Questions and Answers of Elements Of Electromagnetics

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
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
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
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 <
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
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
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°,
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
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,
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
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
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
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
(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
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
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
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)
Determine the electric field intensity required to levitate a body 2 kg in mass and charged with–4 mC.

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