Gradient - Electromagnetic Theory Questions and Answers - Sanfoundry by staff10 This set of Electromagnetic Theory Multiple Choice Questions & Answers (MCQs) focuses on “Gradient”. 1. Gradient of a function is a constant. State True/False. a) True b) False View Answer Answer: b Explanation: Gradient of any scalar function may be defined as a vector. The vector’s magnitude and direction are those of the maximum space rate of change of φ. 2. The mathematical perception of the gradient is said to be a) Tangent b) Chord c) Slope d) Arc View Answer Answer: c Explanation: The gradient is the rate of change of space of flux in electromagnetics. This is analogous to the slope in mathematics. 3. Divergence of gradient of a vector function is equivalent to a) Laplacian operation b) Curl operation c) Double gradient operation d) Null vector View Answer Answer: a Explanation: Div (Grad V) = (Del)2V, which is the Laplacian operation. A function is said to be harmonic in nature, when its Laplacian tends to zero. 4. The gradient of xi + yj + zk is a) 0 b) 1 c) 2

d) 3 View Answer Answer: d Explanation: Grad (xi + yj + zk) = 1 + 1 + 1 = 3. In other words, the gradient of any position vector is 3. 5. Find the gradient of t = x2y+ ez at the point p(1,5,-2) a) i + 10j + 0.135k b) 10i + j + 0.135k c) i + 0.135j + 10k d) 10i + 0.135j + k View Answer Answer: b Explanation: Grad(t) = 2xy i + x2 j + ez k. On substituting p(1,5,-2), we get 10i + j + 0.135k. 6. Curl of gradient of a vector is a) Unity b) Zero c) Null vector d) Depends on the constants of the vector View Answer Answer: c Explanation: Gradient of any function leads to a vector. Similarly curl of that vector gives another vector, which is always zero for all constants of the vector. A zero value in vector is always termed as null vector(not simply a zero). 7. Find the gradient of the function given by, x2 + y2 + z2 at (1,1,1) a) i + j + k b) 2i + 2j + 2k c) 2xi + 2yj + 2zk d) 4xi + 2yj + 4zk View Answer Answer: b Explanation: Grad(x2+y2+z2) = 2xi + 2yj + 2zk. Put x=1, y=1, z=1, the gradient will be 2i + 2j + 2k. 8. The gradient can be replaced by which of the following? a) Maxwell equation b) Volume integral c) Differential equation d) Surface integral View Answer

Answer: c Explanation: Since gradient is the maximum space rate of change of flux, it can be replaced by differential equations. 9. When gradient of a function is zero, the function lies parallel to the x-axis. State True/False. a) True b) False View Answer Answer: a Explanation: Gradient of a function is zero implies slope is zero. When slope is zero, the function will be parallel to x-axis or y value is constant. 10. Find the gradient of the function sin x + cos y. a) cos x i – sin y j b) cos x i + sin y j c) sin x i – cos y j d) sin x i + cos y j View Answer Answer: a Explanation: Grad (sin x + cos y) gives partial differentiation of sin x+ cos y with respect to x and partial differentiation of sin x + cos y with respect to y and similarly with respect to z. This gives cos x i – sin y j + 0 k = cos x i – sin y j.

Divergence - Electromagnetic Theory Questions and Answers - Sanfoundry by staff10 This set of Electromagnetic Theory Multiple Choice Questions & Answers (MCQs) focuses on “Divergence”. 1. The divergence of a vector is a scalar. State True/False. a) True b) False View Answer Answer: a Explanation: Divergence can be computed only for a vector. Since it is the measure of outward flow of flux from a small closed surface as the volume shrinks to zero, the result will be directionless (scalar).

2. The divergence concept can be illustrated using Pascal’s law. State True/False. a) True b) False View Answer Answer: a Explanation: Consider the illustration of Pascal’s law, wherein a ball is pricked with holes all over its body. After water is filled in it and pressure is applied on it, the water flows out the holes uniformly. This is analogous to the flux flowing outside a closed surface as the volume reduces. 3. Compute the divergence of the vector xi + yj + zk. a) 0 b) 1 c) 2 d) 3 View Answer Answer: d Explanation: The vector given is a position vector. The divergence of any position vector is always 3. 4. Find the divergence of the vector yi + zj + xk. a) -1 b) 0 c) 1 d) 3 View Answer Answer: b Explanation: Div (yi + zj + xk) = Dx(y) + Dy(z) + Dz(x), which is zero. Here D refers to partial differentiation. 5. Given D = e-xsin y i – e-xcos y j Find divergence of D. a) 3 b) 2 c) 1 d) 0 View Answer Answer: d Explanation: Div (D) = Dx(e-xsin y) + Dy(-e-xcos y ) = -e-xsin y + e-xsin y = 0. 6. Find the divergence of the vector F= xe-x i + y j – xz k a) (1 – x)(1 + e-x) b) (x – 1)(1 + e-x)

c) (1 – x)(1 – e) d) (x – 1)(1 – e) View Answer Answer: a Explanation: Div(F) = Dx(xe-x) + Dy(y)+Dz(-xz) = -xe-x + e-x + 1 – x = e-x(1 – x) + (1 – x) = (1 – x)(1 + e-x). 7. Determine the divergence of F = 30 i + 2xy j + 5xz2 k at (1,1,-0.2) and state the nature of the field. a) 1, solenoidal b) 0, solenoidal c) 1, divergent d) 0, divergent View Answer Answer: b Explanation: Div(F) = Dx(30) + Dy(2xy) + Dz(5xz2) = 0 + 2x + 10xz = 2x + 10xz Divergence at (1,1,-0.2) will give zero. As the divergence is zero, field is solenoidal. Alternate/Shortcut: Without calculation, we can easily choose option b, as by theory when the divergence is zero, the vector is solenoidal. Option b is the only one which is satisfying this condition. 8. Find whether the vector is solenoidal, E = yz i + xz j + xy k a) Yes, solenoidal b) No, non-solenoidal c) Solenoidal with negative divergence d) Variable divergence View Answer Answer: a Explanation: Div(E) = Dx(yz) + Dy(xz) + Dz(xy) = 0. The divergence is zero, thus vector is divergentless or solenoidal. 9. Find the divergence of the field, P = x2yz i + xz k a) xyz + 2x b) 2xyz + x c) xyz + 2z d) 2xyz + z View Answer Answer: b Explanation: Div(P) = Dx(x2yz) + Dy(0) + Dz(xz) = 2xyz + x, which is option b. For different values of x,y,z the divergence of the field varies.

10. Identify the nature of the field, if the divergence is zero and curl is also zero. a) Solenoidal, irrotational b) Divergent, rotational c) Solenoidal, irrotational d) Divergent, rotational View Answer Answer: c Explanation: Since the vector field does not diverge (moves in a straight path), the divergence is zero. Also, the path does not possess any curls, so the field is irrotational.

Curl - Electromagnetic Theory Questions and Answers - Sanfoundry by staff10 This set of Electromagnetic Theory Multiple Choice Questions & Answers (MCQs) focuses on “Curl”. 1. Curl is defined as the angular velocity at every point of the vector field. State True/False. a) True b) False View Answer Answer: a Explanation: Curl is defined as the circulation of a vector per unit area. It is the cross product of the del operator and any vector field. Circulation implies the angular at every point of the vector field. It is obtained by multiplying the component of the vector parallel to the specified closed path at each point along it, by the differential path length and summing the results. 2. The curl of curl of a vector is given by, a) Div(Grad V) – (Del)2V b) Grad(Div V) – (Del)2V c) (Del)2V – Div(Grad V) d) (Del)2V – Grad(Div V) View Answer Answer: b Explanation: Curl (Curl V) = Grad (Div V) – (Del)2V is a standard result of the curl operation. 3. Which of the following theorem use the curl operation? a) Green’s theorem b) Gauss Divergence theorem c) Stoke’s theorem

d) Maxwell equation View Answer Answer: c Explanation: The Stoke’s theorem is given by ∫ A.dl = ∫Curl(A).ds, which uses the curl operation. There can be confusion with Maxwell equation also, but it uses curl in electromagnetics specifically, whereas the Stoke’s theorem uses it in a generalised manner. Thus the best option is c. 4. The curl of a curl of a vector gives a a) Scalar b) Vector c) Zero value d) Non zero value View Answer Answer: b Explanation: Curl is always defined for vectors only. The curl of a vector is a vector only. The curl of the resultant vector is also a vector only. 5. Find the curl of the vector and state its nature at (1,1,-0.2) F = 30 i + 2xy j + 5xz2 k a) √4.01 b) √4.02 c) √4.03 d) √4.04 View Answer Answer: d Explanation: Curl F = -5z2 j + 2y k. At (1,1,-0.2), Curl F = -0.2 j + 2 k. |Curl F| = √(-0.22+22) = √4.04. 6. Is the vector is irrotational. E = yz i + xz j + xy k a) Yes b) No View Answer Answer: a Explanation: Curl E = i(Dy(xy) – Dz(xz)) – j (Dx(xy) – Dz(yz)) + k(Dx(xz) – Dy(yz)) = i(x – x) – j(y – y) + k(z – z) = 0 Since the curl is zero, the vector is irrotational or curl-free. 7. Find the curl of A = (y cos ax)i + (y + ex)k a) 2i – ex j – cos ax k b) i – ex j – cos ax k c) 2i – ex j + cos ax k

d) i – ex j + cos ax k View Answer Answer: b Explanation: Curl A = i(Dy(y + ex)) – j (Dx(y + ex) – Dz(y cos ax)) + k(-Dy(y cos ax)) = 1.i – j(ex) – k cos ax = i – ex j – cos ax k. 8. Find the curl of the vector A = yz i + 4xy j + y k a) xi + j + (4y – z)k b) xi + yj + (z – 4y)k c) i + j + (4y – z)k d) i + yj + (4y – z)k View Answer Answer: d Explanation: Curl A = i(Dy(y) – Dz(0)) – j (Dx(0) – Dz(yz)) + k(Dx(4xy) – Dy(yz)) = i + y j + (4y – z)k, which is option d. 9. Curl cannot be employed in which one of the following? a) Directional coupler b) Magic Tee c) Isolator and Terminator d) Waveguides View Answer Answer: d Explanation: In the options a, b, c, the EM waves travel both in linear and angular motion, which involves curl too. But in waveguides, as the name suggests, only guided propagation occurs (no bending or curl of waves). 10. Which of the following Maxwell equations use curl operation? a) Maxwell 1st and 2nd equation b) Maxwell 3rd and 4th equation c) All the four equations d) None of the equations View Answer Answer: a Explanation: Maxwell 1st equation, Curl (H) = J (Ampere law) Maxwell 2nd equation, Curl (E) = -D(B)/Dt (Faraday’s law) Maxwell 3rd equation, Div (D) = Q (Gauss law for electric field) Maxwell 4th equation, Div (B) = 0(Gauss law for magnetic field) It is clear that only 1st and 2nd equations use the curl operation.

Line Integral - Electromagnetic Theory Questions and Answers - Sanfoundry by staff10 This set of Electromagnetic Theory Multiple Choice Questions & Answers (MCQs) focuses on “Line Integral”. 1. An electric field is given as E = 6y2z i + 12xyz j + 6xy2 k. An incremental path is given by dl = -3 i + 5 j – 2 k mm. The work done in moving a 2mC charge along the path if the location of the path is at p(0,2,5) is (in Joule) a) 0.64 b) 0.72 c) 0.78 d) 0.80 View Answer Answer: b Explanation: W = -Q E.dl W = -2 X 10-3 X (6y2z i + 12xyz j + 6xy2 k) . (-3 i + 5 j -2 k) At p(0,2,5), W = -2(-18.22.5) X 10-3 = 0.72 J. 2. The integral form of potential and field relation is given by line integral. State True/False a) True b) False View Answer Answer: a Explanation: Vab = -∫ E.dl is the relation between potential and field. It is clear that it is given by line integral. 3. If V = 2x2y – 5z, find its electric field at point (-4,3,6) a) 47.905 b) 57.905 c) 67.905 d) 77.905 View Answer Answer: b Explanation: E = -Grad (V) = -4xy i – 2×2 j + 5k At (-4,3,6), E = 48 i – 32 j + 5 k, |E| = √3353 = 57.905 units. 4. Find the potential between two points p(1,-1,0) and q(2,1,3) with E = 40xy i + 20x2 j + 2 k a) 104

b) 105 c) 106 d) 107 View Answer Answer: c Explanation: V = -∫ E.dl = -∫ (40xy dx + 20x2 dy + 2 dz) , from q to p. On integrating, we get 106 volts. 5. Find the potential between a(-7,2,1) and b(4,1,2). Given E = (-6y/x2 )i + ( 6/x) j + 5 k. a) -8.014 b) -8.114 c) -8.214 d) -8.314 View Answer Answer: c Explanation: V = -∫ E.dl = -∫ (-6y/x2 )dx + ( 6/x)dy + 5 dz, from b to a. On integrating, we get -8.214 volts. 6. The potential of a uniformly charged line with density λ is given by, λ/(2πε) ln(b/a). State True/False. a) True b) False View Answer Answer: a Explanation: The electric field intensity is given by, E = λ/(2πεr) Vab = -∫ E.dr = -∫ λ/(2πεr). On integrating from b to a, we get λ/(2πε) ln(b/a). 7. A field in which a test charge around any closed surface in static path is zero is called a) Solenoidal b) Rotational c) Irrotational d) Conservative View Answer Answer: d Explanation: Work done in moving a charge in a closed path is zero. It is expressed as, ∫ E.dl = 0. The field having this property is called conservative or lamellar field. 8. The potential in a lamellar field is a) 1 b) 0 c) -1

d) ∞ View Answer Answer: b Explanation: Work done in a lamellar field is zero. ∫ E.dl = 0,thus ∑V = 0. The potential will be zero. 9. Line integral is used to calculate a) Force b) Area c) Volume d) Length View Answer Answer: d Explanation: Length is a linear quantity, whereas area is two dimensional and volume is three dimensional. Thus single or line integral can be used to find length in general. 10. The energy stored in the inductor 100mH with a current of 2A is a) 0.2 b) 0.4 c) 0.6 d) 0.8 View Answer Answer: a Explanation: dw = ei dt = Li di, W = L∫ i.di Energy E = 0.5LI2 = 0.5 X 0.1 X 22 = 0.2 Joule.

Surface Integral - Electromagnetic Theory Questions and Answers - Sanfoundry by staff10 This set of Electromagnetic Theory Multiple Choice Questions & Answers (MCQs) focuses on “Surface Integral”. 1. Gauss law for electric field uses surface integral. State True/False a) True b) False View Answer

Answer: a Explanation: Gauss law states that the electric flux ing through any closed surface is equal to the total charge enclosed by the surface. Thus the charge is defined as a surface integral. 2. Surface integral is used to compute a) Surface b) Area c) Volume d) density View Answer Answer: b Explanation: Surface integral is used to compute area, which is the product of two quantities length and breadth. Thus it is two dimensional integral. 3. Coulomb’s law can be derived from Gauss law. State True/ False a) True b) False View Answer Answer: a Explanation: Gauss law, Q = ∫∫D.ds By considering area of a sphere, ds = r2sin θ dθ dφ. On integrating, we get Q = 4πr2D and D = εE, where E = F/Q. Thus, we get Coulomb’s law F = Q1 x Q2/4∏εR2. 4. Evaluate Gauss law for D = 5r2/4 i in spherical coordinates with r = 4m and θ = π/2. a) 600 b) 599.8 c) 588.9 d) 577.8 View Answer Answer: c Explanation: ∫∫ ( 5r2/4) . (r2 sin θ dθ dφ), which is the integral to be evaluated. Put r = 4m and substitute θ = 0→ π/4 and φ = 0→ 2π, the integral evaluates to 588.9. 5. Compute the Gauss law for D= 10ρ3/4 i, in cylindrical coordinates with ρ= 4m, z=0 and z=5. a) 6100 π b) 6200 π c) 6300 π d) 6400 π View Answer

Answer: d Explanation: ∫∫ D.ds = ∫∫ (10ρ3/4).(ρ dφ dz), which is the integral to be evaluated. Put ρ = 4m, z = 0→5 and φ = 0→2π, the integral evaluates to 6400π. 6. Compute divergence theorem for D= 5r2/4 i in spherical coordinates between r=1 and r=2. a) 80π b) 5π c) 75π d) 85π View Answer Answer: c Explanation: ∫∫ ( 5r2/4) . (r2 sin θ dθ dφ), which is the integral to be evaluated. Since it is double integral, we need to keep only two variables and one constant compulsorily. Evaluate it as two integrals keeping r = 1 for the first integral and r = 2 for the second integral, with φ = 0→2π and θ = 0→ π. The first integral value is 80π, whereas second integral gives -5π. On summing both integrals, we get 75π. 7. Find the value of divergence theorem for A = xy2 i + y3 j + y2z k for a cuboid given by 0<x<1, 0 <1 and 0 <1. a) 1 b) 4/3 c) 5/3 d) 2 View Answer Answer: c Explanation: A cuboid has six faces. ∫∫A.ds = ∫∫Ax=0 dy dz + ∫∫Ax=1 dy dz + ∫∫Ay=0 dx dz + ∫∫Ay=1 dx dz + ∫∫Az=0 dy dx + ∫∫Az=1 dy dx. Substituting A and integrating we get (1/3) + 1 + (1/3) = 5/3. 8. The ultimate result of the divergence theorem evaluates which one of the following? a) Field intensity b) Field density c) Potential d) Charge and flux View Answer Answer: d Explanation: Gauss law states that the electric flux ing through any closed surface is equal to the total charge enclosed by the surface. Thus, it is given by, ψ = ∫∫ D.ds= Q, where the divergence theorem computes the charge and flux, which are both the same. 9. Find the value of divergence theorem for the field D = 2xy i + x2 j for the rectangular parallelepiped given by x = 0 and 1, y = 0 and 2, z = 0 and 3. a) 10

b) 12 c) 14 d) 16 View Answer Answer: b Explanation: While evaluating surface integral, there has to be two variables and one constant compulsorily. ∫∫D.ds = ∫∫Dx=0 dy dz + ∫∫Dx=1 dy dz + ∫∫Dy=0 dx dz + ∫∫Dy=2 dx dz + ∫∫Dz=0 dy dx + ∫∫Dz=3 dy dx. Put D in equation, the integral value we get is 12. 10. If D = 2xy i + 3yz j + 4xz k, how much flux es through x = 3 plane for which -1 <2 and 0 <4? a) 12 b) 24 c) 36 d) 48 View Answer Answer: c Explanation: By Gauss law, ψ = ∫∫ D.ds, where ds = dydz i at the x-plane. Put x = 3 and integrate at -1 <2 and 0 <4, we get 12 X 3 = 36.

Volume Integral - Electromagnetic Theory Questions and Answers - Sanfoundry by staff10 This set of Electromagnetic Theory Multiple Choice Questions & Answers (MCQs) focuses on “Volume Integral”. 1. The divergence theorem converts a) Line to surface integral b) Surface to volume integral c) Volume to line integral d) Surface to line integral View Answer Answer: b Explanation: The divergence theorem is given by, ∫∫ D.ds = ∫∫∫ Div (D) dv. It is clear that it converts surface (double) integral to volume(triple) integral. 2. The triple integral is used to compute volume. State True/False a) True

b) False View Answer Answer: a Explanation: The triple integral, as the name suggests integrates the function/quantity three times. This gives volume which is the product of three independent quantities. 3. The volume integral is three dimensional. State True/False a) True b) False View Answer Answer: a Explanation: Volume integral integrates the independent quantities by three times. Thus it is said to be three dimensional integral or triple integral. 4. Find the charged enclosed by a sphere of charge density ρ and radius a. a) ρ (4πa2) b) ρ(4πa3/3) c) ρ(2πa2) d) ρ(2πa3/3) View Answer Answer: b Explanation: The charge enclosed by the sphere is Q = ∫∫∫ ρ dv. Where, dv = r2 sin θ dr dθ dφ and on integrating with r = 0->a, φ = 0->2π and θ = 0->π, we get Q = ρ(4πa3/3). 5. Evaluate Gauss law for D = 5r2/4 i in spherical coordinates with r = 4m and θ = π/2 as volume integral. a) 600 b) 588.9 c) 577.8 d) 599.7 View Answer Answer: b Explanation: ∫∫ D.ds = ∫∫∫ Div (D) dv, where RHS needs to be computed. The divergence of D given is, Div(D) = 5r and dv = r2 sin θ dr dθ dφ. On integrating, r = 0->4, φ = 0->2π and θ = 0->π/4, we get Q = 588.9. 6. Compute divergence theorem for D = 5r2/4 i in spherical coordinates between r = 1 and r = 2 in volume integral. a) 80 π b) 5 π c) 75 π

d) 85 π View Answer Answer: c Explanation: D.ds = ∫∫∫ Div (D) dv, where RHS needs to be computed. The divergence of D given is, Div(D) = 5r and dv = r2 sin θ dr dθ dφ. On integrating, r = 1->2, φ = 0->2π and θ = 0->π, we get Q = 75 π. 7. Compute the Gauss law for D = 10ρ3/4 i, in cylindrical coordinates with ρ = 4m, z = 0 and z = 5, hence find charge using volume integral. a) 6100 π b) 6200 π c) 6300 π d) 6400 π View Answer Answer: d Explanation: Q = D.ds = ∫∫∫ Div (D) dv, where RHS needs to be computed. The divergence of D given is, Div(D) = 10 ρ2 and dv = ρ dρ dφ dz. On integrating, ρ = 0->4, φ = 0->2π and z = 0->5, we get Q = 6400 π. 8. Using volume integral, which quantity can be calculated? a) area of cube b) area of cuboid c) volume of cube d) distance of vector View Answer Answer: c Explanation: The volume integral gives the volume of a vector in a region. Thus volume of a cube can be computed. 9. Compute the charge enclosed by a cube of 2m each edge centered at the origin and with the edges parallel to the axes. Given D = 10y3/3 j. a) 20 b) 70/3 c) 80/3 d) 30 View Answer Answer: c Explanation: Div(D) = 10y2 ∫∫∫Div (D) dv = ∫∫∫ 10y2 dx dy dz. On integrating, x = -1->1, y = -1->1 and z = -1->1, we get Q = 80/3.

10. Find the value of divergence theorem for the field D = 2xy i + x2 j for the rectangular parallelepiped given by x = 0 and 1, y = 0 and 2, z = 0 and 3. a) 10 b) 12 c) 14 d) 16 View Answer Answer: b Explanation: Div (D) = 2y ∫∫∫Div (D) dv = ∫∫∫ 2y dx dy dz. On integrating, x = 0->1, y = 0->2 and z = 0->3, we get Q = 12.

Electromagnetic Theory Interview Questions and Answers - Sanfoundry by staff10 This set of Electromagnetic Theory Interview Questions and Answers focuses on “Laplacian Operator”. 1. The point form of Gauss law is given by, Div(V) = ρv State True/False. a) True b) False View Answer Answer: a Explanation: The integral form of Gauss law is ∫∫∫ ρv dv = V. Thus differential or point form will be Div(V) = ρv. 2. If a function is said to be harmonic, then a) Curl(Grad V) = 0 b) Div(Curl V) = 0 c) Div(Grad V) = 0 d) Grad(Curl V) = 0 View Answer Answer: c Explanation: Though option a & b are also correct, for harmonic fields, the Laplacian of electric potential is zero. Now, Laplacian refers to Div(Grad V), which is zero for harmonic fields. 3. The Poisson equation cannot be determined from Laplace equation. State True/False. a) True

b) False View Answer Answer: b Explanation: The Poisson equation is a general case for Laplace equation. If volume charge density exists for a field, then (Del)2V= -ρv/ε, which is called Poisson equation. 4. Given the potential V = 25 sin θ, in free space, determine whether V satisfies Laplace’s equation. a) Yes b) No c) Data sufficient d) Potential is not defined View Answer Answer: a Explanation: (Del)2V = 0 (Del)2V = (Del)2(25 sin θ), which is not equal to zero. Thus the field does not satisfy Laplace equation. 5. If a potential V is 2V at x = 1mm and is zero at x=0 and volume charge density is -106εo, constant throughout the free space region between x = 0 and x = 1mm. Calculate V at x = 0.5mm. a) 0.875 b) 0.675 c) 0.475 d) 0.275 View Answer Answer: d Explanation: Del2(V) = -ρv/εo= +106 On integrating twice with respect to x, V = 106. (x2/2) + C1x + C2. Substitute the boundary conditions, x = 0, V = 0 and x = 1mm, V = 2V in V, C1 = 1500 and C2 = 0. At x = 0.5mm, we get, V = 0.875V. 6. Find the Laplace equation value of the following potential field V = x2 – y2 + z2 a) 0 b) 2 c) 4 d) 6 View Answer Answer: b Explanation: (Del) V = 2x – 2y + 2z (Del)2 V = 2 – 2 + 2= 2, which is non zero value. Thus it doesn’t satisfy Laplace equation.

7. Find the Laplace equation value of the following potential field V = ρ cosφ + z a) 0 b) 1 c) 2 d) 3 View Answer Answer: a Explanation: (Del)2 (ρ cosφ + z)= (cos φ/r) – (cos φ/r) + 0 = 0, this satisfies Laplace equation. The value is 0. 8. Find the Laplace equation value of the following potential field V = r cos θ + φ a) 3 b) 2 c) 1 d) 0 View Answer Answer: d Explanation: (Del)2 (r cos θ + φ) = (2 cosθ/r) – (2 cosθ/r) + 0 = 0, this satisfies Laplace equation. This value is 0. 9. The Laplacian operator cannot be used in which one the following? a) Two dimensional heat equation b) Two dimensional wave equation c) Poisson equation d) Maxwell equation View Answer Answer: d Explanation: The first three options are general cases of Laplacian equation. Maxwell equation uses only divergence and curl, which is first order differential equation, whereas Laplacian operator is second order differential equation. Thus Maxwell equation will not employ Laplacian operator. 10. When a potential satisfies Laplace equation, then it is said to be a) Solenoidal b) Divergent c) Lamellar d) Harmonic View Answer Answer: d Explanation: A field satisfying the Laplace equation is termed as harmonic field.

Stokes Theorem - Electromagnetic Theory Questions and Answers - Sanfoundry by staff10 This set of Electromagnetic Theory Multiple Choice Questions & Answers (MCQs) focuses on “Stoke’s Theorem”. 1. Find the value of Stoke’s theorem for y i + z j + x k. a) i + j b) j + k c) i + j + k d) –i – j – k View Answer Answer: d Explanation: The curl of y i + z j + x k is i(0-1) – j(1-0) + k(0-1) = -i –j –k. Since the curl is zero, the value of Stoke’s theorem is zero. The function is said to be irrotational. 2. The Stoke’s theorem uses which of the following operation? a) Divergence b) Gradient c) Curl d) Laplacian View Answer Answer: c Explanation: ∫A.dl = ∫∫ Curl (A).ds is the expression for Stoke’s theorem. It is clear that the theorem uses curl operation. 3. Which of the following theorem convert line integral to surface integral? a) Gauss divergence and Stoke’s theorem b) Stoke’s theorem only c) Green’ s theorem only d) Stoke’s and Green’s theorem View Answer Answer: d Explanation: The Stoke’s theorem is given by ∫A.dl = ∫∫ Curl (A).ds. Green’s theorem is given by, ∫ F dx + G dy = ∫∫ (dG/dx – dF/dy) dx dy. It is clear that both the theorems convert line to surface integral.

4. Find the value of Stoke’s theorem for A = x i + y j + z k. The state of the function will be a) Solenoidal b) Divergent c) Rotational d) Curl free View Answer Answer: Since curl is required, we need not bother about divergence property. The curl of the function will be i(0-0) – j(0-0) + k(0-0) = 0. The curl is zero, thus the function is said to be irrotational or curl free. 5. The Stoke’s theorem can be used to find which of the following? a) Area enclosed by a function in the given region b) Volume enclosed by a function in the given region c) Linear distance d) Curl of the function View Answer Answer: a Explanation: It states that the line integral of a function gives the surface area of the function enclosed by the given region. This is computed using the double integral of the curl of the function. 6. The energy stored in an inductor 2H and current 4A is a) 4 b) 8 c) 12 d) 16 View Answer Answer: d Explanation: From Stoke’s theorem, we can calculate energy stored in an inductor as 0.5Li2. E = 0.5 X 2 X 42 = 16 units. 7. The voltage of a capacitor 12F with a rating of 2J energy is a) 0.57 b) 5.7 c) 57 d) 570 View Answer Answer: a Explanation: We can compute the energy stored in a capacitor from Stoke’s theorem as 0.5Cv2. Thus given energy is 0.5 X 12 X v2. We get v = 0.57 volts.

8. Find the power, given energy E = 2J and current density J = x2 varies from x = 0 and x = 1. a) 1/3 b) 2/3 c) 1 d) 4/3 View Answer Answer: b Explanation: From Stoke’s theorem, we can calculate P = E X I = ∫ E. J ds = 2∫ x2 dx as x = 0->1. We get P = 2/3 units. 9. The conductivity of a material with current density 1 unit and electric field 200 μV is a) 2000 b) 3000 c) 4000 d) 5000 View Answer Answer: d Explanation: The current density is given by, J = σE. To find conductivity, σ = J/E = 1/200 X 106 = 5000. 10. The resistivity of a material with resistance 200 ohm, length 10m and area twice that of the length is a) 200 b) 300 c) 400 d) 500 View Answer Answer: c Explanation: Resistance calculated from Ohm’s law and Stoke’s theorem will be R = ρL/A. To get resistivity, ρ = RA/L = 200 X 20/10 = 400.

Green's Theorem - Electromagnetic Theory Questions and Answers - Sanfoundry by staff10 This set of Electromagnetic Theory Multiple Choice Questions & Answers (MCQs) focuses on “Green’s Theorem”. 1. Mathematically, the functions in Green’s theorem will be a) Continuous derivatives

b) Discrete derivatives c) Continuous partial derivatives d) Discrete partial derivatives View Answer Answer: c Explanation: The Green’s theorem states that if L and M are functions of (x,y) in an open region containing D and having continuous partial derivatives then, ∫ (F dx + G dy) = ∫∫(dG/dx – dF/dy)dx dy, with path taken anticlockwise. 2. Find the value of Green’s theorem for F = x2 and G = y2 is a) 0 b) 1 c) 2 d) 3 View Answer Answer: a Explanation: ∫∫(dG/dx – dF/dy)dx dy = ∫∫(0 – 0)dx dy = 0. The value of Green’s theorem gives zero for the functions given. 3. Which of the following is not an application of Green’s theorem? a) Solving two dimensional flow integrals b) Area surveying c) Volume of plane figures d) Centroid of plane figures View Answer Answer: c Explanation: In physics, Green’s theorem is used to find the two dimensional flow integrals. In plane geometry, it is used to find the area and centroid of plane figures. 4. The path traversal in calculating the Green’s theorem is a) Clockwise b) Anticlockwise c) Inwards d) Outwards View Answer Answer: b Explanation: The Green’s theorem calculates the area traversed by the functions in the region in the anticlockwise direction. This converts the line integral to surface integral. 5. Calculate the Green’s value for the functions F = y2 and G = x2 for the region x = 1 and y = 2 from origin. a) 0

b) 2 c) -2 d) 1 View Answer Answer: c Explanation: ∫∫(dG/dx – dF/dy)dx dy = ∫∫(2x – 2y)dx dy. On integrating for x = 0->1 and y = 0>2, we get Green’s value as -2. 6. If two functions A and B are discrete, their Green’s value for a region of circle of radius a in the positive quadrant is a) ∞ b) -∞ c) 0 d) Does not exist View Answer Answer: d Explanation: Green’s theorem is valid only for continuous functions. Since the given functions are discrete, the theorem is invalid or does not exist. 7. Applications of Green’s theorem are meant to be in a) One dimensional b) Two dimensional c) Three dimensional d) Four dimensional View Answer Answer: b Explanation: Since Green’s theorem converts line integral to surface integral, we get the value as two dimensional. In other words the functions are variable with respect to x,y, which is two dimensional. 8. The Green’s theorem can be related to which of the following theorems mathematically? a) Gauss divergence theorem b) Stoke’s theorem c) Euler’s theorem d) Leibnitz’s theorem View Answer Answer: b Explanation: The Green’s theorem is a special case of the Kelvin- Stokes theorem, when applied to a region in the x-y plane. It is a widely used theorem in mathematics and physics. 9. The Shoelace formula is a shortcut for the Green’s theorem. State True/False. a) True

b) False View Answer Answer: a Explanation: The Shoelace theorem is used to find the area of polygon using cross multiples. This can be verified by dividing the polygon into triangles. It is a special case of Green’s theorem. 10. Find the area of a right angled triangle with sides of 90 degree unit and the functions described by L = cos y and M = sin x. a) 0 b) 45 c) 90 d) 180 View Answer Answer: d Explanation: dM/dx = cos x and dL/dy = -sin y ∫∫(dM/dx – dL/dy)dx dy = ∫∫ (cos x + sin y)dx dy. On integrating with x = 0->90 and y = 0->90, we get area of right angled triangle as -180 units (taken in clockwise direction). Since area cannot be negative, we take 180 units.

Electromagnetic Theory Questions and Answers for Freshers - Sanfoundry by staff10 This set of Electromagnetic Theory Questions and Answers for Freshers focuses on “Gauss Divergence Theorem”. 1. Gauss theorem uses which of the following operations? a) Gradient b) Curl c) Divergence d) Laplacian View Answer Answer: c Explanation: The Gauss divergence theorem uses divergence operator to convert surface to volume integral. It is used to calculate the volume of the function enclosing the region given. 2. Evaluate the surface integral ∫∫ (3x i + 2y j). dS, where S is the sphere given by x 2 + y2 + z2 = 9. a) 120π

b) 180π c) 240π d) 300π View Answer Answer: b Explanation: We could parameterise surface and find surface integral, but it is wise to use divergence theorem to get faster results. The divergence theorem is given by ∫∫ F.dS = ∫∫∫ Div (F).dV Div (3x i + 2y j) = 3 + 2 = 5. Now the volume integral will be ∫∫∫ 5.dV, where dV is the volume of the sphere 4πr3/3 and r = 3units.Thus we get 180π. 3. The Gauss divergence theorem converts a) line to surface integral b) line to volume integral c) surface to line integral d) surface to volume integral View Answer Answer: d Explanation: The divergence theorem for a function F is given by ∫∫ F.dS = ∫∫∫ Div (F).dV. Thus it converts surface to volume integral. 4. The divergence theorem for a surface consisting of a sphere is computed in which coordinate system? a) Cartesian b) Cylindrical c) Spherical d) Depends on the function View Answer Answer: d Explanation: Seeing the surface as sphere, we would immediately choose spherical system, but it is wrong. The divergence operation is performed in that coordinate system in which the function belongs to. It is independent of the surface region. 5. Find the Gauss value for a position vector in Cartesian system from the origin to one unit in three dimensions. a) 0 b) 3 c) -3 d) 1 View Answer Answer: b Explanation: The position vector in Cartesian system is given by R = x i + y j + z k. Div(R) = 1 +

1 + 1 = 3. By divergence theorem, ∫∫∫3.dV, where V is a cube with x = 0->1, y = 0->1 and z = 0>1. On integrating, we get 3 units. 6. The divergence theorem value for the function x2 + y2 + z2 at a distance of one unit from the origin is a) 0 b) 1 c) 2 d) 3 View Answer Answer: d Explanation: Div (F) = 2x + 2y + 2z. The triple integral of the divergence of the function is ∫∫∫(2x + 2y + 2z)dx dy dz, where x = 0->1, y = 0->1 and z = 0->1. On integrating, we get 3 units. 7. If a function is described by F = (3x + z, y2 − sin x2z, xz + yex5), then the divergence theorem value in the region 0<x<1, 0 <3 and 0 <2 will be a) 13 b) 26 c) 39 d) 51 View Answer Answer: c Explanation: Div (F) = 3 + 2y + x. By divergence theorem, the triple integral of Div F in the region is ∫∫∫ (3 + 2y + x) dx dy dz. On integrating from x = 0->1, y = 0->3 and z = 0->2, we get 39 units. 8. Find the divergence theorem value for the function given by (ez, sin x, y2) a) 1 b) 0 c) -1 d) 2 View Answer Answer: b Explanation: Since the divergence of the function is zero, the triple integral leads to zero. The Gauss theorem gives zero value. 9. For a function given by F = 4x i + 7y j +z k, the divergence theorem evaluates to which of the values given, if the surface considered is a cone of radius 1/2π m and height 4π2 m. a) 1 b) 2 c) 3 d) 4 View Answer

Answer: b Explanation: Div (F) = 4 + 7 + 1 = 12. The divergence theorem gives ∫∫∫(12).dV, where dV is the volume of the cone πr3h/3, where r = 1/2π m and h = 4π2 m. On substituting the radius and height in the triple integral, we get 2 units. 10. Divergence theorem computes to zero for a solenoidal function. State True/False. a) True b) False View Answer Answer: a Explanation: The divergence theorem is given by, ∫∫ F.dS = ∫∫∫ Div (F).dV, for a function F. If the function is solenoidal, its divergence will be zero. Thus the theorem computes to zero.