10−3 ρ v 0, what is the conductivity of the material?

2. *6.20 If the current density in a conducting medium is given by

   J(x,y,z;t)=(x ˆ z−y ˆ 4 y 2+z ˆ 2 x)cos ω t,

   determine the corresponding charge distribution ρ v(x,y,z;t).

3. 6.21 In a certain medium, the direction of current density J points in the radial direction in cylindrical coordinates, and its magnitude is independent of both ϕ and z. Determine J given that the charge density in the medium is

   ρ v=ρ 0 r cos ω t (C/m 3).

4. 6.22 If we were to characterize how good a material is as an insulator by its resistance to dissipating charge, which of the following two materials is the better insulator?

   Dry Soil:ε r=2.5 σ=10−4 (S/m)Fresh Water:ε r=80 σ=10−3 (S/m)

# Sections 6.11: Electromagnetic Potentials

1. 6.23 The electric field of an electromagnetic wave propagating in air is given by

   E(z,t)=x ˆ 4 cos(6×10 8 t−2 z) +y ˆ 3 sin(6×10 8 t−2 z) (V/m).

   Find the associated magnetic field H(z,t).

2. *6.24 The magnetic field in a dielectric material with ε=4 ε 0, μ=μ 0, and σ=0 is given by

   H(y,t)=x ˆ 5 cos(2 π×10 7 t+k y) (A/m).

   Find k and the associated electric field E.

3. 6.25 Given an electric field

   E=x ˆ E 0 sin a y cos(ω t−k z),

   where E 0, a, ω, and k are constants, find H.

4. *6.26 The electric field radiated by a short dipole antenna is given in spherical coordinates by

   E(R,θ;t)=θ ˆ 2×10−2 R sin θ cos(6 π×10 8 t−2 π R)(V/m).

   Find H(R,θ;t).

5. 6.27 A Hertzian dipole is a short conducting wire carrying an approximately constant current over its length l. If such a dipole is placed along the z axis with its midpoint at the origin and if the current flowing through it is i(t)=I 0 cos ω t, find the following:

   1. The retarded vector potential A˜(R,θ,ϕ) at an observation point Q(R,θ,ϕ) in a spherical coordinate system.

   2. The magnetic field phasor H˜(R,θ,ϕ).

   Assume l to be sufficiently small so that the observation point is approximately equidistant to all points on the dipole; that is, assume R′≈R.

6. 6.28 In free space, the magnetic field is given by

   H=ϕ^36 r cos(6×10 9 t−k z) (m A/m).

   1. *a. Determine k.

   2. b.Determine E.

   3. c.Determine J d.

7. 6.29 The magnetic field in a given dielectric medium is given by

   H=y ˆ 6 cos 2 z sin(2×10 7 t−0.1 x) (A/m),

   where x and z are in meters.

   1. Determine E.

   2. Determine the displacement current density J d.

   3. Determine the charge density ρ v.

​Test Your Understanding of Chapter 6:
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