Electromagnetic Theory

Question 1
As shown in the figure below, two concentric conducting spherical shells, centered at r=0 and having radii r=c and r=d are maintained at potentials such that the potential V(r) at r=c is V_1 and V(r) at r=d is V_2. Assume that V(r) depends only on r , where r is the radial distance. The expression for V(r) in the region between r=c and r=d is

A
V(r)=\frac{cd(V_2-V_1)}{(d-c)r}-\frac{V_1c+V_2d-2V_1d}{d-c}
B
V(r)=\frac{cd(V_1-V_2)}{(d-c)r}-\frac{V_2d-V_1c}{d-c}
C
V(r)=\frac{cd(V_1-V_2)}{(d-c)r}-\frac{V_1c-V_2c}{d-c}
D
V(r)=\frac{cd(V_2-V_1)}{(d-c)r}-\frac{V_2c-V_1c}{d-c}
GATE EE 2022      Electrostatic Fields
Question 1 Explanation: 
We have, Laplace equation
\triangledown ^2V=0 ...(1)
where, \triangledown ^2V=\frac{1}{r^2}\frac{\partial }{\partial r}\left ( r^2\frac{\partial V}{\partial r} \right )+\frac{1}{r^2 \sin \theta }\frac{\partial }{\partial \theta }\left ( \sin \theta \frac{\partial V}{\partial \theta } \right )+ \frac{1}{r^2 \sin ^2\theta } \frac{\partial^2 V}{\partial \phi ^2}
V will have only radial component.
From equation (1)
\frac{1}{r^2}\frac{\partial }{\partial r}\left ( r^2\frac{\partial V}{\partial r} \right )=0
Integrate both side r^2\frac{\partial V}{\partial r} =P
Again integrate, V=-\frac{P}{r}+Q
Here, P and Q are constant.
Given: at r=c, V=V_1
V_1=-\frac{P}{r}+Q ...(2)
at r=d, V=V_2
V_2=-\frac{P}{d}+Q ...(3)
From eq. (2) & (3)
P=\frac{(V_2-V_1)cd}{d-c}
Q=V_1+\frac{d(V_2-V_1)}{d-c}
From eq. (1),
V=-\frac{(V_2-V_1)cd}{r(d-c)}+V_1+\frac{(V_2-V_1)cd}{(d-c)}
V=\frac{(V_1-V_2)cd}{r(d-c)}+\frac{(V_2d-V_1c)}{(d-c)}
Question 2
If the magnetic field intensity (H) in a conducting region is given by the expression, H=x^2\hat{i}+x^2y^2\hat{j}+x^2y^2z^2\hat{k}\;A/m. The magnitude of the current density, in A/m^2 , at x=1m, y=2m and z=1m , is
A
8
B
12
C
16
D
20
GATE EE 2022      Magnetostatic Fields
Question 2 Explanation: 
We have,
\begin{aligned} \vec{J}&=\triangledown \times \vec{H}\\ \vec{J}&=\begin{vmatrix} \hat{a}_x &\hat{a}_y & \hat{a}_z\\ \frac{\partial }{\partial x}& \frac{\partial }{\partial y} &\frac{\partial }{\partial z} \\ x^2& x^2y^2 & x^2y^2z^2 \end{vmatrix} \\ &=\hat{a}_x(2x^2yz^2-0)-\hat{a}_y(2xy^2z^2-0)+\hat{a}_z(2xy^2-0)\\ &=2x^2yz^2\hat{a}_x-2xy^2z^2\hat{a}_y+2xy^2\hat{a}_z\\ &\text{At point (1,2,1)}\\ \vec{J}&=4\hat{a}_x-8\hat{a}_y+8\hat{a}_z\\ |\vec{J}|&=\sqrt{4^2+8^2+8^2}=12A/m^2 \end{aligned}
Question 3
A long conducting cylinder having a radius 'b' is placed along the z axis. The current density is J=J_ar^3\hat{z} for the region r \lt b where r is the distance in the radial direction. The magnetic field intensity (H) for the region inside the conductor (i.e. for r \lt b) is
A
\frac{J_a}{4}r^4
B
\frac{J_a}{3}r^3
C
\frac{J_a}{5}r^4
D
J_a r^3
GATE EE 2022      Magnetostatic Fields
Question 3 Explanation: 
Using Ampere?s law,
\begin{aligned} \oint \vec{H}\cdot d\vec{l}&=I_{enc}\\ \text{where, }I_{enc}&=\int \int \vec{J}\cdot d\vec{s}\\ d\vec{s}&=rdrd\phi \hat{a}_z\\ \therefore \oint \vec{H}\cdot d\vec{l}&=\int \int \vec{J}\cdot r^3\hat{a}_z\cdot rdrd\phi \hat{a}_z\\ \vec{H}\cdot 2 \pi r&=\frac{J_ar^5}{5}(2 \pi)\\ \Rightarrow \vec{H}&=\frac{J_ar^4}{5}A/m \end{aligned}
Question 4
Consider a large parallel plate capacitor. The gap 'd' between the two plates is filled entirely with a dielectric slab of relative permittivity 5. The plates are initially charged to a potential difference of V volts and then disconnected from the source. If the dielectric slab is pulled out completely, then the ratio of the new electric field E_{2} in the gap to the original electric field E_{1} is ________.
A
2
B
5
C
6
D
8
GATE EE 2021      Electrostatic Fields
Question 4 Explanation: 


If voltage source is removed then in both cases charge O is constant.
\begin{aligned} \text { Case-1: }\left(Q_{1},=Q ; V_{1}=\right.&V) \\ Q_{1} &=C_{1} V_{1} \\ Q &=\frac{\epsilon_{0}(5) A}{d} V_{1} \\ Q &=5\left(\frac{\epsilon_{0} A}{d}\right) V_{1} \\ \text { Case-2: }\left(Q_{2}=Q ; V_{2}\right) & \\ Q_{2} &=C_{2} V_{2} \\ Q &=\frac{\epsilon_{0}(1) A}{d} V_{2} \\ Q &=\left(\frac{\epsilon_{0} A}{d}\right) V_{2} \end{aligned}
Equation (i) is equal to equation (ii)
\begin{aligned} \Rightarrow 5\left(\frac{\epsilon_{0} A}{d}\right) V_{1} &=\left(\frac{\epsilon_{0} A}{d}\right) V_{2} \\ \Rightarrow \qquad \qquad 5 V_{1} &=V_{2} \\ \frac{V_{2}}{V_{1}} &=5 \\ E_{1} &=\frac{V_{1}}{d} ; E_{2}=\frac{V_{2}}{d} \\ \frac{E_{2}}{E_{1}} &=\frac{V_{2} / d}{V_{1} / d} \\ \Rightarrow \qquad \qquad \frac{E_{2}}{E_{1}} &=\frac{V_{2}}{V_{1}} \end{aligned}
Put equation (iii) in equation (iv),
\Rightarrow \Rightarrow \qquad \qquad \frac{E_{2}}{E_{1}}=5
Question 5
One coulomb of point charge moving with a uniform velocity 10\:\hat{X} \text{m/s} enters the region x\geq 0 having a magnetic flux density \overrightarrow{B}=\left ( 10y\:\hat{X} + 10x\:\hat{Y}+10\:\hat{Z}\right )T. The magnitude of force on the charge at x=0^{+} is _____________ N.
(\hat{X}, \hat{Y}, and \hat{Z} are unit vectors along x-axis, y-axis and z-axis, respectively.)
A
80
B
125
C
112
D
100
GATE EE 2021      Magnetostatic Fields
Question 5 Explanation: 
Force on a charge moving with \vec{v} velocity due to magnetic field is
\begin{aligned} \vec{F} &=Q(\vec{v} \times \vec{B}) \\ &=1[10 \hat{x} \times(10 y \hat{x}+10 x \hat{y}+10 \hat{z}]\\ &=10(10 x) \hat{z}+10(10)(-\hat{y}) \\ &=100 x \hat{z}-100 \hat{y} \\ \left.\vec{F}\right|_{x=0^{+}} &=-100 \hat{y} \\ |\vec{F}| &=\sqrt{(-100)^{2}}=100 \text { Newton } \end{aligned}

Question 6
A 1\;\mu C point charge is held at the origin of a cartesian coordinate system. If a second point charge of 10\;\mu C is moved from \left ( 0,10,0 \right ) to \left ( 5,5,5\right ) and subsequently to \left ( 5,0,0\right ), then the total work done is ___________\text{mJ}. (Round off to 2 decimal places).
Take \dfrac{1}{4\pi \varepsilon _{0}}=9\times 10^{9} in SI units. All coordinates are in meters.
A
5.25
B
9
C
8.62
D
2.47
GATE EE 2021      Electrostatic Fields
Question 6 Explanation: 


In this case work done is independent of type of path but depends an intial and final point.
\begin{array}{ccccccccc} x & y & z & & r & \theta & \phi \\ (0, & 10, & 0) & \rightarrow & (10, & 90^{\circ}, & \left.90^{\circ}\right) & \text { Initial point }\\ x & y & z & & r & \theta & \phi & \\ (5, & 5, & 5) & \rightarrow & (5 \sqrt{3}, & 54.73^{\circ}, & \left.45^{\circ}\right) & \text { Intermediate point } \\ x & y & z & & r & \theta & \phi & \\ (5, & 0, & 0) & \rightarrow & (5, & 90^{\circ}, & \left.0^{\circ}\right) & \text { Final point } \end{array}
Work done (by external source) in moving Q-charge (10 \mu C) in the presence of electric field \vec{E} is
\begin{aligned} W&=-Q \int_{\text {initial point }}^{\text {final point }} \vec{E} \cdot d \vec{l}=-\left(10 \times 10^{-6}\right) \int_{r=10}^{r=5} \frac{9 \times 10^{3}}{r^{2}} \hat{a}_{r} \cdot d r \hat{a}_{r} \\ W&=-10 \times 10^{-6}\left(9 \times 10^{3}\right)\left(-\frac{1}{r}\right)_{r=10}^{5} \\ &=9 \times 10^{-2}\left[\frac{1}{5}-\frac{1}{10}\right]=9 \times 10^{-2}\left[\frac{10-5}{50}\right]=9 \times 10^{-2}\left(10^{-1}\right) \\ W&=9 \mathrm{~mJ} \end{aligned}
Question 7
Which one of the following vector functions represents a magnetic field \overrightarrow{B}?
(\hat{X}, \hat{Y} and \hat{Z} are unit vectors along x-axis, y-axis, and z-axis, respectively)
A
10x\hat{X}+20y\hat{Y}-30z\hat{Z}
B
10y\hat{X}+20x\hat{Y}-10z\hat{Z}
C
10z\hat{X}+20y\hat{Y}-30x\hat{Z}
D
10x\hat{X}-30z\hat{Y}+20y\hat{Z}
GATE EE 2021      Magnetostatic Fields
Question 7 Explanation: 
If \vec{B} is magnetic flux density then \vec{\nabla} \cdot \vec{B}=0
\begin{aligned} &\vec{\nabla} \cdot \vec{B}=\frac{\partial B x}{\partial x}+\frac{\partial B y}{\partial y}+\frac{\partial B z}{\partial z}\\ &\frac{\partial}{\partial x}(10 x)+\frac{\partial}{\partial y}(20 y)+\frac{\partial}{\partial z}(-30 z)=\vec{\nabla} \cdot \vec{B}\\ &\qquad \qquad \vec{\nabla} \cdot \vec{B}=10+20-30=0 \end{aligned}
Question 8
Let a_r, a_\phi \; and \; a_z be unit vectors along r, \phi \; and \; z directions, respectively in the cylindrical coordinate system. For the electric flux density given by D = (a_r15 + a_\phi 2r - a_z3rz) Coulomb/m^2, the total electric flux, in Coulomb, emanating from the volume enclosed by a solid cylinder of radius 3 m and height 5 m oriented along the z-axis with its base at the origin is:
A
108 \pi
B
54 \pi
C
90 \pi
D
180 \pi
GATE EE 2020      Electrostatic Fields
Question 8 Explanation: 
\begin{aligned} & \psi / \text{Crossing closed surface} \\ \oint \oint \vec{D}.\vec{ds}&=\int \int \int (\vec{\triangledown }\cdot \vec{D}) dv \\ (\vec{V}.\vec{D}) &=\frac{1}{\rho }\frac{\partial }{\partial p}(\rho D_{\rho })+\frac{1}{\rho }\frac{\partial D_{\phi }}{\partial \phi }+\frac{\partial D_{z}}{\partial z} \\ &=\frac{1}{\rho }\frac{\partial }{\partial p}(\rho 15)+\frac{1}{\rho }\frac{\partial }{\partial \phi }(2\rho )+\frac{\partial }{\partial z}(-3\rho z) \\ &=\frac{1}{\rho }15-3\rho \\ \int \int \int (\vec{\triangledown }\cdot \vec{D})dv&=\int \int \int \left ( \frac{15}{\rho}-3\rho \right )\rho d\rho d\phi dz \\ &=\int \int \int 15d\rho d\phi dz-3\int \int \int \rho ^{2}d\rho d\phi dz \\ &=15\int_{\rho =0}^{3}d\rho \int_{\phi =0}^{2\pi }d\phi \int_{z=0}^{5}-3\int_{\rho =0}^{3}\rho^{2}d\rho \int_{\phi =0}^{2\pi }d\phi \int_{z=0}^{5}dz \\ &= 15(3-0)(2\pi )(5)-3\left ( \frac{3^{3}}{3} \right )\times (2\pi )(5) \\ &=45(10\pi )-27(10\pi )=180\pi C \end{aligned}
Question 9
The static electric field inside a dielectric medium with relative permittivity, \varepsilon _r = 2.25, expressed in cylindrical coordinate system is given by the following expression

E=a_r 2r+a_\varphi \left ( \frac{3}{r} \right )+a_z6

where a_r,a_\varphi ,a_z are unit vectors along r, \varphi \; and \;z directions, respectively. If the above expression represents a valid electrostatic field inside the medium, then the volume charge density associated with this field in terms of free space permittivity, \varepsilon _0, in SI units is given by:
A
3\varepsilon _0
B
4\varepsilon _0
C
5\varepsilon _0
D
9\varepsilon _0
GATE EE 2020      Electrostatic Fields
Question 9 Explanation: 
\vec{D}=\epsilon \vec{E}=\epsilon _{o}\epsilon _{r}\vec{E}
\vec{D}=\epsilon _{o}2.25\left ( 2r\hat{a_{r}}+\frac{3}{r}\hat{a_{\phi}+6\hat{a_{z}}} \right )
\vec{D}=4.5\epsilon _{o}r\hat{a_{r}}+\frac{6.75\epsilon _{o}}{r}\hat{a_{\phi }}+13.5\epsilon _{o}\hat{a_{z}}
volume charge density
\rho _{v}=\vec{\bigtriangledown }\cdot \vec{D}
\rho _{v}=\frac{1}{r}\frac{\partial }{\partial r}(rD_{r})+\frac{1}{r}\frac{\partial D_{\phi }}{\partial \phi }+\frac{\partial D_{z}}{\partial z}
\rho _{v}=\frac{1}{r}\frac{\partial }{\partial r}(r4.5\epsilon _{o}r)+\frac{1}{r}\frac{\partial }{\partial \phi }\left ( \frac{6.75\epsilon _{o}}{r} \right )+\frac{\partial }{\partial z}(13.5\epsilon _{o})
\, \, \, =\frac{1}{r}\frac{\partial }{\partial r}(4.5\epsilon _{o}r^{2})+0+0
\, \, \, =\frac{1}{r}(4.5\epsilon _{o})(2r)=9\epsilon _{o}
Question 10
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      Coordinate Systems and Vector Calculus
Question 10 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}
There are 10 questions to complete.