# Coordinate Systems and Vector Calculus

 Question 1
In the figure, the electric field $E$ and the magnetic field $B$ point to $\mathrm{x}$ and $\mathrm{z}$ 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. 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 $y$.
GATE EE 2023   Electromagnetic Theory
Question 1 Explanation:
As per Answer key of IIT Official : MTA (Marks to All)
Given :
 Question 2
The vector function expressed by

$F = a_x(5y - k_1z) + a_y(3z + k_2x) + a_z(k_3y - 4x)$

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:
 A $k_1=3, k_2=3, k_3=7$ B $k_1=3, k_2=8, k_3=5$ C $k_1=4, k_2=5, k_3=3$ D $k_1=0, k_2=0, k_3=0$
GATE EE 2020   Electromagnetic Theory
Question 2 Explanation:
$\bar{F}=(5y-k_{1}Z)\hat{i}+(3z+k_{2}x)\hat{j}+(k_{3}y-4x)\hat{k}$ is conservative field
$\bar{F}$ is irrotational,
\begin{aligned} \triangledown \times \bar{F}&=0\\ \begin{vmatrix} \hat{i} &\hat{j} &\hat{k} \\ \frac{\partial }{\partial x}&\frac{\partial }{\partial y} &\frac{\partial }{\partial z} \\ 5y-k_{1}z &3z-k_{2}x &k_{3}y -4x \end{vmatrix}&=0\\ \hat{i}(k_{3}-3)-\hat{j}(-4+k_{1})+\hat{k}(k_{2}-5)&=0\\ k_{3}-3&=0\\ 4-k_{1}&=0\\ k_{2}-5&=0\\ k_{1}&=4\\ k_{2}&=5\\ k_{3}&=3 \end{aligned}

 Question 3
The figures show diagrammatic representations of vector fields $\vec{X},\vec{Y}$ and $\vec{Z}$ respectively. Which one of the following choices is true? A $\bigtriangledown \cdot \vec{X}=0$, $\bigtriangledown \times \vec{Y}\neq 0,\bigtriangledown \times \vec{Z}=0$ B $\bigtriangledown \cdot \vec{X} \neq 0$, $\bigtriangledown \times \vec{Y}=0$, $\bigtriangledown \times \vec{Z} \neq 0$ C $\bigtriangledown \cdot \vec{X}\neq 0$, $\bigtriangledown \times \vec{Y}\neq 0$, $\bigtriangledown \times \vec{Z}\neq 0$ D $\bigtriangledown \cdot \vec{X}=0$, $\bigtriangledown \times \vec{Y}= 0$, $\bigtriangledown \times \vec{Z}=0$
GATE EE 2017-SET-2   Electromagnetic Theory
Question 3 Explanation:
$\vec{X}$ is going away so $\vec{\triangledown } \cdot \vec{X}\neq 0$
$\vec{Y}$ is moving circulator direction so $\vec{\triangledown } \cdot \vec{Y}\neq 0$
$\vec{Z}$ has circular rotation so $\vec{\triangledown } \cdot \vec{Z}\neq 0$
 Question 4
The line integral of the vector field $F = 5xz \hat{i}+ (3x^{2} + 2y) \hat{j} + x^{2}z\hat{k}$ along a path from (0,0,0) to (1,1,1) parametrized by (t, $t^{2}$, t) is _____.
 A 4.41 B 2.26 C 6.56 D 8.34
GATE EE 2016-SET-2   Electromagnetic Theory
Question 4 Explanation:
\begin{aligned} E &=5xZ\bar{i}+(3x^2 +2y)\bar{j}+x^2z\bar{k} \\ &=\int _C \bar{F}\bar{d}r \\ &= \int _C 5xzdx +(3x^2+2y)dy +x^2zdz\\ x&=t, \; y=t^2,\; z=t,\; t=0 \; to \; 1 \\ dx&=dt \\ dy &=2tdt,\; dz =dt\\ &=\int_{0}^{1} 5t^2dt+(3t^2+2t^2)2tdt+t^3dt\\ &= \int_{0}^{1} (5t^2+11t^3)dt\\ &=\left [ \frac{5t^3}{3}+\frac{11t^4}{4} \right ]_0^1=\frac{5}{3}+\frac{11}{4}=4.41 \end{aligned}
 Question 5
In cylindrical coordinate system, the potential produced by a uniform ring charge is given by $\psi = f(r, z)$ , where f is a continuous function of r and z. Let $\vec{E}$ be the resulting electric field. Then the magnitude of $\bigtriangledown \times \vec{E}$
 A increases with r. B is 0. C is 3. D decreases with z.
GATE EE 2016-SET-1   Electromagnetic Theory
Question 5 Explanation:
V is given as static field in time invariant.
Hence, $\triangledown \times E=0$

There are 5 questions to complete.