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engineering
elements of electromagnetics

Elements Of Electromagnetics 7th Edition Matthew Sadiku - Solutions

Given that A = ax + αay + az and B = αax + ay + az, if A and B are normal to each other, α is(a) 22(b) -1/2(c) 0(d) 1(e) 2
Let A = 10ax + 5ay – 2az. Find: (a) A × ay, (b) A ∙ az, (c) The angle between A and az.
The component of 6ax + 2ay – 3az along 3ax – 4ay is(a) –12ax – 9ay – 3az(b) 30ax – 40ay (c) 10/7(d) 2(e) 10
Given A = – 6ax + 3ay + 2az, the projection of A along ay is(a) –12(b) –4 (c) 3(d) 7(e) 12
Given thatfind: (a) |P + Q – R|, (b) P ∙ Q X R, (c) Q X P ∙ R, (d) (P X Q) ∙ (Q X R),(e) (P X Q) X (Q X R), (f) cos θPR, (g) sin θPQ. P = 2a, ay-2a₂ Q = 4a, + 3a, + 2a₂ R = -a₂ + a, + 2a₂
If A = 4ax – 6ay + az and B = 2ax + 5az , find:(a) A ∙ B + 2|B|2(b) A unit vector perpendicular to both A and B
Determine the dot product, cross product, and angle betweenP = 2ax – 6ay – 5az and Q = 3ay + az
Prove that vectors P = 2ax + 4ay – 6az and Q = 5ax + 2ay – 3az are orthogonal vectors.
Simplify the following expressions:(a) A X (A X B)(b) A X [A X (A X B)]
A right angle triangle has its corners located at P1(5, –3, 1), P2(1, –2, 4), and P3(3, 3, 5).(a) Which corner is a right angle? (b) Calculate the area of the triangle.
Two points P(2, 4, –1) and Q(12, 16, 9) form a straight line. Calculate the time taken for a sonar signal traveling at 300 m/s to get from the origin to the midpoint of PQ.
Find the area of the parallelogram formed by the vectors D = 4ax + ay + 5az and E = –ax + 2ay + 3az.
A cube of side 1 m has one corner placed at the origin. Determine the angle between the diagonals of the cube.
Given two vectors A and B, show that the vector component of A perpendicular to B is C = A - A B -B B.B
Figure 1.15 shows that A makes specific angles with respect to each axis. For A = –ax – 4ay + 6az, find the direction angles α, β, and ϒ. X Y 8 20 B A
If H = 2xyax –(x + z)ay + z2az, find:(a) A unit vector parallel to H at P(1, 3, –2)(b) The equation of the surface on which ΙHΙ = 10
Let P = 2ax – 4ay + az and Q = ax + 2ay. Find R which has magnitude 4 and is perpendicular to both P and Q.
Let G = x2ax – yay + 2zaz and H = yzax + 3ay – xzaz. At point (1, –2, 3), (a) Calculate the magnitude of G and H, (b) Determine G ∙ H, (c) Find the angle between G and H.
A vector field is given by H = 10yz2ax – 8xyzay + 12y2az(a) Evaluate H at P(–1, 2, 4)(b) Find the component of H along ax – ay at P.
Given two vector fields D = yzax + xzay + xyaz and E = 5xyax + 6(x2+ 3)ay + 8z2az(a) Evaluate C = D + E at point P(–1, 2, 4). (b) Find the angle C makes with the x-axis at P.
Convert the following Cartesian points to cylindrical and spherical coordinates:(a) P(2, 5, 1)(b) Q(–3, 4, 0)(c) R(6, 2, 24)
The ranges of θ and φ as given by eq. (2.17) are not the only possible ones. The following are all alternative ranges of θ and φ, except(a) 0 ≤ θ < 2π, 0 ≤ φ ≤ π(b) 0 ≤ θ < 2π, 0 ≤ φ < 2π(c) - π ≤ θ ≤ π, 0 ≤ φ ≤ π(d) - π/2 ≤ θ ≤ π/2, 0 ≤ φ <
At Cartesian point ( –3, 4, –1), which of these is incorrect?(a) p = -5(b) r = √(26)(c) θ = tan–15/–1(d) φ = tan–14/–3
The rectangular coordinates at point P are (x = 2, y = 6, z = –4). (a) What are its cylindrical coordinates? (b) What are its spherical coordinates?
Which of these is not valid at point (0, 4, 0)?(a) aΦ = –ax(b) aθ = –az (c) ar = 4ay(d) aρ = ay
Given vector A = 2aρ + 3aΦ + 4az,convert A into Cartesian coordinates at point (2,π /2, –1).
Match the items in the left list with those in the right list. Each answer can be used once, more than once, or not at all.(a) θ = π/4(b) φ = 2π/3(c) x = -10(d) r = 1, θ = π/3, φ = π/4(e) ρ = 5(f) ρ = 3, φ = 5π/3(g) ρ = 10, z = 1(h) r = 4, φ = π/6(i) r = 5, θ = π/3(i) Infinite
Let B = xaz. Express B in(a) Cylindrical coordinates,(b) Spherical coordinates.
If the integral is regarded as the work done in moving a particle from A to B, find the work done by the force field F = 2xyax + (x² – z²)ay – 3xz²az on a particle that travels from A(0, 0, 0) to B(2, 1, 3) along (a) The segment (0, 0, 0) → (0, 1, 0) → (2, 1, 0)
Let A = yax + zay + xaz. Find the flux of A through surface y = 1, 0 < x < 1, 0 < z < 2.
Find the gradient of the following scalar fields and evaluate the gradient at the specified point.(a) V(x, y, z) = 10xyz – 2x²z at P(–1, 4, 3)(b) U(p, Φ, z) = 2p sin Φ + pz at Q(2, 90°, –1)(c) W(r, θ, Φ) = 4 / r sin θ cos Φ at R(1, π/6, π/2)sin (a) V(x, y, z) = 10xyz - 2x²z at
Consider the scalar function T = r sin θ cos Φ. Determine the magnitude and direction of the maximum rate of change of T at P(2, 6°, 30°).
Let f = x2y – 2xy2 + z3. Find the directional derivative of f at point (2, 4, –3) in the direction of ax + 2ay – az.
If H = 10 cos θar, evaluate ∫s H∙ dS over a hemisphere defined by r = 1, 0 < Φ < 2π, 0 < θ < π/2.
Let D = 2ρz2aρ + ρcos2 Φ az. Evaluate(a)(b)over the region defined by 2 ≤ ρ ≤ 5, –1 ≤ z ≤ 1,0 < Φ < 2π. f, D. ds S
(a) Prove that for scalar fields V and U,(b) Verify part (a) by assuming that V = 5x2y + 2yz and U = 3xyz. V(UV) = UVV + VVU
Evaluate the divergence of the following vector fields: (a) A = xyax + y²ay - xzaz(b) B = ρz²ap + ρ sin² ΦaΦ + 2ρz sin² Φaz(c) C = rar + r cos²θaΦ
If H = 4aρ - 3aΦ + 5az, at (1, π/2, 0) the component of H parallel to surface ρ = 1 is(a) 4aρ(b) 5az(c) -3aφ(d) -3aφ + 5az(e) 5aφ + 3az
At every point in space, aΦ · aθ = 1.(a) True (b) False
Given point T(10, 60°, 30°) in spherical coordinates, express T in Cartesian and cylindrical coordinates.
A unit normal vector to the cone θ = 30° is:(a) ar(b) aθ(c) aΦ(d) None of these
Given G = 20ar + 50aθ + 40aφ, at (1, π/2, π/6) the component of G perpendicular to surface θ = π/2 is(a) 20ar(b) 50aθ(c) 40aφ(d) 20ar + 40aθ(e) - 40aθ + 20aφ
Let Transform B to cylindrical coordinates. = √x² + y² ax + y √x² + y² ay + za₂.
Where surfaces r = 2 and z = 1 intersect is(a) An infinite plane(b) A semi-infinite plane(c) A circle(d) A cylinder(e) A cone
Which of the following is mathematically defined?(a)(b) (c)(d) VXV.A
Calculate the area of the surface defined by r = 5, 0 < θ < π/4, 0 < Φ, π/2.
Which of the following is zero?(a) Grad div (b) Div grad (c) Curl grad(d) Curl curl
Calculate the volume defined by 2 < ρ < 5, 0 < Φ < 30°, 0 < z < 10.
Let H = xy2ax + x2yay. Evaluate the line integral along the parabola x = y2 joining point P(1, 1, 0) to point Q(16, 4, 0).
Evaluate the line integral ∫L (2x2 – 4xy)dx + 3xy – 2x2y)dy over the straight path L joining point P(1, 21, 2) to Q(3, 1, 2).
A vector field is represented by F = ρ2aρ + zaΦ + cos Φaz Newtons. Evaluate the work done or ∫L F ∙ dl, where L is from P(2, 0°, 0) to Q(2, π/4, 3). Assume that L consists of the arc ρ = 2, 0 < Φ < π/4, z = 0, followed by the line ρ = 2, Φ = π/4, 0 < z
If H = (x – y) ax + (x2 + zy)ay + 5yzaz evaluate ∫L H ∙ dl along the contour of Figure 3.28. 0
Determine the circulation of B = xyax – yzay + xzaz around the path L on the x = 1 plane, shown in Figure 3.29. Z 1 0 1 L x = 1 X
If D = x2 zax + y3ay + yz2az, calculate the flux of D passing through the volume bounded by planes x = –1, x = 1, y = 0, y = 4, z = 1, and z = 3.
A vector field is specified as A = rar – 3aθ + 5ΦaΦ. Find the flux of the field out of the closed surface defined by 0 < r < 4,0 < θ < π/2, 0 < Φ < 3 , π/2.
(a) Evaluate ∫v xy dv, where v is defined by 0 < x < 1< 0 < y < 1< 0 z < 2.(b) Determine ∫v ρz dv, where v is bounded by ρ = 1 < ρ = 3, Φ = 0, Φ = π, z = 0, and z = 2.
Calculate the gradient of:(a) V1 = 6xy – 2xz + z(b) V2 = 10ρ cos Φ – ρz(c) V3 =2 / r cos Φ
If r = xax + yay + zaz is the position vector of point (x, y, z), r = |r|, and n is an integer, show that الي Vr" = nr-²r.
A family of planes is described by F = x – 2y + z. Find a unit normal an to the planes.
(a) Using the gradient concept, prove that the angle between two planesis(b) Calculate the angle between two planes x + 2y + 3z = 5 and x + y = 0. ax + by + cz = d ax + By + y2 = 8
Consider the object shown in Figure 3.8. For the volume element, match the items in the left-hand column with those on the right.(a) dl from A to D(b) dl from E to A(c) dl from A to B(d) dS for face EFGH(e) dS for face AEHD(i) –r² sin θ dθ dΦ ar(ii) -r sin cdr dΦ aθ (iii) r dr
The surface current density J in a rectangular waveguide is plotted in Figure 3.27. It is evident from the figure that J diverges at the top wall of the guide, whereas it is divergencelessat the side wall.(a) True (b) False
Given field A = 3x2yzax + x3zay + (x3y – 2z)az, it can be said that A is(a) Harmonic (b) Divergenceless (c) Solenoidal(d) Rotational(e) Conservative
Stokes’s theorem is applicable only when a closed path exists and the vector field and its derivatives are continuous within the path.(a) True(b) False(c) Not necessarily
If a vector field Q is solenoidal, which of these is true?(a)(b)(c)(d)(e) 0 = IP.d 'f
Given that F = x2yax – yay, find(a)where L is shown in Figure 3.30.(b)where S is the area bounded by L.(c) Is Stokes’s theorem satisfied? fF-di L
Let H = ρ sin Φaρ + ρ cos ΦaΦ – raz ; find V x H and V x V x H.
Let show that A = xa, + ya, + za (x² + y² + 2²) 3/2
Evaluate (a)(b)(c) VXA and V (V x A) if: X
Evaluate the curl of the following vector fields:(a) A = xyax + y2ay – xzaz(b) B = ρz2aρ + ρ sin2 Φ aΦ + 2ρz sin2 Φ az(c) C = rar + r cos2 θ aΦ
Let A = ρ sin Φ aρ + ρ2aΦ; evaluate if L is the contour of Figure 3.31. fL A.di
Let A = 4x2 e-yax – 8xe-yay. Determine VX [V(V.A)].
LetDetermine(a) ∇V,(b) ∇ X ∇V,(c) ∇ ∙ ∇V V = sin cos o r
If F = 2ρzap + 3z sin Φ aΦ – 4r cos Φ az, verify Stokes’s theorem for the open surface defined by z = 1, 0 < ρ < 2, 0 < Φ < 45°.
A rigid body spins about a fixed axis through its center with angular velocity ω. If u is the velocity at any point in the body, show that w = 1/2 V X u.
Given that H = 2xzax + 5xyzay + 8(y + z)az, find (a) ∇ · H.(b) ∇ × H.
Let B = r2ar + 4r cos 2θaθ. Find the divergence and curl of B.
For a vector field A and a scalar field V, show in Cartesian coordinates that(a)(b) V. (VVV) = VV²V + |VV|² 2
If B = x2yax + (2x2 + y)ay – (y – z)az, find(a) ∇ ∙ B(b) ∇ × B(c) ∇ (∇ ∙ B)(d) ∇ × ∇ × B
Find ∇2V for each of the following scalar fields:(a) V1 = x3 + y3 + z3(b) V2 = ρz2 sin 2Φ(c) V3 = r2 (1 + cos θ sin Φ)
If r = xax + yay + zaz is the position vector of point (x, y, z), r = ΙrΙ, show that:(a)(b) V(In r) r p²
Find the Laplacian of the following scalar fields and compute the value at the specified point.(a) U = x3y2 exz, (1, –1, 1)(b) V = ρ2z(cos Φ + sin Φ), (5, π/6, –2)(c) W = e–r sin θ cos Φ, (1, π/3, π/6)
(a) If U(x, y, z) = xy2z3, find ∇U and ∇2U.(b) If V(ρ, Φ, z) = sin Φ / ρ, find ∇V and ∇2 V.(c) If W(r, θ, Φ,) = r2 sin θ cos Φ, find ∇W and ∇2W.
In cylindrical coordinates,If G = 2ρ sin Φaρ + 4ρ cos ΦaΦ + (z2 + 1)paz, find = ∇2G. А VA = (VA, -20), + (14+ 2 ) + + Рад = ap 2 дар Аф р² аф p² a+ az раф
Given that V = r2z cos Φ, find ∇V and ∇2V.
If V = 5 cos Φ / r2 , find: (a) ∇V, (b) ∇ ∙ ∇V, (c) ∇ X ∇V.
The electric field due to a line charge is given bywhere l is a constant. Show that E is solenoidal. Show that it is also conservative. E = – λ ρ Σπερ
According to eq. (3.64), ∇ X (∇ X A) = ∇ (∇ ∙ A) – ∇2A. Show that A = xzax + z2ay + yzaz satisfies this vector identity. V²A = V(VA) - VXVXA
Let U = 4xyz2 + 10yz. Show that ∇2U = ∇ ∙ ∇U.
A vector field is given by H = 10 / r2 ar. Show that for any closed path L. ΦΗ Ε H• di = 0
The field of an electric dipole is given bywhere k is a constant. Show that E is conservative. (2cosea, sinoa.) E = k- p3
Consider the following vector fields:A = xax + yay + zazB = 2ρ cos Φaρ – 4ρ sin ΦaΦ + 3azC = sin θ ar + r sin θaΦWhich of these fields are (a) Solenoidal and (b) Irrotational?
Show that the vector field B = (3x2 z + y2)ax + 2xyay + x3az is conservative.
Show that the vector field D = (3ρ + 1) sin Φaz is solenoidal.
Point charges Q1 = 1 nC and Q2 = 2 nC are at a distance apart. Which of the following statements are incorrect?(a) The force on Q1 is repulsive.(b) The force on Q2 is the same in magnitude as that on Q1.(c) As the distance between them decreases, the force on Q1 increases
Plane z = 10 m carries charge 20 nC/m2. The electric field intensity at the origin is(a) –10 az V/m(b) –18π az V/m (c) –72π az V/m(d) –360π az V/m
Point charges 30 nC, 220 nC, and 10 nC are located at 121, 0, 22, 10, 0, 02, and 11, 5, 212, respectively. The total flux leaving a cube of side 6 m centered at the origin is(a) –20 nC(b) 10 nC (c) 20 nC(d) 30 nC(e) 60 nC
A point Q is located at (a, 0, 0), while another charge –Q is at (–a, 0, 0). Find E at:(a) (0, 0, 0) (b) (0, a, 0), (c) (a, 0, a).
The electric flux density on a spherical surface r = b is the same for a point charge Q located at the origin and for charge Q uniformly distributed on surface r = a(a < b).(a) Yes (b) No(c) Not necessarily
Determine the electric field intensity required to levitate a body 2 kg in mass and charged with–4 mC.

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