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engineering
elements of electromagnetics

Elements Of Electromagnetics 7th Edition Matthew Sadiku - Solutions

A conductor located at 0 < y < 1.6 m moves with velocity 2ax m/s in a magnetic field, B =10 cos βyaz Wb/m2 where b is a constant. Determine the induced voltage.
A circuit conducting loop lies in the xy-plane as shown in Figure 9.21. The loop has a radius of 0.2 m and resistance R = 4Ω. If B = 40 sin 104 taz mWb/m2, find the currrent. X R
The concept of displacement current was a major contribution attributed to(a) Faraday(b) Lenz(c) Maxwell(d) Lorenz(e) Your professor
Two conducting coils 1 and 2 (identical except that 2 is split) are placed in a uniform magnetic field that decreases at a constant rate as in Figure 9.18. If the plane of the coils is perpendicular to the field lines, which of the following statements is true?(a) An emf is induced in both
Assuming that each loop is stationary and the time-varying magnetic field B induces current I, which of the configurations in Figure 9.17 are incorrect? Increasing B (a) Decreasing B (c) (b) Decreasing B Increasing B (d)
A conducting rod has one end grounded at the origin, while the other end is free to move in the z = 0 plane. The rod rotates at 30 rad/s in a static magnetic field B = 60az mWb/m2. If the rod is 8 cm long, find the voltage induced in the rod.
A rectangular coil has a cross-sectional area of 30 cm2 and 50 turns. If the coil rotates at 60 rad/s in a magnetic field of 0.2 Wb/m2 such that its axis of rotation is perpendicular to the direction of the field, determine the induced emf in the coil.
If Es = 10 ej4x ay, which of these is not a correct representation of E?(a) Re (Esejωt)(b) Re (Ese–jωt)(c) Im(Esejωt)(d) 10 cos (ωt + j4x) ay(e) 10 sin1vt 1 4x2 ay
An electromagnetic relay is modeled as shown in Figure 8.45. What force is on the armature (moving part) of the relay if the flux in the air gap is 2 mWb? The area of the gap is 0.3 cm2, and its length 1.5 mm. D 177 N
A loop is rotating about the y-axis in a magnetic field B = Bo sin ωt ax Wb/m2. The voltage induced in the loop is due to(a) Motional emf(b) Transformer emf(c) A combination of motional and transformer emf(d) None of the above
(a) If the cross section of the toroid of Figure 7.15 is a square of side a, show that the self-inductance of the toroid is(b) If the toroid has a circular cross section as in Figure 7.15, show thatwhereρo >> a. L= = MoN²a 2πT In 2p. + a 2p. a
A loop resides outside the region between two parallel long wires carrying currents in opposite directions as shown in Figure 8.41. Find the total flux linking the loop. I b W h
The flux through each turn of a 100-turn coil is (t3 – 2t) mWb, where t is in seconds. The induced emf at t = 2 s is(a) 1 V(b) –1 V(c) 4 mV(d) 0.4 V(e) –0.4 V
An air gap in an electric machine has length 4.4 mm and area 4.82 x 10–2 m2. Find the reluctance of the gap.
A coaxial cable consists of an inner conductor of radius 1.2 cm and an outer conductor of radius 1.8 cm. The two conductors are separated by an insulating medium (μ = 4 μo). If the cable is 3 m long and carries 25 mA current, calculate the energy stored in the medium.
Medium 1 is free space and is defined by r < a, while medium 2 is a magnetic material with permeability μ2 and defined by r > a. The magnetic flux densities in the media are:Find μ2. 0.8a³ B₁ = Bu[(1 + 1.6²) cos 0a, - (1 - 0,84² )sin 0a, p³ p3 B₂= Bo₂(cosa, - sin @a,)
The magnetic field in a material space (μ = 15μo) is given by B = 4ax + 12ay mWb/m2 Calculate the energy stored in region 0 < x < 2, 0 < y < 3, 0 < z < 4.
A coaxial cable has an internal inductance that is twice the external inductance. If the inner radius is 6.5 mm, calculate the outer radius.
A wire of radius 2 mm is 40 m long. Calculate its inductance. Assume μ = μo.
An air-filled toroid of square cross section has inner radius 3 cm, outer radius 5 cm, and height 2 cm. How many turns are required to produce an inductance of 45 μH?
The plane z = 0 separates air (z ≥ 0, μ = μo) from iron (z ≤ 0, μ = 200μo). Given that H = 10ax + 15ay – 3az A/m in air, find B in iron and the angle it makes with the interface.
Inside a right circular cylinder, m15 800 mo, while the exterior is free space. Given that B1 = μo (22aρ + 45aΦ) Wb/m2, determine B2 just outside the cylinder
If μ1 = 2μo for region 1 (0 < Φ < π) and μ2 = 5μo for region 2 (π < Φ < 2π) and B2 = 10aρ + 15aΦ – 20az mWb/m2. Calculate (a) B1, (b) The energy densities in the two media.
Suppose space is divided into region 1 (y < 0, μ1 = μoμr1) and region 2 (y < 0, μ1 = μoμr2). If H1 = αax + βay + δaz A/m, find H2.
A current sheet with K =12ay A/m is placed at x = 0, which separates region 1 x < 0, μ = 2μo and region 2, x > 0, μ = 4μo. If H1 = 10ax + 6az, A/m, find H2.
In region x < 0 < μ = μo, a uniform magnetic field makes angle 42° with the normal to the interface. Calculate the angle the field makes with the normal in region x > 0, μ = 6.5μo.
In medium 1 (z < 0) μ1 = 5μo, while in medium 2 (z < 0) μ2 = 2μo. If B1 = 4ax –10ay + 12az mWb/m2, find B2 and the energy density in medium 2.
Region 1, for which μ1 = 2.5μo, is defined by z < 0, while region 2, for which μ2 = 4μo, is defined by z > 0. If B1 = 6ax – 4.2ay + 1.8az mWb/m2, find H2 and the angle H2 makes with the interface.
An electromagnet is made of a ferromagnetic material whose magnetization curve can be approximated by B(H) = BoH/(Ho + H) mWb/m2 where Bo = 2 Wb/m2 and Ho = 100 A/m Find μr when H = 250 A/m.
For a linear, isotropic, and homogeneous magnetic medium, show that M = Xm μ(1 + Xm) -B.
A triangular loop is placed in the x-z plane, as shown in Figure 8.39. Assume that a dc current I = 2 A flows in the loop and that B = 30az m Wb/m exists in the region. Find the forces and torque on the loop. 5 AB I 5 X
In a ferromagnetic material (μ = 80μo), B = 20xay mWb/m2. Determine: (a) μr, (b) Xm, (c) H, (d) M, (e) Jb.
In a magnetic material, with Xm = 6.5, the magnetization is M = 24y2az A/m. Find μr, H, and J at y = 2 cm.
High-current circuit breakers typically consist of coils that generate a magnetic field to blow out the arc formed when the contacts open. An arc 30 mm long carries a current of 520 A in a direction perpendicular to a magnetic flux density of 0.4 mWb/m2. Determine the magnetic force on the arc.
A loop with 50 turns and surface area of 12 cm2 carries a current of 4 A. If the loop rotates in a uniform magnetic field of 100 mWb/m2, find the torque exerted on the loop.
A current sheet with K = 10ax A/m lies in free space in the z = 2 m plane. A filamentary conductor on the x-axis carries a current of 2.5 A in the ax-direction. Determine the force per unit length on the conductor.
The earth has a magnetic moment of about 8 x 1022 A · m2 and its radius is 6370 km. Imagine that there is a loop around the equator and determine how much current in the loop would result in the same magnetic moment.
A 60-turn coil carries a current of 2 A and lies in the plane x + 2y – 5z = 12 such that the magnetic moment m of the coil is directed away from the origin. Calculate m, assuming that the area of the coil is 8 cm2.
The magnetic field in a certain region is B = 40 ax mWb/m2. A conductor that is 2 m in length lies in the z-axis and carries a current of 5 A in the az-direction. Calculate the force on the conductor.
Two infinitely long parallel wires are separated by a distance of 20 cm. If the wires carry current of 10 A in opposite directions, calculate the force on the wires.
A multilayer coil of 2000 turns of fine wire is 20 mm long and has a thickness 5 mm of winding. If the coil carries a current of 5 mA, the mmf generated is(a) 10 A · t(b) 500 A · t (c) 2000 A · t(d) None of the above
The force on differential length dl at point P in the conducting circular loop in Figure 8.33 is(a) Outward along OP(b) Inward along OP(c) In the direction of the magnetic field(d) Tangential to the loop at P Ов 0 p² Po dl
Determine |B| that will produce the same force on a charged particle moving at 140 m/s that an electric field of 12 kV/m produces.
Each of the following pairs consists of an electric circuit term and the corresponding magnetic circuit term. Which pairs are not corresponding?(a) V and F(b) G and P(c) ε and μ(d) ∑ I = 0 and ∑ Ψ = 0
The resultant force on the circular loop in Figure 8.33 has the magnitude of(a) 2πρoIB(b) πρ2oIB (c) 2ρoIB(d) Zero ОВ 0 P Р Po I dl
A straight conductor 0.2 m long carries a current 4.5 A along ax. If the conductor lies in the magnetic field B = 2.5(ay + az) mWb/m2, calculate the force on the conductor.
Which of these formulas is wrong?(a) B1n = B2n(b) B2 = √B22n + B22t(c) H1 = H1n + H1f(d) an21 x (H1 – H2) = K, where an21 is a unit vector normal to the interface and directed from region 2 to region 1.
Two large conducting plates are 8 cm apart and have a potential difference 12 kV. A drop of oil with mass 0.4 g is suspended in space between the plates. Find the charge on the drop.
The magnetic field intensity in a certain conducting medium is H = xy2ax + x2 zay – y2zaz A/m(a) Calculate the current density at point P (2, –1, 3).(b) opv What is at P? at
Which of these materials requires the lowest value of magnetic field strength to magnetize it?(a) Nickel (b) Silver (c) Tungsten(d) Sodium chloride
What is the unit of magnetic charge?(a) Ampere-meter squared(b) Coulomb (c) Ampere(d) Ampere-meter
An electron (m = 9.11 X 10–31kg) moves in a circular orbit of radius 0.4 X 10–10 m with an angular velocity of 2 X 1016 rad/s. Find the centripetal force required to hold the electron.
The magnetic vector potential at a distant point from a small circular loop is given by where Ao is a constant. Determine the magnetic flux density B. A -sir -sin @a, Wb/m
An infinitely long conductor of radius a carries a uniform current with J = Jo az. Show that the magnetic vector potential for r < a is A = 1 Hop²a=
Two thin parallel wires carry currents along the same direction. The force experienced by one due to the other is(a) Parallel to the lines(b) Perpendicular to the lines and attractive(c) Perpendicular to the lines and repulsive(d) Zero
In free space, a small circular loop of current produceswhere k is a constant. Find B. k A = sina
Find the B field corresponding to the magnetic vector potentil π.Χ. пу A = sin cos a₂ 2 2
A 4 mC charge has velocity u = 1.4ax – 3.2ay – az m/s at point P(2, 5, –3) in the presence of E = 2xyzax + x2 zay + x2yaz V/m and B = y2ax + z2ay + x2az Wb/m2. Find the force on the charge at P.
Which of the following statements are not true about electric force Fe and magnetic force Fm on a charged particle?(a) E and Fe are parallel to each other, whereas B and Fm are perpendicular to each other.(b) Both Fe and Fm depend on the velocity of the charged particle.(c) Both Fe and Fm can
Let A = 10ρ2az mWb/m.(a) Find H and J.(b) Determine the total current crossing the surface z = 1, 0 ≤ ρ ≤ 2, ≤ Φ ≤ 2π.
Given that A = 10 / r sinθaΦWb/m, find H at point (4, 60o, 30o).
Given that Wb/m exists in free space.(a) Show that = ∇ · A = 0(b) Find B at point T(1, 30°, 60°) A= = 2cose + sine
In free space, A = 10 sin π yax + (4 + cos π x)az Wb/m. Find H and J.
In free space, B = 20 / ρ sin2 Φaz Wb/m2. Determine the magnetic flux crossing the strip z = 0, 1 < ρ < 2 m, 0 < Φ < π/4.
If B = 2 / r3 cos θar + 1/ r3 sin θaθ Wb/m2, find the magnetic flux through the spherical cap r = 1,θ < π/3.
An electron beam forms a current of density(a) Determine the total current.(b) Find the magnetic field intensity everywhere. J= {J(1-par Τα P p>α p
A cylindrical conductor of radius a = 1 cm carries current I which produces H = 4ρaΦ A/m. Find I.
Assume a conductor, H = 103ρ2aΦA/m. (a) Find J. (b) Calculate the current through the surface 0 < ρ 2, 0 < Φ < 2π, z = 0.
Let H = y2ax + x2ay A/m. Find J at (1, –4, 7).
Let H = ko (ρ / a)aΦ, ρ < a, where ko is a constant. (a) Find J for ρ < a. (b) Find H for ρ > a.
An infinitely long cylindrical conductor of radius a is placed along the z-axis. If the current density is J = Jo / az, where Jo is constant, find H everywhere.
A conducting cylinder of radius a carries current I along + az.(a) Use Ampère’s law to find H for ρ < a and ρ > a. (b) Find J.
The z = 0 plane carries current K = 10ax A/m, while current filament situated at y = 0, z = 6 carries current I along ax. Find I such that H(0, 0, 3) = 0.
Two identical loops are parallel and separated by distance d as shown in Figure 7.35. Calculate H at (0, 0, d) assuming that a = 3 cm, d = 4 cm, and I = 10 A.
A conducting filament carries current I from point A(0, 0, a) to point B(0, 0, b). Show that at point P(x, y, 0), H I b a = √² + y²[√x + 47√x² + y²³√x² + y² + b² √√√x² + y² +₁ ] ² a
For the currents and closed paths of Figure 7.26, calculate the value of f, H-dl. L
The z-axis carries filamentary current of 10π A along az. Which of these is incorrect?(a) H = –ax A/m at (0, 5, 0)(b) H = aФA/m at 15, π/4, 02(c) H = –0.8ax – 0.6ay at (–3, 4, 0)(d) H = –aФ at (5, 3π/2, 0)
Which one of these equations is not Maxwell’s equation for a static electromagnetic field in a linear homogeneous medium?(a) ∇ · B = 0(b) ∇ x 3 D = 0(c) (d) (e) fL B-dl= μl
Plane y = 0 carries a uniform current of 30az mA/m. At (1, 10, –2), the magnetic field intensity is(a) –15ax mA/m(b) 15ax mA/m (c) 477.5ay mA/m(d) 18.85ay μA/m(e) None of the above
Two bar magnets with their north poles having strength Qm1 = A · m and Qm2 = 10 A · m (magnetic charges) are placed inside a volume as shown in Figure 7.27. The magnetic flux leaving the volume is(a) 200 Wb(b) 30 Wb(c) 10 Wb(d) 0 Wb(e) –10 Wb N S 1 2 S N -Volume
Which of these statements is not characteristic of a static magnetic field?(a) It is solenoidal.(b) It is conservative.(c) It has no sinks or sources.(d) Magnetic flux lines are always closed.(e) The total number of flux lines entering a given region is equal to the total number of flux lines
Two identical coaxial circular coils carry the same current I but in opposite directions. The magnitude of the magnetic field B at a point on the axis midway between the coils is (a) Zero(b) The same as that produced by one coil(c) Twice that produced by one coil(d) Half that produced by one
Two current elements I1dl1 = 4 x 10–5 ax A.m at (0, 0, 0) and I2dl2 = 6 x 10–5 ay A.m at (0, 0, 1) are in free space. Find H at (3, 1, –2).
Determine whether each of the following potentials satisfies Laplace’s equation.(a) V1 = 3xyz + y – z2(b) V2 = 10sinΦ / ρ(c) V3 = 5sinΦ / r
Given V = x3y + yz + cz2, find c such that V satifies Laplace’s equation.
Two conducting coaxial cylinders are located at ρ = 1 cm and ρ = 1.5 cm. The inner conductor is maintained at 50 V while the outer one is grounded. If the cylinders are separated by a dielectric material with ε = 4εo, find the surface charge density on the inner conductor.
In cylindrical coordinates, V = 50 V on plane Φ = π/2 and V = 0 on plane Φ = 0. Assuming that the planes are insulated along the z-axis, determine E between the planes.
(a) Show that V = Vo(1 – a2/ρ2) ρ sin Φ (where Vo is constant) satisfies Laplace’s equation.(b) Determine E for ρ2 >>a2.
Two conducting planes are located at x = 0 and x = 50 mm. The zero voltage reference is at x = 20 mm. Given that E = –110ax V/m, calculate the conductor voltages.
A 10-nC point charge is located at point (0, 0, 10 m) above a grounded conducting plane z = 0. (a) Find the surface charge density at point (2, –4, 0). (b) Calculate the total charge on the plate.
The current density isDetermine the current through the surface r = 3, π/4 < θ < π/2, 0 < Φ < 2π. J= 20 cose r+ 3 a, A/m²
A conducting sphere of radius a has a total charge Q uniformly distributed on its surface.(a) If the sphere is embedded in a medium with permittivity ε, find the energy stored.(b) Repeat part (a) if the permittivity varies as ε = εo E = 1 + &o (1 +. r 2
Evaluate both sides of the divergence theorem for the vector field H = 2xyax + +x2 + z2 )ay + 2yzaz and the rectangular region defined by 0 < x < 1,1 < y < 2 < –1 < z < 3.
Let and ε = εo. (a) Find E at point P(1, 60°, 30°). (b) Determine pv at P. V= 10 cos 0 sin b r
In cylindrical coordinates, the equation is called(a) Maxwell’s equation (b) Laplace’s equation(c) Poisson’s equation(d) Helmholtz’s equation(e) Lorentz’s equation дела 1 де ap² pip + дене az² + 10 = 0
Which of the following statements are incorrect?(a) The conductivities of conductors and insulators vary with temperature and frequency.(b) A conductor is an equipotential body in steady state, and E is always tangential to the conductor.(c) Nonpolar molecules have no permanent dipoles.(d) In a
The relaxation time, Tr = e/s, of a material is the time taken by a charge placed in its interior to decrease by a factor of e–1 or to ≈ 37% of its original magnitude.
A long wire with circular cross section has a diameter of 4 mm. The wire is 5 m long and it carries 2 A when a 12 V voltage is applied across its ends. Determine the conductivity of the wire.
Boundary conditions must be satisfied by an electric field existing in two different media separated by an interface. For a dielectric–dielectric interfaceE1t = E2tD1n – D2n = ρS or D1s = D2n if ρS = 0
Given V = 5x3y2 z and ε = 2.25εo, find (a) E at point P(–3, 1, 2), (b) ρv at P.

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