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physics
electrodynamics

Elements of Electromagnetics 3rd Edition Matthew - Solutions

A parallel-plate capacitor has its plates at x = 0, d and the space between the plates is filled with an inhomogeneous material with permittivity ε = ε0(1 + x/d). If the plate at x = d is maintained at Vo while the plate at x = 0 is grounded, find:(a) V and E(b) P(c) ρps at x
A spherical capacitor has inner radius a?and outer radius b?and filled with an inhomogeneous dielectric with ? = ?ok/r2. Show that the capacitance of the capacitor is
The y- and z-axes, respectively, carry filamentary currents 10 A along ay and 20 A along – az. Find H at (– 3, 4, 5).
A conducting filament carries current I from point A(0, 0, ?) to point 5(0, 0, b). Show that at point P(x, y, 0),
Consider AB in figure as part of an electric circuit. Find H at the origin due toAB.
Repeat Problem 7.3 for the conductor AB in figure.
Line x = 0, y = 0, 0 < z < 10m carries current 2 A along az. Calculate H at points (a) (5, 0, 0) (b) (5, 5, 0) (c) (5, 15, 0) (d) (5, – 15, 0)
(a) Find H at (0, 0, 5) due to side 2 of the triangular loop in figure (a).(b) Find H at (0, 0, 5) due to the entire loop.
An infinitely long conductor is bent into an L shape as shown in figure. If a direct current of 5 A flows in the current, find the magnetic field intensity at? (a) (2, 2, 0), (b) (0, ? 2, 0), and? (c) (0, 0, 2).
Find H at the center C of an equilateral triangular loop of side 4 m carrying 5 A of current as infigure.
A rectangular loop carrying 10 A of current is placed on z = 0 plane as shown in figure. Evaluate H at(a) (2, 2, 0)(b) (4, 2, 0)(c) (4, 8, 0)(d) (0, 0,2)
A square conducting loop of side 2a lies in the z = 0 plane and carries a current I in the counterclockwise direction. Show that at the center of theloop
(a) A filamentary loop carrying current I is bent to assume the shape of a regular polygon of n sides. Show that at the center of the polygon where r is the radius of the circle circumscribed by the polygon.(b) Apply this to cases when n = 3 and n = 4 and see if your results agree with those for
For the filamentary loop shown in figure find the magnetic field strength at O.
Two identical current loops have their centers at (0, 0, 0) and (0, 0, 4) and their axes the same as the z-axis (so that the "Helmholtz coil" is formed). If each loop has radius 2 m and carries current 5 A in a^, calculate H at (a) (0, 0, 0) (b) (0, 0, 2)
A 3-cm-long solenoid carries a current of 400mA. If the solenoid is to produce a magnetic flux density of 5mWb/m2, how many turns of wire are needed?
A solenoid of radius 4 mm and length 2 cm has 150turns/m and carries current 500mA. Find: (a) |H| at the center, (b) |H| at the ends of the solenoid.
Plane x = 10 carries current 100mA/m along az while line x = 1, y = –2 carries filamentary current 20π mA along az. Determine H at (4, 3, 2).
A hollow conducting cylinder has inner radius a and outer radius b and carries current I along the positive z-direction. Find H everywhere.
(a) An infinitely long solid conductor of radius a is placed along the z-axis. If the conductor carries current I in the + z direction, show thatwithin the conductor. Find the corresponding current density,(b) If I = 3 A and a = 2 cm in part (a), find H at (0, 1 cm, 0) and (0, 4 cm, 0).
If H = yax – xay A/m on plane z = 0, (a) Determine the current density and (b) Verify Ampere's law by taking the circulation of H around the edge of the rectangle Z = 0, 0 < x < 3, – 1 < y < 4.
In a certain conducting region, (a) Determine J at (5, 2, ?3) (b) Find the current passing through x =??1, 0 (c) Show that ? ? B = 0
An infinitely long filamentary wire carries a current of 2 A in the + z-direction. Calculate (a) B at (– 3, 4, 7) (b) The flux through the square loop described by 2 < ρ < 6, 0 < z < 4, Φ = 90°
The electric motor shown in figure has fieldCalculate the flux per pole passing through the air gap if the axial length of the pole is 20 cm.
Consider the two-wire transmission line whose cross section is illustrated in figure. Each wire is of radius 2 cm and the wires are separated 10 cm. The wire centered at (0, 0) carries current 5 A while the other centered at (10 cm, 0) carries the return current. Find H at(a) (5 cm, 0)(b) (10 cm,
Determine the magnetic flux through a rectangular loop (? X b) due to an infinitely long conductor carrying current I as shown in figure. The loop and the straight conductors are separated by distance d.
A brass ring with triangular cross section encircles a very long straight wire concentrically as in figure. If the wire carries a current I, show that the total number of magnetic flux lines in the ring is calculate ? if ? = 30 cm, b = 10 cm, h = 5 cm, and I = 10 A.
Consider the following arbitrary fields. Find out which of them can possibly represent electrostatic or magnetostatic field in free space. (a) A = y cos αxax + (y + e–x)az (b) B = 20/ρ aρ (c) C = r2 sin θ aΦ
Reconsider the previous problem for the following fields.
For a current distribution in free space, A = {2x2y + yz)ax + {xy2 – xz3)ay – (6xyz ~ 2x2y2 )az Wb/m (a) Calculate B. (b) Find the magnetic flux through a loop described by x = 1, 0 < y, z < 2. (c) Show that ∆ ∙ A = 0 and ∆ ∙ B = 0.
The magnetic vector potential of a current distribution in free space is given by A = 15e–p sin Φ az Wb/m Find H at (3, π/4, – 10). Calculate the flux through ρ = 5, 0 < Φ < π/2, 0 < z < 10.
A conductor of radius a carries a uniform current with J = Joaz. Show that the magnetic vector potential for p > ais
An infinitely long conductor of radius a is placed such that its axis is along the z-axis. The vector magnetic potential, due to a direct current Io flowing along az, in the conductor, is given byFind the corresponding H. Confirm your result using Ampere'slaw.
The magnetic vector potential of two parallel infinite straight current filaments in free space carrying equal current I in opposite direction iswhere d is the separation distance between the filaments (with one filament placed along the z-axis). Find the corresponding magnetic flux density B.
Find the current density J toin freespace.
Prove that the magnetic scalar potential at (0, 0, z) due to a circular loop of radius a shown in figure (a)is
A coaxial transmission line is constructed such that the radius of the inner conductor is a and the outer conductor has radii 3a and 4a. Find the vector magnetic potential within the outer conductor. Assume Az = 0 for ρ = 3a.
The z-axis carries a filamentary current 12 A along az. Calculate Vm at (4, 30°, – 2 ) if Vm = O at (10, 60°, 7).
Plane z = – 2 carries a current of 50ay A/m. If Vm = 0 at the origin, find Vm at(a) (– 2, 0, 5)(b) (10, 3, 1)
Prove in cylindrical coordinates that(a) ∆ × (∆V) = 0(b) ∆ ∙ (∆ × A) = 0
If R = r ? r' and? = |R|, show that where ? and ?? are del operators with respect to (x,?y, z)?and (x', y',?z), respectively.
An electron with velocity u = (3ax + 12ay – 4az) X 105m/s experiences no net force at a point in a magnetic field B = 10ax + 20ay + 30az mWb/m2. Find E at that point.
A charged particle of mass 1 kg and charge 2 C starts at the origin with velocity 10az m/s in a magnetic field B = 1ax, Wb/m2. Find the location and the kinetic energy of the particle at t = 2 s.
A particle with mass 1 kg and charge 2 C starts from rest at point (2, 3, –4) in a region where E = –4ay V/m and B = 5a Wb/m2. Calculate(a) The location of the particle at t = 1s(b) Its velocity and K.E. at that location
A – 2-mC charge starts at point (0, 1, 2) with a velocity of 5ax m/s in a magnetic field B = 6ay Wb/m2. Determine the position and velocity of the particle after 10 s assuming that the mass of the charge is 1 gram. Describe the motion of the charge.
By injecting an electron beam normally to the plane edge of a uniform field Boaz, electrons can be dispersed according to their velocity as in figure.(a) Show that the electrons would be ejected out of the field in paths parallel to the input beam as shown.(b) Derive an expression for the exit
Given that B = 6xax ? 9yay + 3zaz Wb/m2, find the total force experienced by the rectangular loop (on z = 0 plane) shown in figure.
A current element of length 2 cm is located at the origin in free space and carries current 12mA along ax. A filamentary current of 15az A is located along x = 3, y = 4. Find the force on the current filament.
Three infinite lines L1, L2, and L3 defined by x = 0, y = 0; x = 0, y = 4; x = 3, y = 4, respectively, carry filamentary currents –100 A, 200 A, and 300 A along az. Find the force per unit length on(a) L2 due to L1,(b) L1 due to L2(c) L3 due to L1(d) L3 due to L1 and L2. State whether each force
A conductor 2 m long carrying 3A is placed parallel to the z-axis at distance ?0 = 10 cm as shown in figure. If the field in the region is cos (?/3) a? Wb/m2, how much work is required to rotate the conductor one revolution about the z-axis?
A conducting triangular loop carrying a current of 2 A is located close to an infinitely long, straight conductor with a current of 5 A, as shown in figure. Calculate(a) The force on side 1 of the triangular loop and(b) The total force on theloop.
A three-phase transmission line consists of three conductors that are supported at points A, B, and C to form an equilateral triangle as shown in figure. At one instant, conductors A and B both carry a current of 75 A while conductor C carries a return current of 150 A. Find the force per meter on
An infinitely long tube of inner radius a and outer radius b is made of a conducting magnetic material. The tube carries a total current I and is placed along the z-axis. If it is exposed to a constant magnetic field Boap, determine the force per unit length acting on the tube.
An infinitely long conductor is buried but insulated from an iron mass (? = 2000?o) as shown in figure. Using image theory, estimate the magnetic flux density at point P.
A galvanometer has a rectangular coil of side 10 by 30 mm pivoted about the center of the shorter side. It is mounted in radial magnetic field so that a constant magnetic field of 0.4 Wb/m2 always acts across the plane of the coil. If the coil has 1000 turns and carries current 2mA, find the torque
A small magnet placed at the origin produces B = – 0.5az, mWb/m2 at(10, 0, 0). Find B at(a) (0, 3, 0)(b) (3, 4, 0)(c) (1, 1, – 1)
A block of iron (μ = 5000μo) is placed in a uniform magnetic field with 1.5 Wb/m2. If iron consists of 8.5 X 1028 atoms/m3 calculate: (a) The magnetization M, (b) The average magnetic current.
In a certain material for which μ = 6.5/μo, H = 10ax + 25ay – 40az A/m find (a) The magnetic susceptibility xm of the material (b) The magnetic flux density B (c) The magnetization M, (d) The magnetic energy density
In a ferromagnetic material (μ = 4.5/μo), calculate: B = 4yaz mWb/m2 (a) Xm, (b) H, (c) M, (d) Jb.
The magnetic field intensity is H = 1200 A/m in a material when B = 2 Wb/m2. When H is reduced to 400 A/m, B = 1.4 Wb/m2. Calculate the change in the magnetization M.
An infinitely long cylindrical conductor of radius a and permeability μoμr is placed along the z-axis. If the conductor carries a uniformly distributed current / along az find M and Jb for 0 < p < a.
(a) For the boundary between two magnetic media such as is shown in figure, show that the boundary conditions on the magnetization vector are(b) If the boundary is not current free, show that instead of eq. (8.49), weobtain
If μ1 = 2μo for region 1 (0 < Φ < π) and μ2 = 5μo for region 2 (π < μ < Φ 2π) and B2 = 10ap + 15aΦ – 20az mWb/m2. Calculate: (a) B1 (b) The energy densities in the two media.
The interface 2x + y = 8 between two media carries no current. If medium 1 (2x + y > 8) is nonmagnetic with H1 = – 4ax + 3ay – az A/m. Find: (a) The magnetic energy density in medium 1, (b) M2 and B2 in medium 2 (2x + y < 8) with μ = 10μo, (c) The angles H1 and H2 make with the
The interface 4x – 5z = 0 between two magnetic media carries current 35ay A/m. If H1 = 25ax – 30a0 + 45az, A/m in region 4x – 5z < 0 where μr1 = 5, calculate H2 in region 4x – 5z > 0 where μr2 = 10.
The plane z = 0 separates air (z > 0, μ, = μo) from iron (z < 0, μ = 200μo). Given that H = 10ax + 15ay – 3az, A/min air, find B in iron and the angle it makes with the interface.
Region 0 ? z ? 2 m is filled with an infinite slab of magnetic material (??= 2.5?o). If the surfaces of the slab at z?= 0 and z = 2,?respectively, carry surface currents 30ax A/m and ? 40ax A/m as in figure, calculate H and B for (a) z (b) 0 (c) z > 2
In a certain region for which xm = 19,H = 5x2yzax + 10xy2zay – 15xyz2az A/mHow much energy is stored in 0 < x < 1, 0 < v < 2, – 1 < z < 2?
The magnetization curve for an iron alloy is approximately given by B = ⅓ H + H2μ Wb/m2. Find: (a) μr when H = 210 A/m, (b) The energy stored per unit volume in the alloy as H increases from 0 to 210 A/m.
(a) If the cross section of the toroid of figure 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, show thatwhere ?o >> a.
When two parallel identical wires are separated by 3 m, the inductance per unit length is 2.5μH/m. Calculate the diameter of each wire.
The core of a toroid is 12 cm2 and is made of material with μr = 200. If the mean radius of the toroid is 50 cm, calculate the number of turns needed to obtain an inductance of 2.5 H.
Show that the mutual inductance between the rectangular loop and the infinite line current of figure isCalculate M12 when ? = b = ?o = 1 m.
Prove that the mutual inductance between the closed wound coaxial solenoids of length ?1 and ?2 (?1 >> X ?2) turns N1 and N2, and radii r1, and r2 with r1 = r2 is
A cobalt ring (μr = 600) has a mean radius of 30 cm. If a coil wound on the ring carries 12 A, calculate the number of turns required to establish an average magnetic flux density of 1.5 Wb/m in the ring.
Refer to figure. If the current in the coil is 0.5 A, find the mmf and the magnetic field intensity in the air gap, assume that μ = 500/no and that all branches have the same cross-sectional area of 10 cm2.
The magnetic circuit of figure has current 10 A in the coil of 2000 turns. Assume that all branches have the same cross section of 2 cm2 and that the material of the core is iron with nr = 1500. Calculate R, F, and ? for (a) The core (b) The air gap
Consider the magnetic circuit in figure. Assuming that the core (? = 1000?o) has a uniform cross section of 4 cm2, determine the flux density in the air gap.
A toroid with air gap, shown in figure, has a square cross section. A long conductor carrying current I2 is inserted in the air gap. If I1 = 200mA, N = 750, ?o = 10 cm, ? = 5 mm, and ?? = 1 mm, calculate(a) The force across the gap when I2 = 0 and the relative permeability of the toroid is 300.(b)
A section of an electromagnet with a plate below it carrying a load is shown in figure. The electromagnet has a contact area of 200 cm2 per pole with the middle pole having a winding of 1000 turns with I = 3 A. Calculate the maximum mass that can be lifted. Assume that the reluctance of the
Figure shows the cross section of an electromechanical system in which the plunger moves freely between two nonmagnetic sleeves. Assuming that all legs have the same cross-sectional area S, showthat
A conducting circular loop of radius 20 cm lies in the z = 0 plane in a magnetic field B = 10cos 377 t az mWb/m2. Calculate the induced voltage in the loop.
A rod of length ℓ rotates about the z-axis with an angular velocity w. If B = Boaz, calculate the voltage induced on the conductor.
A 30-cm by 40-cm rectangular loop rotates at 130rad/s in a magnetic field 0.06 Wb/m2 normal to the axis of rotation. If the loop has 50 turns, determine the induced voltage in the loop.
Figure shows a conducting loop of area 20 cm2 and resistance 4?. If B = 40 cos104taz mWb/m2, find the induced current in the loop and indicate its direction.
Find the induced emf in the V-shaped loop of figure.(a) Take B = 0.1z, Wb/m2 and u = 2ax m/s and assume that the sliding rod starts at the origin when t = 0.(b) Repeat part (a) if B = 0.5xazWb/m2.
A square loop of side a?recedes with a uniform velocity uoav from an infinitely long filament carrying current I along az?as shown in figure. Assuming that ? = ?o at time t = 0, show that the emf induced in the loop at t > 0 is
A conducting rod moves with a constant velocity of 3az m/s parallel to a long straight wire carrying current 15 A as in figure. Calculate the emf induced in the rod and state which end is at higherpotential.
A conducting bar is connected via flexible leads to a pair of rails in a magnetic field B = 6 cos 10t ax mWb/m2 as in figure. If the z-axis is the equilibrium position of the bar and its velocity is 2 cos 10t ay m/s, find the voltage induced init.
A car travels at 120 km/hr. If the earth's magnetic field is 4.3 X 10–5 Wb/m2, find the induced voltage in the car bumper of length 1.6 m. Assume that the angle between the earth magnetic field and the normal to the car is 65°.
If the area of the loop in figure is 10 cm2, calculate V1 andV2.
As portrayed in figure, a bar magnet is thrust toward the center of a coil of 10 turns and resistance 15?. If the magnetic flux through the coil changes from 0.45 Wb to 0.64 Wb in 0.02 s, what is the magnitude and direction (as viewed from the side near the magnet) of the induced current?
The cross section of a homopolar generator disk is shown in figure. The disk has inner radius ?1 = 2 cm and outer radius ?2 = 10 cm and rotates in a uniform magnetic field 15 mWb/m2 at a speed of 60rad/s. Calculate the induced voltage.
A 50-V voltage generator at 20 MHz is connected to the plates of an air dielectric parallel-plate capacitor with plate area 2.8 cm2 and separation distance 0.2 mm. Find the maximum value of displacement current density and displacement current.
The ratio J/Jd (conduction current density to displacement current density) is very important at high frequencies. Calculate the ratio at 1 GHz for: (a) Distilled water (μ = μo, ε = 81εo, a = 2 X 10–3 S/m) (b) Sea water (μ, = μo, ε = 81εo, a = 25
Assuming that sea water has μ = μo, ε = 81εo, σ = 20 S/m, determine the frequency at which the conduction current density is 10 times the displacement current density in magnitude.
A conductor with cross-sectional area of 10 cm2 carries a conduction current 0.2 sin 109tmA. Given that σ = 2.5 × 106 S/m and εr = 6, calculate the magnitude of the displacement current density.
(a) Write Maxwell's equations for a linear, homogeneous medium in terms of Es and Hs only assuming the time factor e–jwt.(b) In Cartesian coordinates, write the point form of Maxwell's equations in Table 9.2 as eight scalar equations.
Show that in a source-free region (J = 0, ρv = 0), Maxwell's equations can be reduced to two. Identify the two all-embracing equations.
In a linear homogeneous and isotropic conductor, show that the charge density pvsatisfies

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