Q1: In the (x, y, z) coordinate system, three point-charges Q, Q and αQ are located in free space at (−1, 0, 0),(1, 0, 0) and (0, −1, 0) respectively. The value of α for the electric field to be zero at (0, 0.5, 0) is _____ (rounded off to 1 decimal places). (2024)
(a) -1.61
(b) -2.36
(c) -2.87
(d) -3.25
Ans: (a)
Sol: From the figure,
Q2: In the figure, the electric field E and the magnetic field B point to x and z directions, respectively, and have constant magnitudes. A positive charge 'q' is released from rest at the origin. Which of the following statement(s) is/ are true. (2023)
(a) The charge will move in the direction of z with constant velocity.
(b) The charge will al ways move on the y-z plane only.
(c) The trajectory of the charge will be a cycle.
(d) The charge will progress in the direction of y.
Ans: (a)
Sol: As per Answer key of IIT Official : MTA (Marks to All)
Given :
Q3: The vector function expressed by
represents a conservative field, where a_{x}, a_{y}, a_{z} are unit vectors along x, y and z directions, respectively. The values of constants k_{1}, k_{2}, k_{3} are given by: (2020)
(a) $\mathit{k}1=3,\mathit{k}2=3,\mathit{k}3=7$k_{1} = 3, k_{2} = 3, k_{3} = 7
(b) k_{1} = 3, k_{2} = 8, k_{3} = 5
(c) $\mathit{k}1=4,\mathit{k}2=5,\mathit{k}3=3$k_{1} = 4, k_{2} = 5, k_{3} = 3
(d) k_{1}= 0, k_{2 }= 0, k_{3 }= 0
Ans: (c)
Sol:
Q4: The figures show diagrammatic representations of vector fields respectively. Which one of the following choices is true? (SET-2 (2017))
(a) (b) (c) (d) Ans: (c)
Sol:
Q5: The line integral of the vector field along a path from (0, 0, 0) to (1, 1, 1) parametrized by (t, t^{2}, t) is _____. (SET-2 (2016))
(a) 4.41
(b) 2.26
(c) 6.56
(d) 8.34
Ans: (a)
Sol:
Q6: In cylindrical coordinate system, the potential produced by a uniform ring charge is given by ψ = f(r, z) , where f is a continuous function of r and z. Let be the resulting electric field. Then the magnitude of (SET-1 (2016))
(a) increases with r.
(b) is 0.
(c) is 3.
(d) decreases with z.
Ans: (b)
Sol: V is given as static field in time invariant.
Hence, ▽ × E = 0
Q7: Match the following. (SET-2 (2015))
(a) P-2 Q-1 R-4 S-3
(b) P-4 Q-1 R-3 S-2
(c) P-4 Q-3 R-1 S-2
(d) P-3 Q-4 R-2 S-1
Ans: (b)
Sol: Stokes theorem Gauss theorem Divergence theorem
Cauchy integral theorem
Q8: Consider a function where r is the distance from the origin and is the unit vector in the radial direction. The divergence of this function over a sphere of radius R, which includes the origin, is (SET-1 (2015))
(a) 0
(b) 2π
(c) 4π
(d) Rπ
Ans: (c)
Sol: From divergence theorem as we know,
Q9: The direction of vector A is radially outward from the origin, with ∣A∣ = kr^{n}, where r^{2 }= x^{2}+y^{2}+z^{2} and k is a constant. The value of n for which ▽⋅A = 0 is (2012)
(a) -2
(b) 2
(c) 1
(d) 0
Ans: (a)
Sol:
Q10: Divergence of the vector field
is (2007)
(a) $2\mathit{z}\mathrm{cos}\mathit{z}2$2z cos z^{2}
(b) $\mathrm{sin}\mathit{x}\mathit{y}+2\mathit{z}\mathrm{cos}\mathit{z}2$sinxy + 2z cos z^{2 }
(c) xsinxy − cosz
(d) None of these
Ans: (a)
Sol: Divergence
Q11: Consider the following statements with reference to the equation δp/δt
- This is a point form of the continuity equation.
- Divergence of current density is equal to the decrease of charge per unit volume per unit at every point.
- This is Max well's divergence equation
- This represents the conservation of charge
Select the correct answer. (2006)
(a) Only 2 and 4 are true
(b) 1, 2 and 3 are true
(c) 2, 3 and 4 are true
(d) 1, 2 and 4 are true
Ans: (d)
Q12: If is the electric intensity, is equal to (2005)
(a) (b) (c) null vector
(d) Zero
Ans: (d)
Sol: Divergence of a curl field is always zero.
i.e.
Q13: Given a vector field the divergence theorem states that (2002)
(a) (b) (c) (d) Ans: (a)