# Problems

# Section 5.1: Magnetic Forces and Torques

1. *5.1 An electron with a speed of 8×10 6 m/s is projected along the positive x direction into a medium containing a uniform magnetic flux density B=(x ˆ 4−z ˆ 3) T. Given that e=1.6×10−19 C and the mass of an electron is m e=9.1×10−31 kg, determine the initial acceleration vector of the electron (at the moment it is projected into the medium).

2. 5.2 When a particle with charge q and mass m is introduced into a medium with a uniform field B such that the initial velocity of the particle u is perpendicular to B (Fig. P5.2), the magnetic force exerted on the particle causes it to move in a circle of radius a. By equating F m to the centripetal force on the particle, determine a in terms of q, m, u, and B.

   # Figure P5.2 Particle of charge q projected with velocity u into a medium with a uniform field B perpendicular to u (Problem 5.2).

   Figure P5.2 Full Alternative Text

3. 5.3 The circuit shown in Fig. P5.3 uses two identical springs to support a 10 cm long horizontal wire with a mass of 20 g. In the absence of a magnetic field, the weight of the wire causes the springs to stretch a distance of 0.2 cm each. When a uniform magnetic field is turned on in the region containing the horizontal wire, the springs are observed to stretch an additional 0.5 cm each. What is the intensity of the magnetic flux density B? The force equation for a spring is F=k d, where k is the spring constant and d is the distance it has been stretched.

   # Figure P5.3 Configuration of Problem 5.3.

4. *5.4 The rectangular loop shown in Fig. P5.4 consists of 20 closely wrapped turns and is hinged along the z axis. The plane of the loop makes an angle of 30° with the y axis, and the current in the windings is 0.5 A. What is the magnitude of the torque exerted on the loop in the presence of a uniform field B=y ˆ 2.4 T? When viewed from above, is the expected direction of rotation clockwise or counterclockwise?

   # Figure P5.4 Hinged rectangular loop of Problem 5.4.

   Figure P5.4 Full Alternative Text

5. 5.5 In a cylindrical coordinate system, a 2-m long straight wire carrying a current of 5 A in the positive z direction is located at r=4 cm, ϕ=π/2, and −1 m≤z≤1 m.

   1. *a. If B=r ˆ 0.2 cos ϕ (T), what is the magnetic force acting on the wire?

   2. b.How much work is required to rotate the wire once about the z axis in the negative ϕ direction (while maintaining r=4 cm)?

   3. c.At what angle ϕ is the force a maximum?

6. 5.6 A 20-turn rectangular coil with sides l=30 cm and w=10 cm is placed in the y−z plane as shown in Fig. P5.6.

   # Figure P5.6 Rectangular loop of Problem 5.6.

   Figure P5.6 Full Alternative Text

   1. If the coil, which carries a current I=10 A, is in the presence of a magnetic flux density

      B=2×10−2(x ˆ+y ˆ 2) (T).

      Determine the torque acting on the coil.

   2. At what angle ϕ is the torque zero?

   3. At what angle ϕ is the torque maximum? Determine its value.

# Section 5.2: The Biot–Savart Law

1. *5.7 An 8 cm × 12 cm rectangular loop of wire is situated in the x−y plane with the center of the loop at the origin and its long sides parallel to the x axis. The loop has a current of 50 A flowing clockwise (when viewed from above). Determine the magnetic flux density at the center of the loop.

2. 5.8 Use the approach outlined in Example 5-2 to develop an expression for the magnetic field H at an arbitrary point P due to the linear conductor defined by the geometry shown in Fig. P5.8. If the conductor extends between z 1=3 m and z 2=7 m and carries a current I=15 A, find H at P=(2,ϕ,0).

   # Figure P5.8 Current-carrying linear conductor of Problem 5.8.

   Figure P5.8 Full Alternative Text

3. *5.9 The loop shown in Fig. P5.9 consists of radial lines and segments of circles whose centers are at point P. Determine the magnetic field H at P in terms of a, b, θ, and I.

   # Figure P5.9 Configuration of Problem 5.9.

   Figure P5.9 Full Alternative Text

4. 5.10 An infinitely long, thin conducting sheet defined over the space 0≤x≤w and −∞≤y≤∞ is carrying a current with a uniform surface current density J s=y ˆ 5 (A/m). Obtain an expression for the magnetic field at point P=(0,0,z) in Cartesian coordinates.

5. *5.11 An infinitely long wire carrying a 25 A current in the positive x direction is placed along the x axis in the vicinity of a 20-turn circular loop located in the x−y plane (Fig. P5.11). If the magnetic field at the center of the loop is zero, what is the direction and magnitude of the current flowing in the loop?

   # Figure P5.11 Circular loop next to a linear current (Problem 5.11).

6. 5.12 Two infinitely long, parallel wires are carrying 6 A currents in opposite directions. Determine the magnetic flux density at point P in Fig. P5.12.

   # Figure P5.12 Arrangement for Problem 5.12.

7. *5.13 A long, East–West-oriented power cable carrying an unknown current I is at a height of 8 m above the Earth’s surface. If the magnetic flux density recorded by a magnetic-field meter placed at the surface is 15 μT when the current is flowing through the cable and 20 μT when the current is zero, what is the magnitude of I?

8. 5.14 Two parallel, circular loops carrying a current of 40 A each are arranged as shown in Fig. P5.14. The first loop is situated in the x−y plane with its center at the origin, and the second loop’s center is at z=2 m. If the two loops have the same radius a=3 m, determine the magnetic field at

   1. z=0

   2. z=1 m

   3. z=2 m

   # Figure P5.14 Parallel circular loops of Problem 5.14.

   Figure P5.14 Full Alternative Text

9. 5.15 A circular loop of radius a carrying current I 1 is located in the x−y plane as shown in Fig. P5.15. In addition, an infinitely long wire carrying current I 2 in a direction parallel with the z axis is located at y=y 0.

   1. Determine H at P=(0,0,h).

   2. Evaluate H for a=3 cm, y 0=10 cm, h=4 cm, I 1=10 A, and I 2=20 A.

   # Figure P5.15 Problem 5.15.

   Figure P5.15 Full Alternative Text

10. *5.16 The long, straight conductor shown in Fig. P5.16 lies in the plane of the rectangular loop at a distance d=0.1 m. The loop has dimensions a=0.2 m and b=0.5 m, and the currents are I 1=20 A and I 2=30 A. Determine the net magnetic force acting on the loop.

    # Figure P5.16 Current loop next to a conducting wire (Problem 5.16).

    Figure P5.16 Full Alternative Text

11. 5.17 In the arrangement shown in Fig. P5.17, each of the two long, parallel conductors carries a current I, is supported by 8-cm long strings, and has a mass per unit length of 1.2 g/cm. Due to the repulsive force acting on the conductors, the angle θ between the supporting strings is 10°. Determine the magnitude of I and the relative directions of the currents in the two conductors.

    # Figure P5.17 Parallel conductors supported by strings (Problem 5.17).

    Figure P5.17 Full Alternative Text

12. 5.18 An infinitely long, thin conducting sheet of width w along the x direction lies in the x−y plane and carries a current I in the −y direction. Determine the following:

    1. *a. The magnetic field at a point P midway between the edges of the sheet and at a height h above it (Fig. P5.18).

       # Figure P5.18 A linear current source above a current sheet (Problem 5.18).

       Figure P5.18 Full Alternative Text

    2. b.The force per unit length exerted on an infinitely long wire passing through point P and parallel to the sheet if the current through the wire is equal in magnitude but opposite in direction to that carried by the sheet.

13. 5.19 Three long, parallel wires are arranged as shown in Fig. P5.19. Determine the force per unit length acting on the wire carrying I 3.

    # Figure P5.19 Three parallel wires of Problem 5.19.

    Figure P5.19 Full Alternative Text

14. *5.20 A square loop placed as shown in Fig. P5.20 has 2 m sides and carries a current I 1=5 A. If a straight, long conductor carrying a current I 2=10 A is introduced and placed just above the midpoints of two of the loop’s sides, determine the net force acting on the loop.

    # Figure P5.20 Long wire carrying current I 2, just above a square loop carrying I 1 (Problem 5.20).

    Figure P5.20 Full Alternative Text

# Section 5.3: Maxwell’s Magnetostatic Equations

1. 5.21 Current I flows along the positive z direction in the inner conductor of a long coaxial cable and returns through the outer conductor. The inner conductor has radius a, and the inner and outer radii of the outer conductor are b and c, respectively.

   1. Determine the magnetic field in each of the following regions: 0≤r≤a, a≤r≤b, b≤r≤c, and r≥c.

   2. Plot the magnitude of H as a function of r over the range from r=0 to r=10 cm given that I=10 A, a=2 cm, b=4 cm, and c=5 cm.

2. 5.22 A long cylindrical conductor whose axis is coincident with the z axis has a radius a and carries a current characterized by a current density J=z ˆ J 0/r, where J 0 is a constant and r is the radial distance from the cylinder’s axis. Obtain an expression for the magnetic field H for

   1. 0≤r≤a

   2. r>a

3. 5.23 Repeat Problem 5.22 for a current density J=z ˆ J 0 e−r.

4. *5.24 In a certain conducting region, the magnetic field is given in cylindrical coordinates by

   H=ϕ ˆ 4 r[1−(1+3 r)e−3 r].

   Find the current density J.

5. 5.25 A cylindrical conductor whose axis is coincident with the z axis has an internal magnetic field given by

   H=ϕ ˆ 2 r[1−(4 r+1)e−4 r] (A/m) for r≤a,

   where a is the conductor’s radius. If a=5 cm, what is the total current flowing in the conductor?

# Section 5.4: Vector Magnetic Potential

1. 5.26 With reference to Fig. 5-10:

   1. *a. Derive an expression for the vector magnetic potential A at a point P located at a distance r from the wire in the x−y plane.

   2. b.Derive B from A. Show that your result is identical with the expression given by Eq. (5.29), which was derived by applying the Biot–Savart law.

2. 5.27 In a given region of space, the vector magnetic potential is given by A=x ˆ 5 cos π y+z ˆ(2+sin π x) (Wb/m).

   1. *a. Determine B.

   2. b.Use Eq. (5.66) to calculate the magnetic flux passing through a square loop with 0.25-m long edges if the loop is in the x−y plane, its center is at the origin, and its edges are parallel to the x and y axes.

   3. c.Calculate Φ again using Eq. (5.67).

3. 5.28 A uniform current density given by

   J=z ˆ J 0 (A/m 2)

   gives rise to a vector magnetic potential

   A=−z ˆ μ 0 J 0 4(x 2+y 2) (Wb/m).

   1. Apply the vector Poisson’s equation to confirm the above statement.

   2. Use the expression for A to find H.

   3. Use the expression for J in conjunction with Ampère’s law to find H. Compare your result with that obtained in part (b).

4. *5.29 A thin current element extending between z=−L/2 and z=L/2 carries a current I along +z ˆ through a circular cross-section of radius a.

   1. Find A at a point P located very far from the origin (assume R is so much larger than L that point P may be considered to be at approximately the same distance from every point along the current element).

   2. Determine the corresponding H.

# Section 5.5: Magnetic Properties of Materials

1. 5.30 In the model of the hydrogen atom proposed by Bohr in 1913, the electron moves around the nucleus at a speed of 2×10 6 m/s in a circular orbit of radius 5×10−11 m. What is the magnitude of the magnetic moment generated by the electron’s motion?

2. *5.31 Iron contains 8.5×10 28 atoms/m 3. At saturation, the alignment of the electrons’ spin magnetic moments in iron can contribute 1.5 T to the total magnetic flux density B. If the spin magnetic moment of a single electron is 9.27×10−24 (A⋅m 2), how many electrons per atom contribute to the saturated field?

# Section 5.6: Magnetic Boundary Conditions

1. 5.32 The x−y plane separates two magnetic media with magnetic permeabilities μ 1 and μ 2 (Fig. P5.32). If there is no surface current at the interface and the magnetic field in medium 1 is

   H 1=x ˆ H 1 x+y ˆ H 1 y+z ˆ H 1 z,

   find:

   1. H 2

   2. θ 1 and θ 2

   3. Evaluate H 2, θ 1, and θ 2 for H 1 x=2 (A/m), H 1 y=0, H 1 z=4 u (A/m), μ 1=μ 0, and μ 2=4 μ 0

   # Figure P5.32 Adjacent magnetic media (Problem 5.32).

   Figure P5.32 Full Alternative Text

2. *5.33 Given that a current sheet with surface current density J s=x ˆ 8 (A/m) exists at y=0, the interface between two magnetic media, and H 1=z ˆ 11 (A/m) in medium 1 (y>0), determine H 2 in medium 2 (y<0).

3. 5.34 In Fig P5.34, the plane defined by x−y=1 separates medium 1 of permeability μ 1 from medium 2 of permeability μ 2. If no surface current exists on the boundary and

   B 1=x ˆ 2+y ˆ 3 (T),

   find B 2 and then evaluate your result for μ 1=5 μ 2. Hint: Start by deriving the equation for the unit vector normal to the given plane.

   # Figure P5.34 Magnetic media separated by the plane x−y=1 (Problem 5.34).

   Figure P5.34 Full Alternative Text

4. *5.35 The plane boundary defined by z=0 separates air from a block of iron. If B 1=x ˆ 4−y ˆ 6+z ˆ 8 in air (z≥0), find B 2 in iron (z≤0) given that μ=5000 μ 0 for iron.

5. 5.36 Show that, if no surface current densities exist at the parallel interfaces shown in Fig. P5.36, the relationship between θ 4 and θ 1 is independent of μ 2.

   # Figure P5.36 Three magnetic media with parallel interfaces (Problem 5.36).

   Figure P5.36 Full Alternative Text

# Sections 5.7 and 5.8: Inductance and Magnetic Energy

1. *5.37 Obtain an expression for the self-inductance per unit length for the parallel wire transmission line of Fig. 5-28(a) in terms of a, d, and μ, where a is the radius of the wires, d is the axis-to-axis distance between the wires, and μ is the permeability of the medium in which they reside.

2. 5.38 A solenoid with a length of 20 cm and a radius of 5 cm consists of 400 turns and carries a current of 12 A. If z=0 represents the midpoint of the solenoid, generate a plot for |H(z)| as a function of z along the axis of the solenoid for the range −20 cm≤z≤20 cm in 1 cm steps.

3. 5.39 In terms of the dc current I, how much magnetic energy is stored in the insulating medium of a 3-m long, air-filled section of a coaxial transmission line given that the radius of the inner conductor is 5 cm and the inner radius of the outer conductor is 10 cm?

4. *5.40 Determine the mutual inductance between the circular loop and the linear current shown in Fig. P5.40.

   # Figure P5.40 Linear conductor with current I 1 next to a circular loop of radius a at distance d (Problem 5.40).

   Figure P5.40 Full Alternative Text

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