If then the value of at (2, 2, 0) will be
Given,
If then the value of is
Thus,
What is the value of constant b so that the vector
is solenoidal?
Since vector is solenoidal, therefore
= 0
or, [1 + 1 + b] = 0 or b = 2
Match ListI with Listll and select the correct answer using the codes given below the lists:
ListI
A. Gauss’s divergence theorem
B. Stroke’s theorem
C. The divergence
D. The curl
Listll
Assertion (A): Vector differential operator is a vector quantity and it signifies that certain operations of a differentiation are to be carried out on the scalar function following it.
Reason (R): Vector differential operator posses properties similar to ordinary vectors
Vector differential operator (‘∇’) is not a vector quantity. Hence, assertion is a false statement.
Consider the following statements:
1. Divergence of a vector function at each point gives the rate per unit volume at which the physical entity is issuing from that point.
2. If a vector function ϕ represents temperature, then grad ϕ or ∇ϕ will represents rate of change of temperature with distance.
3. The curl of a vector function A gives the measure of the angular velocity at every point of the vector field.
All the given statements are correct.
Assertion (A): The Gauss’s divergence theorem permits us to express certain integrals by means of surface integrals.
Reason (R): Gauss’s divergence theorem states that “the surface integral of the curl of a vector field taken over any surface s is equal to the line integral of the vector field around the closed periphery (contour) of the surface.
Reason is a statement of stroke’s theorem not that of Gauss's divergence theorem.
If is any vector field in cartesian coordinates system, then
Let, be any vector field in cartesian coordinate system then, we can prove that
Also, Div. Curl
If and , then
Given,
∴
Also,
If S is any closed surface enclosing a volume V and then the value of (is a unit vector) will be equal to
Assertion (A): The laplacian operator of a scalar function ϕ can be defined as “Gradient of the divergence of the scalar ϕ”
Reason (R): Laplacian operator may be a “scalar laplacian" or a “vector laplacian'’ depending upon whether it is operated with a scalar function or a vector, respectively.
Assertion is not true because the laplacian operator (∇^{2})of a scalar function ϕ can be defined as “Divergence of the gradient of the scalar ϕ”. i.e. ∇.∇ϕ
Match ListI (Terms) with ListII (Type) and select the correct answer using the codes given below the lists:
ListI
A. Curl= 0
B. Div = 0
C. Div grad (ϕ) = 0
D. Div div (ϕ) = 0
Listll
1. Laplace equation
2. Irrotational
3. Solenoidal
4. Not defined
Codes:
A B C D
(a) 2 3 1 4
(b) 4 1 3 2
(c) 2 1 3 4
(d) 4 3 1 2
Which of the following relations are not correct?
(A x B)^{2} = A^{2}B^{2}  (A·B)^{2}
If uF = ∇_{v}, where u and v are scalar fields and F is a vector field, then F. curl F is equal to
Given,
∴
or,
Hence,
A vector field which has a vanishing divergence is called as ____________
By the definition: A vector field whose divergence comes out to be zero or Vanishes is called as a Solenoidal Vector Field.
i.e.
is a Solenoidal Vector field.
Which of the following statements is not true of a phasor?
A phasor is always a vector quantity.
Match ListI (Physical quantities) with Listll (Dimensions) and select the correct answer using the codes given below the lists:
ListI
A. Electric potential
B. Magnetic flux
C. Magnetic field intensity
D. Magnetic flux density
Listll
1. MT^{2}I^{1}
2. ML^{2}T^{3}I^{1 }
3. IL^{1}
4. ML^{2}T^{2}I^{1}
Codes:
A B C D
(a) 2 4 3 1
(b) 4 2 3 1
(c) 1 2 1 3
(d) 4 2 1 3
Which of the following statements is not true regarding vector algebra?
Option (c) is not correct because cross product of two unlike vectors is a third unit vector having positive sign for normal rotation and negative for reverse rotation while cross product of two like unit vectors is zero.
A rigid body is rotating with an angular velocity of ω where, and v is the line velocity. If is the position vector given by then the value of curl will be equal to
Taking the curl, we have:
(Since )
or,
∴
If then which of the following relation will hold true?
We have:
= 1.1 + 1.1 + 1.1 = 1 + 1 + 1 = 3
Also,
Thus,
Hence, both (a) and (b) will hold true.
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