1. Curl is defined as the angular velocity at every point of the vector field. State True/False.
a) True
b) False
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)
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
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 Stoke’s theorem.
4. The curl of a curl of a vector gives a
a) Scalar
b) Vector
c) Zero value
d) Non zero value
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
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
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
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
Explanation: Curl A = i(Dy(y) – Dz(0)) – j (Dx(0) – Dz(yz)) + k(Dx(4xy) – Dy(yz)) =
i + y j + (4y – z)k.
9. Curl cannot be employed in which one of the following?
a) Directional coupler
b) Magic Tee
c) Isolator and Terminator
d) Waveguides
Explanation: In the Directional coupler, Magic Tee, Isolator and Terminator 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
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.