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engineering
elements of electromagnetics
Elements Of Electromagnetics 7th Edition Matthew Sadiku - Solutions
The work done by the force F = 4ax – 3ay + 2az N in giving a 1 nC charge a displacement of 10ax + 2ay – 7az m is(a) 103 nJ(b) 60 nJ(c) 64 nJ(d) 20 nJ
By saying that the electrostatic field is conservative, we do not mean that(a) It is the gradient of a scalar potential.(b) Its circulation is identically zero.(c) Its curl is identically zero.(d) The work done in a closed path inside the field is zero.(e) The potential difference between any two
A charge Q is uniformly distributed throughout a sphere of radius a. Taking the potential at infinity as zero, the potential at r = b < a is(a)(b)(c)(d) Qr -La dr 4πε.a³ 3 8
A cube is defined by 0 < x < a, 0 < y < a, and 0 < z < a. If it is charged with ρv = ρox / a, where ρo is a constant, calculate the total charge in the cube.
Suppose a uniform electric field exists in the room in which you are working, such that the lines of force are horizontal and at right angles to one wall. As you walk toward the wall from which the lines of force emerge into the room, are you walking toward (a) Points of higher potential?(b)
Given that ρs = 6xy C/m2, calculate the total charge on the triangular region in Figure 4.25. 0 +2 X
A volume charge with density ρv = ρ2z mC/m3 exists in a region defined by 0 < ρ <2, 0 < z < 1, 30° < Φ < 90°. Calculate the total charge in the region.
A potential field is given by V = 3x2y – yz. Which of the following is not true?(a) At point (1, 0, –1), V and E vanish.(b) x2y = 1 is an equipotential line on the xy-plane.(c) The equipotential surface V = 28 passes through point P + 2, –1, 4).(d) The electric field at P is 12ax – 8ay
A wedge-shaped surface has its corners located at (0, 0, 4), (2, 0, 4), and (2, 3, 4). If the surface has charge distribution with ρs = 10x2 yz mC/m2, find the total charge on the surface.
An electric potential field is produced by point charges 1 μC and 4 μC located at (–2, 1, 5) and (1, 3, –1), respectively. The energy stored in the field is(a) 2.57 mJ(b) 5.14 mJ (c) 10.28 mJ(d) None of the above
Given that ρv = 4ρ2 z cos Φ nC/m3, find the total charge contained in a wedge defined by 0 < ρ < 2, 0 < Φ < π/4, 0 < z < 1.
Line 0 < x < 1 m is charged with density 12x2 nC/m. (a) Find the total charge.(b) Determine the electric field intensity at (0, 0, 1000 m).
A uniform charge 12 μC/m is formed on a loop described by x2 + y2 = 9 on the z = 0 plane. Determine the force exerted on a 4 mC point charge at (0, 0, 4).
An annular disk of inner radius a and outer radius b is placed on the xy-plane and centered at the origin. If the disk carries uniform charge with density ρs, find E at (0, 0, h).
(a) An infinite sheet at z = 0 has a uniform charge density 12 nC/m2. Find E on both sides of the planar sheet.(b) A second sheet at z = 4 has a uniform charge density of –12 nC/m2. Show that E exists only between the planar sheets. Find E.
Plane x + 2y = 5 carries charge ρS = 6 nC/m2. Determine E at (–1, 0, 1).
Plane x = 0 has a uniform charge density ρs, while plane x = a has –ρs. Determine the electric field intensity in regions (a) x < 0, (b) 0 < x < a, (c) x > a.
Three surface charge distributions are located in free space as follows: 10 μC/m2 at x = 2, –20 μC/m2 at y = –3, and 30 μC/m2 at z = 5. Determine E at (a) P (5,–1, 4),(b) R(0, –2, 1), (c) Q (3, –4, 10).
The gravitation force between two bodies of masses m1 and m2 iswhere G = 6.67 X 10–11 N(m/kg2). Find the ratio of the electrostatic and gravitational forces between two electrons. F = Gm₁m₂ p²2² -a,
A sphere of radius a is centered at the origin. IfDetermine E everywhere. Pv -{ Spilz, 0, 0
Three point charges are located in the z = 0 plane: a charge +Q at point (–1, 0), a charge +Q at point (1, 0), and a charge –2Q at point (0, 1). Determine the electric flux density at (0, 0).
The electric flux density in free space is given by D = y2ax + 2xyay – 4zaz nC/m2(a) Find the volume charge density. (b) Determine the flux through surface x = 3, 0 < y < 6, 0 < z < 5.
In free space, E = 12ρzcos Φaρ – 6ρzsin ΦaΦ + 6ρ2cos Φaz V/m. Find the electric flux through surface Φ = 90°0 < ρ < 2, 0 < z < 5.
If D = sinθsinΦaT + cossθsinΦaθ + cosΦaΦ nC/m², find: (a) The charge density at (2, 30°, 60°), (b) The flux through r = 2, 0 < θ < 30°, 0 < Φ < 60°.
Let(a) Find the net flux crossing surface r = 2 m and r = 6 m.(b) Determine D at r = 1 m and r = 5 m. Pv 10 12² 0, mC/m³, 1
If spherical surfaces r = 1 m and r = 2 m, respectively, carry uniform surface charge densities 8 nC/m2 and –6 mC/m2, find D at r = 3 m.
The volume charge density inside an atomic nucleus of radius a is where ρo is a constant.(a) Calculate the total charge.(b) Determine E and V outside the nucleus.(c) Determine E and V inside the nucleus.(d) Prove that E is maximum at r = 0.745a. Pv = Po 1-²) a²
Let charge Q be uniformly distributed on a circular ring defined by a < r < b and shown in Figure 4.26. Find D at (0, 0, h). b y X
Let D = 2xyax + x2 ay C/m2 and find(a) The volume charge density ρv.(b) The flux through surface 0 < x < 1, 0 < z < 1, y = 1.(c) The total charge contained in the region 0 < x < 1 < 0 < y < 1, 0 < z <1.
A long coaxial cable has an inner conductor with radius a and outer conductor with radius b. If the inner conductor has ps = po/p, where po is a constant, determine E everywhere.
Two point charges Q = 2 nC and Q = –4 nC are located at (1, 0, 3) and (–2, 1, 5), respectively. Determine the potential at P(1, –2, 3).
A charge of 8 nC is placed at each of the four corners of a square of sides 4 cm long. Calculate the electrical potential at the point 3 cm above the center of the square.
The potential distribution in free space is given by V = ρ2e–z sinΦ V Calculate the electric field strength at (4, π/4, –1).
In free space, an electric field is given byCalculate the volume charge density. E = Eoplakaps O
Find V = x2y(z + 3) V.(a) E at (3, 4, –6)(b) The charge within the cube 0 < x < 1, 0 < y < 1, 0 < z < 1.
If D = 4xax–10y2ay + z2az C/m2, find the charge density at P(1, 2, 3).
A 10 nC charge is uniformly distributed over a spherical shell r = 3 cm, and a –5 nC charge is uniformly distributed over another spherical shell r = 5 cm. Find D for regions r < 3 cm < 3 cm < r < 5 cm, r > 5 cm.
The electric field intensity in free space is given by E = 2 xyzax + x2z ay + x2y az V/m. Calculate the amount of work necessary to move a 2 μC charge from (2, 1, –1) to (5, 1, 2).
Let E = 10 / r2 a2 V/m. Find VAB, where A is (1,π/4, π/2) and B is (5,π, 0).
In free space, E = 20xax + 40yay – 10zaz V/m. Calculate the work done in transferring a 2 mC charge along the arc ρ = 2, 0 < Φ < π/2 in the z = 0 plane.
A sheet of charge with density ρs = 40 nC/m2 occupies the x = 0 plane. Determine the work done in moving a 10 μC charge from point A(3, 4, –1) to point B(1, 2, 6).
Let V = ρe–zsin Φ. (a) Find E. (b) Show that E is conservative.
In free space, V = 1 / r3 sin θ cos θ . Find D at (1, 30°, 60°).
Each of two concentric spherical shells has inner radius a and outer radius b. If the inner shell carries charge Q, while the outer shell carries charge –Q, determine the potential difference Vab between the shells.
A uniform surface charge with density ρs exists on a hemispherical surface with r = a and θ ≤ π/2. Calculate the electric potential at the center.
If determine whether D is a genuine electric flux density. Determine the flux crossing ρ = 1, 0 ≤ Φ ≤ π/4, 0 < z < 1. D = 2p sin da,- cos o 2p a C/m²,
A dipole has dipole moment p = 2ax + 6ay – 4az μC ∙ m. If the dipole is located in free space at (2, 3, –1), find the potential at (4, 0, 1).
Determine the amount of work needed to transfer two charges of 40 nC and –50 nC from infinity to locations (0, 0, 1) and (2, 0, 0), respectively.
If V = 2x2 + 6y2 V in free space, find the energy stored in a volume defined by –1 ≤ x ≤ 1, –1 ≤ y ≤ 1, and –1 ≤ z ≤ 1.
In free space, E = y2ax + 2xyay – 4zaz V/m. Determine the energy stored in the region defined by 0 < x < 2, –1 < y < 1, 0 < z < 4.
In free space, V = re–z sin f. (a) Find E. (b) Determine the energy stored in the region 0 < ρ < 1, 0 < Φ < 2π, 0 < z < 2.
A spherical region of radius a has total charge Q. If the charge is uniformly distributed, apply Gauss's law to find D both inside and outside the sphere.
Given that E = 12ρzcosΦaρ – 6ρzsinΦaΦ + 6ρ2cosΦaz (a) Find the volume charge density at A(2, 180º, –1), (b) Calculate the work done in transferring a 10 μC charge from A to B(2, 0º, –1).
At point P(2,0,21), calculate the value of the following dot products: (a) aρ ∙ ax, (b) aΦ ∙ ay, (c) ar ∙ az
Show that the vector fieldsA = ρ sin Φaρ + ρ cos ΦaΦ + ρazB = ρ sin Φaρ + ρ cosΦaΦ – ρazare perpendicular to each other at any point.
If H = ρ2 cos Φaρ – r sin ΦaΦ, find H ∙ ax at point P(2, 60°, –1).
Prove the following expressions:(a)(b) a = az = аф ad X az - ap аф ap X ар аф az X ap
(a) Show that point transformation between cylindrical and spherical coordinates is obtained usingor(b) Show that vector transformation between cylindrical and spherical coordinates is obtained usingor r = Vp² + 2², 0 = tan¯¹ Ф = ф
A wedge is described by z = 0, 30° < φ < 60°. Which of the following is incorrect:(a) The wedge lies in the xy-plane.(b) It is infinitely long.(c) On the wedge, 0 < ρ < ∞.(d) A unit normal to the wedge is ± az.(e) The wedge includes neither the x-axis nor the y-axis.
If B = r sin θ ar – r2 cos ΦaΦ, (a) Find B at (2, π/2, 3π/2), (b) Convert B to Cartersian coordinates.
Given that A = 3aρ + 2aΦ + az and B = 5aρ – 8az , find:(a) A + B, (b) A ∙ B, (c) A X B, (d) The angle between A and B.
Given that G = 3ρaρ + ρ cosΦaΦ – z2az, find the component of G along ax at point Q(3,–4,6).
Let G = yzax + xzay + xyaz. Transform G to cylindrical coordinates.
Calculate the distance between points P(4, 30°, 0°) and Q(6, 90°, 180°).
At point (0, 4, –1), express aρ and aΦ in Cartesian coordinates.
Let A = (2z - sin Φ)aρ + (4p + 2 cosΦ)aΦ – 3pzaz, and B = ρ cosΦaρ , + sinΦaΦ + az.(a) Find the minimum angle between A and B at (1, 60°, –1).(b) Determine a unit vector normal to both A and B at (1, 90°, 0).
Given that B = ρ2 sin Φaρ + (z – 1) cos ΦaΦ + z2az, find B ∙ ax at (4, π/4, –1).
If r = x ax + yay + zaz, describe the surface defined by:(a) r · ax + r · ay = 5(b) Ιr x azΙ = 10
Express the following vectors in rectangular coordinates:(a) A = ρ sin Φ aρ + ρ cos Φ aΦ – 2zaz(b) B = 4r cos Φ ar + r aθ
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