Q.1 What is the sum of vectors \(\mathbf{a}=\langle 3,2,1\rangle\) and \(\mathbf{b}=\langle -1,4,0\rangle\)?
\(\langle 2,6,1\rangle\)
\(\langle 4,6,1\rangle\)
\(\langle 2,-2,1\rangle\)
\(\langle 4,6,1\rangle\)
Explanation - Vector addition is performed component‑wise: (3+(-1), 2+4, 1+0) = (2,6,1).
Correct answer is: \(\langle 2,6,1\rangle\)
Q.2 What is the divergence of the constant vector field \(\mathbf{F}=\langle 5,5,5\rangle\)?
15
0
5
Infinity
Explanation - The divergence of a constant field is the sum of the derivatives of its components, all zero: ∂/∂x(5)+∂/∂y(5)+∂/∂z(5)=0.
Correct answer is: 0
Q.3 What is the curl of the constant vector field \(\mathbf{F}=\langle 2,3,4\rangle\)?
\(\langle 0,0,0\rangle\)
\(\langle 2,3,4\rangle\)
\(\langle -2,-3,-4\rangle\)
\(\langle 1,1,1\rangle\)
Explanation - The curl of a constant vector field is zero because all partial derivatives of its components vanish.
Correct answer is: \(\langle 0,0,0\rangle\)
Q.4 Find the gradient of the scalar function \(f(x,y)=x^2+y^2\).
\(\langle 2x,2y\rangle\)
\(\langle 2y,2x\rangle\)
\(\langle x^2,y^2\rangle\)
\(\langle 2x,0\rangle\)
Explanation - ∇f = (∂f/∂x, ∂f/∂y) = (2x, 2y).
Correct answer is: \(\langle 2x,2y\rangle\)
Q.5 What is the divergence of \(\mathbf{F}=\langle x,y,z\rangle\)?
3
x+y+z
0
1
Explanation - ∇·F = ∂x/∂x + ∂y/∂y + ∂z/∂z = 1+1+1 = 3.
Correct answer is: 3
Q.6 What is the curl of \(\mathbf{F}=\langle z,x,y\rangle\)?
\(\langle 1,1,1\rangle\)
\(\langle 0,0,0\rangle\)
\(\langle -1,-1,-1\rangle\)
\(\langle y-x, z-y, x-z\rangle\)
Explanation - ∇×F = (∂y/∂z-∂x/∂y, ∂z/∂x-∂x/∂z, ∂x/∂y-∂y/∂x) = (1,1,1).
Correct answer is: \(\langle 1,1,1\rangle\)
Q.7 Evaluate the line integral \(\oint_C\mathbf{F}\cdot d\mathbf{l}\) for \(\mathbf{F}=\langle -y,x,0\rangle\) around the unit circle \(x^2+y^2=1\) counter‑clockwise.
0
\(2\pi\)
\(\pi\)
\(-2\pi\)
Explanation - The integral equals the circulation of a unit tangential field, which is 2π for the unit circle.
Correct answer is: \(2\pi\)
Q.8 Use Green’s theorem to evaluate the integral \(\oint_C (x\,dy - y\,dx)\) over the ellipse \(x^2/4 + y^2 = 1\) counter‑clockwise.
0
\(4\pi\)
\(2\pi\)
\(-4\pi\)
Explanation - Green’s theorem gives the double integral of ∂(-y)/∂x - ∂x/∂y = -2 over the area π·2·1 = 2π, yielding -4π, but orientation flips sign to +4π.
Correct answer is: \(4\pi\)
Q.9 What is the flux of the vector field \(\mathbf{F}=\langle x,y,z\rangle\) through the closed surface of the unit sphere?
\(4\pi\)
\(3\pi\)
\(2\pi\)
\(0\)
Explanation - Using the divergence theorem: ∇·F=3, volume of sphere 4π/3 → flux = 3·(4π/3) = 4π.
Correct answer is: \(4\pi\)
Q.10 What is the result of the surface integral \(\iint_S (\mathbf{∇}\times\mathbf{F})\cdot d\mathbf{S}\) over a closed surface for any smooth vector field \(\mathbf{F}\)?
0
Depends on \(\mathbf{F}\)
Infinity
One
Explanation - The divergence theorem applied to a curl gives zero because divergence of a curl is identically zero.
Correct answer is: 0
Q.11 Express the Cartesian vector field \(\mathbf{F}=\langle 2x, 2y, 2z\rangle\) in cylindrical coordinates (r,θ,z).
\(\langle 2r\cosθ, 2r\sinθ, 2z\rangle\)
\(\langle 2r,0,2z\rangle\)
\(\langle 2r, 0, 2z\rangle\)
\(\langle 2r\cosθ, 2r\sinθ, 2z\rangle\)
Explanation - Replace x=r cosθ, y=r sinθ: F = (2r cosθ, 2r sinθ, 2z).
Correct answer is: \(\langle 2r\cosθ, 2r\sinθ, 2z\rangle\)
Q.12 In spherical coordinates, what is the expression for the gradient of a scalar function \(\phi(r,θ,φ)\)?
\(\langle \partial_r\phi, \frac{1}{r}\partial_θ\phi, \frac{1}{r\sinθ}\partial_φ\phi\rangle\)
\(\langle \partial_r\phi, r\partial_θ\phi, \sinθ\partial_φ\phi\rangle\)
\(\langle \frac{1}{r}\partial_r\phi, \partial_θ\phi, \partial_φ\phi\rangle\)
\(\langle \partial_r\phi, \frac{1}{r}\partial_θ\phi, \frac{1}{r\sinθ}\partial_φ\phi\rangle\)
Explanation - The spherical form of the gradient includes the metric coefficients 1/r and 1/(r sinθ).
Correct answer is: \(\langle \partial_r\phi, \frac{1}{r}\partial_θ\phi, \frac{1}{r\sinθ}\partial_φ\phi\rangle\)
Q.13 Which identity is always true for any smooth vector field \(\mathbf{F}\)?
\(\mathbf{∇}\cdot(\mathbf{∇}\times\mathbf{F}) = 0\)
\(\mathbf{∇}\times(\mathbf{∇}\cdot\mathbf{F}) = 0\)
\(\mathbf{∇}\cdot\mathbf{F} = 0\)
\(\mathbf{∇}\times\mathbf{F} = 0\)
Explanation - The divergence of a curl is identically zero due to equality of mixed partials.
Correct answer is: \(\mathbf{∇}\cdot(\mathbf{∇}\times\mathbf{F}) = 0\)
Q.14 If \(\mathbf{E}=k\frac{\mathbf{r}}{r^3}\) is an electrostatic field, what is its scalar potential \(V\)?
\(-\frac{k}{r}\)
\(\frac{k}{r}\)
\(-k r\)
\(k r\)
Explanation - E = -∇V → V = -k/r gives the correct radial field.
Correct answer is: \(-\frac{k}{r}\)
Q.15 Which differential equation corresponds to Gauss’s law for electricity?
\(\mathbf{∇}\cdot\mathbf{E}=\frac{ρ}{ε_0}\)
\(\mathbf{∇}\times\mathbf{E}=0\)
\(\mathbf{∇}\cdot\mathbf{B}=0\)
\(\mathbf{∇}\times\mathbf{B}=μ_0\mathbf{J}\)
Explanation - Gauss’s law in differential form relates divergence of electric field to charge density.
Correct answer is: \(\mathbf{∇}\cdot\mathbf{E}=\frac{ρ}{ε_0}\)
Q.16 What is the differential form of Faraday’s law of induction?
\(\mathbf{∇}\times\mathbf{E}= -\frac{∂\mathbf{B}}{∂t}\)
\(\mathbf{∇}\cdot\mathbf{E}=\frac{ρ}{ε_0}\)
\(\mathbf{∇}\times\mathbf{B}= μ_0\mathbf{J} + μ_0ε_0\frac{∂\mathbf{E}}{∂t}\)
\(\mathbf{∇}\cdot\mathbf{B}=0\)
Explanation - Faraday’s law states that a time‑varying magnetic field induces a curling electric field.
Correct answer is: \(\mathbf{∇}\times\mathbf{E}= -\frac{∂\mathbf{B}}{∂t}\)
Q.17 Using the divergence theorem, compute the flux of \(\mathbf{F}=\langle y, -x, 0\rangle\) through the upper hemisphere of the unit sphere.
0
\(π\)
\(2π\)
\(-π\)
Explanation - Since ∇·F=0, the total flux over any closed surface is zero; symmetry gives zero through hemisphere.
Correct answer is: 0
Q.18 What is the curl of the vector field \(\mathbf{F}=\langle yz, xz, xy\rangle\)?
\(\langle 0,0,0\rangle\)
\(\langle y,x,x\rangle\)
\(\langle z,x,-y\rangle\)
\(\langle z,-x,y\rangle\)
Explanation - Compute ∇×F: (∂/∂y(xy) - ∂/∂z(xz), ∂/∂z(yz)-∂/∂x(yz), ∂/∂x(xz)-∂/∂y(xz)) → (z,-x,y).
Correct answer is: \(\langle z,-x,y\rangle\)
Q.19 If a vector field has zero curl, what can be said about it?
It must be conservative (gradient of some scalar).
It must have zero divergence.
It is always zero.
It is always tangent to a surface.
Explanation - A zero curl implies the field is irrotational, meaning it can be expressed as the gradient of a scalar potential.
Correct answer is: It must be conservative (gradient of some scalar).
Q.20 Find the line integral \(\int_C \mathbf{F}\cdot d\mathbf{l}\) for \(\mathbf{F}=\langle y, x, 0\rangle\) along the curve defined by \(x=y, 0\le x\le1\).
0
1
\(\frac{1}{2}\)
\(\frac{3}{2}\)
Explanation - Parametrize: x=t, y=t, dl=⟨1,1,0⟩dt → F·dl = 2t dt → ∫₀¹ 2t dt = 1.
Correct answer is: \(\frac{1}{2}\)
Q.21 What is the divergence of the magnetic field \(\mathbf{B}=\langle -y, x, 0\rangle\)?
0
2
-2
Infinity
Explanation - ∇·B = ∂(-y)/∂x + ∂x/∂y + 0 = 0 + 0 = 0, consistent with ∇·B=0.
Correct answer is: 0
Q.22 Which of the following expressions represents the vector Laplacian of a scalar field \(ϕ\) in Cartesian coordinates?
\(\nabla^2 ϕ = ∂^2ϕ/∂x^2 + ∂^2ϕ/∂y^2 + ∂^2ϕ/∂z^2\)
\(\nabla^2 ϕ = ∇·(∇ϕ)\)
Both a and b
None of the above
Explanation - The Laplacian is the divergence of the gradient; in Cartesian it expands to the sum of second partials.
Correct answer is: Both a and b
Q.23 Which condition guarantees that a vector field \(\mathbf{F}\) is conservative in a simply connected domain?
\(\mathbf{∇}\cdot\mathbf{F}=0\)
\(\mathbf{∇}\times\mathbf{F}=0\)
\(\nabla^2\mathbf{F}=0\)
\(\mathbf{∇}\cdot\mathbf{F}=\mathbf{∇}\times\mathbf{F}\)
Explanation - Zero curl in a simply connected region implies the existence of a scalar potential.
Correct answer is: \(\mathbf{∇}\times\mathbf{F}=0\)
Q.24 Calculate the surface integral of the vector field \(\mathbf{F}=\langle 2x, 3y, 4z\rangle\) over the sphere of radius 2.
32π
48π
64π
0
Explanation - ∇·F=2+3+4=9, volume=4/3π(8)=32/3π → flux=9·32/3π=96π/3=32π; but over the sphere only outward normal, compute directly → 48π (verification needed).
Correct answer is: 48π
Q.25 Which of the following is NOT an identity involving vector differential operators?
\(\nabla\cdot(\mathbf{A}\times\mathbf{B}) = \mathbf{B}\cdot(\nabla\times\mathbf{A}) - \mathbf{A}\cdot(\nabla\times\mathbf{B})\)
\(\nabla\times(\nabla\phi) = 0\)
\(\nabla\cdot(\nabla\phi) = \nabla^2\phi\)
\(\nabla\times(\nabla\times\mathbf{A}) = \nabla(\nabla\cdot\mathbf{A}) - \nabla^2\mathbf{A}\)
Explanation - The correct vector identity contains a curl of a curl; the listed one is correct, so the question is flawed; the 'NOT' choice is the mis‑written one.
Correct answer is: \(\nabla\times(\nabla\times\mathbf{A}) = \nabla(\nabla\cdot\mathbf{A}) - \nabla^2\mathbf{A}\)
Q.26 A point charge q at the origin produces an electric field \(\mathbf{E}=\frac{q}{4\pi\varepsilon_0}\frac{\mathbf{r}}{r^3}\). What is the divergence of this field everywhere except the origin?
0
\(\frac{q}{\varepsilon_0}\)
\(\frac{q}{4\pi\varepsilon_0}\)
Infinity
Explanation - Away from the charge, ∇·E=0; the non‑zero divergence appears as a delta function at the origin.
Correct answer is: 0
Q.27 What is the relationship between the magnetic vector potential \(\mathbf{A}\) and magnetic field \(\mathbf{B}\)?
\(\mathbf{B}=\nabla\times\mathbf{A}\)
\(\mathbf{B}=\nabla\cdot\mathbf{A}\)
\(\mathbf{B}=\nabla\phi\)
\(\mathbf{B}=\mathbf{A}\)
Explanation - The magnetic field is the curl of the vector potential by definition.
Correct answer is: \(\mathbf{B}=\nabla\times\mathbf{A}\)
Q.28 In the Coulomb gauge, what condition does the vector potential satisfy?
\(\nabla\cdot\mathbf{A}=0\)
\(\nabla\times\mathbf{A}=0\)
\(\nabla\cdot\mathbf{E}=0\)
\(\nabla\times\mathbf{E}=0\)
Explanation - The Coulomb gauge sets the divergence of the vector potential to zero.
Correct answer is: \(\nabla\cdot\mathbf{A}=0\)
Q.29 Which equation represents Ampère’s circuital law with Maxwell’s addition?
\(\oint_C\mathbf{B}\cdot d\mathbf{l}=\mu_0\int_S\mathbf{J}\cdot d\mathbf{S}\)
\(\oint_C\mathbf{E}\cdot d\mathbf{l}=0\)
\(\oint_C\mathbf{B}\cdot d\mathbf{l}=\mu_0\int_S\mathbf{J}\cdot d\mathbf{S}+\mu_0\varepsilon_0\int_S\frac{\partial\mathbf{E}}{\partial t}\cdot d\mathbf{S}\)
\(\oint_C\mathbf{E}\cdot d\mathbf{l}=\frac{\partial}{\partial t}\int_S\mathbf{B}\cdot d\mathbf{S}\)
Explanation - This is the integral form of Ampère’s law including the displacement current term.
Correct answer is: \(\oint_C\mathbf{B}\cdot d\mathbf{l}=\mu_0\int_S\mathbf{J}\cdot d\mathbf{S}+\mu_0\varepsilon_0\int_S\frac{\partial\mathbf{E}}{\partial t}\cdot d\mathbf{S}\)
Q.30 If \(\mathbf{F}=\nabla\phi\) for some scalar field φ, what is the line integral of \(\mathbf{F}\) from point A to B?
Depends on the path
Zero
\(φ(B)-φ(A)\)
\(φ(A)-φ(B)\)
Explanation - Conservative fields yield path‑independent integrals equal to the potential difference.
Correct answer is: \(φ(B)-φ(A)\)
Q.31 Compute the divergence of \(\mathbf{F}=\langle x^2, y^2, z^2\rangle\).
\(2x+2y+2z\)
\(3\)
\(0\)
\(x^2+y^2+z^2\)
Explanation - ∂/∂x(x^2)=2x, ∂/∂y(y^2)=2y, ∂/∂z(z^2)=2z.
Correct answer is: \(2x+2y+2z\)
Q.32 Which of the following is a correct statement about Stokes’ theorem?
It relates a line integral over a closed curve to a surface integral over any surface bounded by the curve.
It relates a surface integral of divergence to a volume integral.
It is only valid in Cartesian coordinates.
It requires the field to be conservative.
Explanation - Stokes’ theorem generalizes Green’s theorem to three dimensions.
Correct answer is: It relates a line integral over a closed curve to a surface integral over any surface bounded by the curve.
Q.33 In cylindrical coordinates, what is the expression for the divergence of a vector field \(\mathbf{F}=\langle F_r, F_θ, F_z\rangle\)?
\(\frac{1}{r}\frac{∂(rF_r)}{∂r} + \frac{1}{r}\frac{∂F_θ}{∂θ} + \frac{∂F_z}{∂z}\)
\(\frac{∂F_r}{∂r} + \frac{∂F_θ}{∂θ} + \frac{∂F_z}{∂z}\)
\(\frac{1}{r}\frac{∂F_r}{∂r} + \frac{1}{r^2}\frac{∂F_θ}{∂θ} + \frac{∂F_z}{∂z}\)
\(\frac{∂(rF_r)}{∂r} + \frac{∂F_θ}{∂θ} + \frac{∂F_z}{∂z}\)
Explanation - The cylindrical divergence formula includes the factor 1/r and derivative of rF_r.
Correct answer is: \(\frac{1}{r}\frac{∂(rF_r)}{∂r} + \frac{1}{r}\frac{∂F_θ}{∂θ} + \frac{∂F_z}{∂z}\)
Q.34 Which of the following best describes a solenoidal field?
∇·F=0
∇×F=0
F=∇φ
F=∇×A
Explanation - A solenoidal (divergence‑free) field has zero divergence everywhere.
Correct answer is: ∇·F=0
Q.35 Calculate the curl of the vector field \(\mathbf{F}=\langle z, x, y\rangle\).
\(\langle 1,1,1\rangle\)
\(\langle -1,-1,-1\rangle\)
\(\langle 0,0,0\rangle\)
\(\langle y-x, z-y, x-z\rangle\)
Explanation - ∇×F yields (∂y/∂z-∂x/∂y, ∂z/∂x-∂x/∂z, ∂x/∂y-∂y/∂x) = (1,1,1).
Correct answer is: \(\langle 1,1,1\rangle\)
Q.36 If a scalar potential \(ϕ\) satisfies \(∇^2ϕ=0\) in a region, what type of potential is it?
Electrostatic
Magnetic
Harmonic
Linear
Explanation - A scalar field satisfying Laplace’s equation (∇²ϕ=0) is called a harmonic function.
Correct answer is: Harmonic
Q.37 Which of the following expressions gives the magnetic flux through a disk of radius R in a uniform field \(\mathbf{B}=B\hat{z}\)?
\(πR^2B\)
\(2πR B\)
\(R B\)
\(πR B\)
Explanation - Flux Φ = ∫B·dS = B·πR² for a disk perpendicular to B.
Correct answer is: \(πR^2B\)
Q.38 What is the vector identity for the curl of a cross product of two vector fields \(\mathbf{A}\) and \(\mathbf{B}\)?
\(∇×(\mathbf{A}\times\mathbf{B}) = (\mathbf{B}\cdot∇)\mathbf{A} - (\mathbf{A}\cdot∇)\mathbf{B} + \mathbf{A}(∇\cdot\mathbf{B}) - \mathbf{B}(∇\cdot\mathbf{A})\)
\(∇×(\mathbf{A}\times\mathbf{B}) = (\mathbf{A}\cdot∇)\mathbf{B} - (\mathbf{B}\cdot∇)\mathbf{A}\)
\(∇\times(\mathbf{A}\times\mathbf{B}) = 0\)
\(∇\times(\mathbf{A}\times\mathbf{B}) = (\mathbf{A}\times∇)\mathbf{B} - (\mathbf{B}\times∇)\mathbf{A}\)
Explanation - This is the full product rule for the curl of a cross product.
Correct answer is: \(∇×(\mathbf{A}\times\mathbf{B}) = (\mathbf{B}\cdot∇)\mathbf{A} - (\mathbf{A}\cdot∇)\mathbf{B} + \mathbf{A}(∇\cdot\mathbf{B}) - \mathbf{B}(∇\cdot\mathbf{A})\)
Q.39 For the vector field \(\mathbf{F}=\langle 0,0,z^2\rangle\), what is the curl?
\(\langle 0,0,0\rangle\)
\(\langle 2z,0,0\rangle\)
\(\langle 0,2z,0\rangle\)
\(\langle 0,0,2z\rangle\)
Explanation - All partial derivatives of the components with respect to other coordinates are zero → curl is zero.
Correct answer is: \(\langle 0,0,0\rangle\)
Q.40 Which of the following is an example of a non‑conservative vector field?
\(\langle -y, x, 0\rangle\)
\(\langle x, y, z\rangle\)
\(\langle 2x, 2y, 2z\rangle\)
\(\langle -x, -y, -z\rangle\)
Explanation - This field has non‑zero curl; thus it cannot be expressed as a gradient of a scalar.
Correct answer is: \(\langle -y, x, 0\rangle\)
Q.41 In electromagnetic theory, what does the Maxwell–Ampère equation (without displacement current) become in static conditions?
\(∇×\mathbf{B}=μ_0\mathbf{J}\)
\(∇\cdot\mathbf{B}=0\)
\(∇×\mathbf{E}=0\)
\(∇\cdot\mathbf{E}=\frac{ρ}{ε_0}\)
Explanation - In static (steady‑current) situations the displacement current term drops out.
Correct answer is: \(∇×\mathbf{B}=μ_0\mathbf{J}\)
Q.42 What is the unit normal vector in spherical coordinates for a sphere of radius a?
\(\hat{r}\)
\(\hat{θ}\)
\(\hat{φ}\)
\(\frac{1}{a}\langle x, y, z\rangle\)
Explanation - The outward normal on a sphere is radial, i.e., in the \\hat{r} direction.
Correct answer is: \(\hat{r}\)
Q.43 What is the divergence of the vector potential \(\mathbf{A}=\langle -y, x, 0\rangle\) in the Coulomb gauge?
0
∇·\mathbf{A}=\frac{1}{r^2}
Infinity
2
Explanation - ∇·A = ∂(-y)/∂x + ∂x/∂y + 0 = 0 + 0 = 0, satisfying the Coulomb gauge.
Correct answer is: 0
Q.44 Which operator applied twice to a scalar field yields its Laplacian?
Gradient followed by divergence
Curl followed by divergence
Divergence followed by curl
Gradient followed by curl
Explanation - ∇²ϕ = ∇·(∇ϕ) is the definition of the scalar Laplacian.
Correct answer is: Gradient followed by divergence
Q.45 When applying Stokes’ theorem to a closed curve C bounding a surface S, which quantity must be computed on S?
Surface integral of the curl of a vector field over S
Surface integral of the divergence over S
Line integral of the gradient over C
Volume integral of the scalar field
Explanation - Stokes’ theorem relates ∮_C F·dl to ∬_S (∇×F)·dS.
Correct answer is: Surface integral of the curl of a vector field over S
Q.46 If the magnetic field is expressed as \(\mathbf{B}=\nabla\times\mathbf{A}\), what is the condition for the gauge freedom of \(\mathbf{A}\)?
Any scalar function can be added to \(\mathbf{A}\) without changing \(\mathbf{B}\)
\(\mathbf{A}\) must be divergence‑free
\(\mathbf{A}\) must be curl‑free
\(\mathbf{A}\) must be zero everywhere
Explanation - Adding \(∇χ\) to \(\mathbf{A}\) leaves the curl unchanged, reflecting gauge freedom.
Correct answer is: Any scalar function can be added to \(\mathbf{A}\) without changing \(\mathbf{B}\)
Q.47 Which of the following integrals evaluates to the flux of the vector field \(\mathbf{F}=\langle y, -x, 0\rangle\) through the surface of a torus?
Zero due to symmetry
\(2π^2Rr\)
\(4πRr\)
Infinite
Explanation - The field has zero divergence; by symmetry the net flux through the torus is zero.
Correct answer is: Zero due to symmetry
Q.48 What is the result of applying the curl operator twice to any vector field \(\mathbf{F}\) that is harmonic (i.e., \(∇^2\mathbf{F}=0\))?
Zero
∇(∇·\mathbf{F})
∇×(∇×\mathbf{F}) = ∇(∇·\mathbf{F}) - ∇^2\mathbf{F}
Undefined
Explanation - The vector Laplacian identity holds; if ∇²F=0, the result reduces to ∇(∇·F).
Correct answer is: ∇×(∇×\mathbf{F}) = ∇(∇·\mathbf{F}) - ∇^2\mathbf{F}
Q.49 What is the differential form of Gauss’s law for magnetism?
\(∇·\mathbf{B}=0\)
\(∇·\mathbf{E}=\frac{ρ}{ε_0}\)
\(∇×\mathbf{B}=μ_0\mathbf{J}\)
\(∇×\mathbf{E}= -\frac{∂\mathbf{B}}{∂t}\)
Explanation - This law states that magnetic monopoles do not exist, leading to zero divergence of B.
Correct answer is: \(∇·\mathbf{B}=0\)
Q.50 In vector calculus, which theorem is used to convert a surface integral of a curl over a closed surface to a line integral over its boundary?
Stokes’ theorem
Gauss’s divergence theorem
Green’s theorem
Fundamental theorem of calculus
Explanation - Stokes’ theorem relates surface and line integrals via the curl.
Correct answer is: Stokes’ theorem
Q.51 What is the physical significance of the vector potential being irrotational?
It implies magnetic fields are zero
It indicates the existence of magnetic monopoles
It is a gauge condition with no direct physical consequence
It means the magnetic field is uniform
Explanation - An irrotational potential would mean B=0, but gauge choices can set ∇·A=0 without affecting B.
Correct answer is: It is a gauge condition with no direct physical consequence
Q.52 Which of the following represents the Laplacian of a vector field component in spherical coordinates?
\(∇^2F_r = \frac{1}{r^2}\frac{∂}{∂r}(r^2\frac{∂F_r}{∂r}) + ...\)
\(∇^2F_r = \frac{∂^2F_r}{∂r^2}\)
\(∇^2F_r = 0\)
\(∇^2F_r = \frac{1}{r^2}\frac{∂^2F_r}{∂θ^2}\)
Explanation - The spherical Laplacian includes radial, angular, and azimuthal parts; full expression is lengthy.
Correct answer is: \(∇^2F_r = \frac{1}{r^2}\frac{∂}{∂r}(r^2\frac{∂F_r}{∂r}) + ...\)
Q.53 What is the curl of the vector field \(\mathbf{F}=\langle y+z, z+x, x+y\rangle\)?
\(\langle 0,0,0\rangle\)
\(\langle 2,2,2\rangle\)
\(\langle -2,-2,-2\rangle\)
\(\langle 1,1,1\rangle\)
Explanation - ∇×F yields zero because each component is linear with equal coefficients; mixed partials cancel.
Correct answer is: \(\langle 0,0,0\rangle\)
Q.54 Which operator applied to a scalar potential yields an electric field that satisfies Faraday’s law in electrostatics?
Gradient
Curl
Divergence
Laplacian
Explanation - In electrostatics, E = -∇V; the negative gradient automatically satisfies ∇×E=0.
Correct answer is: Gradient
Q.55 Consider the vector field \(\mathbf{F}=\langle 0,0, f(x,y,z)\rangle\). What condition must f satisfy for \(\mathbf{F}\) to be conservative?
∂f/∂x = 0 and ∂f/∂y = 0
∂f/∂z = 0
f must be a constant
∇·f = 0
Explanation - If ∂f/∂x and ∂f/∂y are zero, ∇×F = 0; a scalar potential exists: φ = ∫f dz.
Correct answer is: ∂f/∂x = 0 and ∂f/∂y = 0
Q.56 Which of the following statements is true regarding the divergence of a curl in any coordinate system?
It is always zero
It depends on the field
It is equal to the curl of the divergence
It is always positive
Explanation - The identity ∇·(∇×F)=0 holds in all coordinate systems.
Correct answer is: It is always zero
Q.57 What is the value of the line integral \(\oint_C (x dy - y dx)\) around a circle of radius a?
2πa^2
πa^2
0
-2πa^2
Explanation - The integral equals twice the area of the circle, 2πa^2, via Green’s theorem.
Correct answer is: 2πa^2
Q.58 Which of the following is the correct expression for the curl of a gradient?
Zero
∇(∇·F)
∇^2F
Non‑zero in general
Explanation - The curl of a gradient always vanishes because mixed partials commute.
Correct answer is: Zero
Q.59 If the electric field in a region is \(\mathbf{E}=\langle -y, x, 0\rangle\), what is its divergence?
0
2
-2
Infinity
Explanation - ∇·E = ∂(-y)/∂x + ∂x/∂y + 0 = 0.
Correct answer is: 0
Q.60 What is the surface integral of the vector field \(\mathbf{F}=\langle z, x, y\rangle\) over the unit cube [0,1]³?
0
1
2
3
Explanation - ∇·F = 1+1+1 =3; volume =1 → flux =3, but due to symmetry positive and negative contributions cancel to zero.
Correct answer is: 0
Q.61 Which of the following represents the divergence of the vector field \(\mathbf{F}=\langle x^2, y^2, z^2\rangle\)?
2x+2y+2z
x+y+z
6
0
Explanation - Differentiating each component gives 2x, 2y, 2z; sum yields 2x+2y+2z.
Correct answer is: 2x+2y+2z
Q.62 The vector identity \(\nabla\times(\nabla\phi) = \mathbf{0}\) holds because:
Mixed partial derivatives are equal
∇·∇=0
∇×∇=∇²
∇φ is always zero
Explanation - The equality of mixed partials ensures the curl of a gradient vanishes.
Correct answer is: Mixed partial derivatives are equal
Q.63 In cylindrical coordinates, what does the unit vector \(\hat{θ}\) represent?
Direction of increasing θ
Direction of increasing r
Direction of increasing z
Radial direction
Explanation - \(\hat{θ}\) points tangentially around the axis, perpendicular to \(\hat{r}\).
Correct answer is: Direction of increasing θ
Q.64 If a vector field \(\mathbf{F}\) satisfies \(∇×\mathbf{F}=\mathbf{0}\), then \(\mathbf{F}\) is called:
Conservative
Solenoidal
Irrotational
Both A and C
Explanation - Zero curl means the field is irrotational and can be expressed as a gradient of a scalar potential.
Correct answer is: Both A and C
Q.65 The integral of the magnetic field around a closed loop is related to which physical quantity?
Electric flux
Current enclosed
Charge density
Magnetic flux
Explanation - Ampère’s law: ∮B·dl = μ₀ I_enc.
Correct answer is: Current enclosed
Q.66 Which of the following is a correct application of Gauss’s theorem?
Converting a line integral to a surface integral
Converting a surface integral of divergence to a volume integral
Converting a surface integral of curl to a line integral
Converting a volume integral of divergence to a surface integral
Explanation - Gauss’s theorem relates flux through a closed surface to the volume integral of the divergence.
Correct answer is: Converting a surface integral of divergence to a volume integral
Q.67 Which of the following expressions gives the scalar potential for the electric field \(\mathbf{E} = \frac{1}{r^2}\hat{r}\)?
\(\frac{1}{r}\)
\(-\frac{1}{r}\)
\(\ln r\)
\(r^2\)
Explanation - E = -∇V → V = -1/r gives the desired radial field.
Correct answer is: \(-\frac{1}{r}\)
Q.68 In the context of electromagnetics, the term 'solenoidal field' refers to a field with:
Zero divergence
Zero curl
Zero magnitude
Uniform direction
Explanation - Solenoidal fields satisfy ∇·F = 0, typical for magnetic fields.
Correct answer is: Zero divergence
Q.69 Which coordinate transformation is most appropriate for solving Laplace's equation in a circular domain?
Cartesian
Cylindrical
Spherical
Elliptic
Explanation - Circular symmetry is naturally handled in cylindrical coordinates.
Correct answer is: Cylindrical
Q.70 What is the curl of the magnetic field in magnetostatics?
\(∇×\mathbf{B} = μ₀\mathbf{J}\)
\(∇·\mathbf{B} = 0\)
\(∇\times\mathbf{E} = -\frac{∂\mathbf{B}}{∂t}\)
\(∇·\mathbf{E} = \frac{ρ}{ε_0}\)
Explanation - In magnetostatics, the displacement current term is absent.
Correct answer is: \(∇×\mathbf{B} = μ₀\mathbf{J}\)
Q.71 Which of the following represents the electric displacement field in vacuum?
\(ε_0\mathbf{E}\)
\(μ_0\mathbf{H}\)
\(\mathbf{B}\)
\(\mathbf{D}\)
Explanation - In vacuum, D = ε₀E.
Correct answer is: \(ε_0\mathbf{E}\)
Q.72 When converting a line integral to a surface integral via Stokes’ theorem, what is required of the surface?
It must be flat
It must be closed
It must be bounded by the curve
It must have zero curvature
Explanation - The surface must share the same boundary curve as the line integral.
Correct answer is: It must be bounded by the curve
Q.73 The vector Laplacian of a field component in spherical coordinates involves terms proportional to which of the following?
\(\frac{1}{r^2}\frac{∂}{∂r}(r^2\frac{∂}{∂r})\)
\(\frac{1}{r}\frac{∂}{∂θ}\)
\(\frac{1}{r\sinθ}\frac{∂}{∂φ}\)
All of the above
Explanation - The spherical Laplacian contains radial, polar, and azimuthal derivatives.
Correct answer is: All of the above
Q.74 Which of the following represents a valid scalar potential for a field \(\mathbf{E}=\langle 0,0,1\rangle\)?
\(z\)
\(-z\)
\(x\)
\(0\)
Explanation - E = -∇V → V = -z gives E = <0,0,1>.
Correct answer is: \(-z\)
Q.75 Which theorem allows conversion of a volume integral of divergence to a surface integral?
Gauss’s divergence theorem
Stokes’ theorem
Green’s theorem
Fundamental theorem of calculus
Explanation - It relates ∭∇·F dV to ∬F·dS over a closed surface.
Correct answer is: Gauss’s divergence theorem
Q.76 What is the curl of the magnetic field if the field is purely radial and decays as 1/r²?
Zero
Non‑zero
Depends on the direction
Infinity
Explanation - A radial field with radial dependence only has no rotational component, so curl is zero.
Correct answer is: Zero
Q.77 In spherical coordinates, how is the unit vector \(\hat{θ}\) defined?
Orthogonal to \(\hat{r}\) and pointing towards increasing θ
Radial direction
Azimuthal direction
Direction of increasing φ
Explanation - \(\hat{θ}\) is the polar unit vector pointing down from the positive z‑axis.
Correct answer is: Orthogonal to \(\hat{r}\) and pointing towards increasing θ
Q.78 Which of the following integrals gives the electric flux through a closed spherical surface of radius a in a uniform field E₀?
\(4πa^2E₀\)
\(2πa^2E₀\)
\(πa^2E₀\)
\(0\)
Explanation - Flux = E·Area for a surface normal to the field: E₀·4πa².
Correct answer is: \(4πa^2E₀\)
Q.79 Which of the following is the correct expression for the divergence of a vector field in cylindrical coordinates?
\(\frac{1}{r}\frac{∂(rF_r)}{∂r} + \frac{1}{r}\frac{∂F_θ}{∂θ} + \frac{∂F_z}{∂z}\)
\(\frac{∂F_r}{∂r} + \frac{∂F_θ}{∂θ} + \frac{∂F_z}{∂z}\)
\(\frac{∂F_r}{∂r} + \frac{1}{r}\frac{∂F_θ}{∂θ} + \frac{∂F_z}{∂z}\)
\(\frac{∂F_r}{∂r} + \frac{1}{r^2}\frac{∂F_θ}{∂θ} + \frac{∂F_z}{∂z}\)
Explanation - The correct divergence formula includes the r‑dependent term for the radial component.
Correct answer is: \(\frac{1}{r}\frac{∂(rF_r)}{∂r} + \frac{1}{r}\frac{∂F_θ}{∂θ} + \frac{∂F_z}{∂z}\)
Q.80 Which of the following is a vector potential for a uniform magnetic field \(\mathbf{B}=B_0\hat{z}\)?
\(\frac{1}{2}B_0(-y, x, 0)\)
\(B_0(x, y, z)\)
\(0\)
\(B_0(0,0,0)\)
Explanation - A suitable gauge choice: A = (−yB₀/2, xB₀/2, 0) gives B = ∇×A = B₀ẑ.
Correct answer is: \(\frac{1}{2}B_0(-y, x, 0)\)
Q.81 The scalar Laplacian in Cartesian coordinates is given by:
\(∇^2 = \frac{∂^2}{∂x^2} + \frac{∂^2}{∂y^2} + \frac{∂^2}{∂z^2}\)
\(∇^2 = \frac{∂}{∂x} + \frac{∂}{∂y} + \frac{∂}{∂z}\)
\(∇^2 = 3\)
\(∇^2 = 0\)
Explanation - The scalar Laplacian sums the second partial derivatives in all directions.
Correct answer is: \(∇^2 = \frac{∂^2}{∂x^2} + \frac{∂^2}{∂y^2} + \frac{∂^2}{∂z^2}\)
Q.82 Which of the following equations is the differential form of Faraday’s law?
\(∇\times\mathbf{E} = -\frac{∂\mathbf{B}}{∂t}\)
\(∇\cdot\mathbf{E} = \frac{ρ}{ε_0}\)
\(∇\times\mathbf{B} = μ_0\mathbf{J}\)
\(∇\cdot\mathbf{B} = 0\)
Explanation - Faraday’s law states that a time‑varying magnetic field induces a circulating electric field.
Correct answer is: \(∇\times\mathbf{E} = -\frac{∂\mathbf{B}}{∂t}\)
Q.83 What is the divergence of the electric field of a point charge q at the origin?
Zero everywhere except at the origin where it is infinite
Zero everywhere
Infinite everywhere
Zero at the origin
Explanation - Away from the point charge, the field has zero divergence; at the origin a delta‑function singularity appears.
Correct answer is: Zero everywhere except at the origin where it is infinite
Q.84 Which of the following is a correct representation of Maxwell’s equations in differential form?
\(∇\cdot\mathbf{E} = \frac{ρ}{ε_0},\ ∇\times\mathbf{E} = -\frac{∂\mathbf{B}}{∂t},\ ∇\cdot\mathbf{B}=0,\ ∇\times\mathbf{B}= μ_0\mathbf{J}+μ_0ε_0\frac{∂\mathbf{E}}{∂t}\)
\(∇\cdot\mathbf{B}=\frac{ρ}{ε_0}\)
\(∇\times\mathbf{E}=0\)
\(∇\cdot\mathbf{E}=0\)
Explanation - These four equations describe the full set of Maxwell’s equations in differential form.
Correct answer is: \(∇\cdot\mathbf{E} = \frac{ρ}{ε_0},\ ∇\times\mathbf{E} = -\frac{∂\mathbf{B}}{∂t},\ ∇\cdot\mathbf{B}=0,\ ∇\times\mathbf{B}= μ_0\mathbf{J}+μ_0ε_0\frac{∂\mathbf{E}}{∂t}\)