Electrostatics · Electromagnetic Fields · GATE EE
Marks 1
1
Which one of the following figures represents the radial electric field distribution $E_R$ caused by a spherical cloud of electrons with a volume charge density. $\rho=-3 \rho_0$ for $0 \leq R \leq a$ (both $\rho_0, a$ are positive and $R$ is the radial distance) and $\rho=0$ for $R>a$ ?
GATE EE 2025
2
A positive charge of 1 nC is placed at (0, 0, 0.2) where all dimensions are in metres.
Consider the
x - y
plane to be a conducting ground plane.
Take $${\varepsilon _0} = 8.85 \times {10^{ - 12}}$$ F/m.
The Z component of the E field at (0, 0, 0.1) is closest to
Take $${\varepsilon _0} = 8.85 \times {10^{ - 12}}$$ F/m.
The Z component of the E field at (0, 0, 0.1) is closest to
GATE EE 2018
3
Consider an electron, a neutron and a proton initially at rest and placed along a straight line such
that the neutron is exactly at the center of the line joining the electron and proton. At t=0, the
particles are released but are constrained to move along the same straight line. Which of these
will collide first?
GATE EE 2017 Set 1
4
Consider a solid sphere of radius 5 cm made of a perfect electric conductor. If one million
electrons are added to this sphere, these electrons will be distributed.
GATE EE 2017 Set 2
5
Consider a function $$\overrightarrow f=\frac1{r^2}\widehat r$$ where r is the distance from the origin and $$\widehat r$$ is the unit
vector in the radial direction. The divergence of the function over a sphere of radius R,
which includes the origin, is
GATE EE 2015 Set 1
6
Match the following.
GATE EE 2015 Set 2
7
Two semi-infinite dielectric regions are separated by a plane boundary at y = 0. The dielectric constants of region 1 (y< 0) and region 2 (y>0) are 2 and 5, respectively. Region 1 has uniform electric field $$\overrightarrow E=3{\widehat a}_x\;+\;4{\widehat a}_y\;+\;2{\widehat a}_z$$, where $${\widehat a}_x$$, $${\widehat a}_y$$, and $${\widehat a}_Z$$ are unit vectors along the x, y and z axes, respectively. The electric field in region 2 is
GATE EE 2015 Set 2
8
A hollow metallic sphere of radius r is kept at potential of 1 Volt. The total electric flux
coming out of the concentric spherical surface of radius R(>r) is
GATE EE 2014 Set 3
9
A parallel plate capacitor consisting two dielectric materials is shown in the figure. The
middle dielectric slab is place symmetrically with respect to the plates.
If the potential difference between one of the plates and the nearest surface of dielectric
interface is 2 Volts, then the ratio $$\varepsilon_1\;:\;\varepsilon_2$$ is
If the potential difference between one of the plates and the nearest surface of dielectric
interface is 2 Volts, then the ratio $$\varepsilon_1\;:\;\varepsilon_2$$ isGATE EE 2014 Set 2
10
Divergence of the three-dimentional radial vector field $$\overrightarrow F$$ is
GATE EE 2010
11
Divergence of the vector field
$$$\overrightarrow V\left(x,y,z\right)=-\left(x\cos xy\;+\;y\right)\;\widehat i\;+\;\left(y\cos xy\right)\;\widehat j\;+\;\left(\sin\;z^2\;+\;x^2\;+\;y^2\right)\widehat k$$$ is
GATE EE 2007
12
A parallel plate capacitor is shown in Fig. It is made of two square metal plates of $$400$$ $$mm$$ side. The $$14$$ $$mm$$ space between the plates is filled with two layers of dielectrics of $${\varepsilon _r} = 4,6\,\,mm$$ thick and $${\varepsilon _r} = 2,8\,\,mm$$ thick. Neglecting fringing of fields at the edges, the capacitance is
GATE EE 2004
13
Given a vector field $$\overrightarrow F ,$$ the divergence theorem states that
GATE EE 2002
14
In a uniform electric field, field lines and equipotentials
GATE EE 1997
15
If v, w, q stand for voltage, energy and charge, then v can be expressed
GATE EE 1996
16
In electrostatic field $$\nabla\times\overrightarrow E\equiv0$$
GATE EE 1994
17
An electrostatic potential is given by $$\phi=2x\sqrt y$$ volt in the rectangular co-ordinate system. The magnitude of the electric field at x = 1 m,y = 1 m is _________V/m
GATE EE 1992
18
Which of the following equations represents the Gauss's law in a homogeneous isotropic medium?
GATE EE 1992
Marks 2
1
An air filled cylindrical capacitor (capacitance $C_0$ ) of length $L$, with $a$ and $b$ as its inner and outer radii, respectively, consists of two coaxial conducting surfaces. Its crosssectional view is shown in Figure (i). In order to increase the capacitance, a dielectric material of relative permittivity $\varepsilon_r$ is inserted inside $50 \%$ of the annular region as shown in figure (ii). The value of $\varepsilon_r$ for which the capacitance of the capacitor in figure (ii), becomes $5 C_0$ is
GATE EE 2025
2
In the $(x, y, z)$ coordinate system, three point-charges $Q$, $Q$, and $\alpha Q$ are located in free space at $(-1, 0, 0)$, $(1, 0, 0)$, and $(0, -1, 0)$, respectively. The value of $\alpha$ for the electric field to be zero at $(0, 0.5, 0)$ is _________________ (rounded off to 1 decimal place).
GATE EE 2024
3
The closed curve shown in the figure is described by
$$r = 1 + \cos \theta $$, where $$r = \sqrt {{x^2} + {y^2}} ;x = r\cos \theta ,y = r\sin \theta $$The magnitude of the line integral of the vector field $$F = - y\widehat i + x\widehat j$$ around the closed curve is ___________ (Round off to 2 decimal places).
GATE EE 2023
4
A quadratic function of two variables is given as
$$f({x_1},{x_2}) = x_1^2 + 2x_2^2 + 3{x_1} + 3{x_2} + {x_1}{x_2} + 1$$
The magnitude of the maximum rate of change of the function at the point (1, 1) is ___________ (Round off to the nearest integer).
GATE EE 2023
5
As shown in the figure below, to concentric conducting spherical shells, centred 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 V1 and V(r) at r = d is V2. 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
GATE EE 2022
6
The capacitance of an air-filled parallel-plate capacitor is 60 pF. When a dielectric slab
whose thickness is half the distance between the plates, is placed on one of the plates
covering it entirely, the capacitance becomes 86 pF. Neglecting the fringing effects, the
relative permittivity of the dielectric is _____________ (up to 2 decimal places).
GATE EE 2018
7
A thin soap bubble of radius R = 1 cm, and thickness a = 3.3 µm(a << R), is at a potential of 1
V with respect to a reference point at infinity. The bubble bursts and becomes a single spherical
drop of soap (assuming all the soap is contained in the drop) of radius r. The volume of the soap
in the thin bubble is $$4\mathrm{πR}^2\mathrm a$$ and that of the drop is $$\frac43\mathrm{πr}^3$$. The potential in volts, of the resulting
single spherical drop with respect to the same reference point at infinity is __________. (Give
the answer up to two decimal places.)
GATE EE 2017 Set 2
8
Two electric charges q and -2q are placed at (0,0) and (6,0) on the x-y plane. The equation of the
zero equipotential curve in the x-y plane is
GATE EE 2016 Set 1
9
A parallel plate capacitor is partially filled with glass of dielectric constant 4.0 as shown below. The dielectric strengths of air and glass are 30 kV/cm and 300 kV/cm, respectively. The maximum voltage (in kilovolts), which can be applied across the capacitor without any breakdown, is ______.
GATE EE 2015 Set 1
10
Two semi-infinite conducting sheets are placed at right angles to each other as shown in the figure. A point charge of +𝑄 is placed at a distance of 𝑑 from both sheets. The net force on the charge is $$\frac{Q^2}{4{\mathrm{πε}}_0}\frac{\overrightarrow K}{d^2}$$ , where $$\overrightarrow K$$ is given by
GATE EE 2015 Set 2
11
A perfectly conducting metal plate is placed in x-y plane in a right handed coordinate system.
A charge of $$+32{\mathrm{πε}}_0\sqrt2$$ columbs is placed at coordinate (0, 0, 2). $${\mathrm\varepsilon}_0$$ is the permittivity of free
space. Assume $$\widehat i,\;\widehat j,\;\widehat k$$ to be unit vectors along x, y and z axes respectively. At the coordinate
$$\left(\sqrt2,\sqrt2,0\right)$$, the electric field vector $$\overrightarrow E$$
(Newtons/Columb) will be
GATE EE 2014 Set 3
12
A dielectric slab with 500 mm x 500 mm cross-section is 0.4 m long. The slab is
subjected to a uniform electric field of $$E=6{\widehat a}_x+8{\widehat a}_y$$ kV /mm. The relative
permittivity of the dielectric material is equal to 2. The value of constant $$\varepsilon_0$$ is
$$8.85\times10^{-12}\;F/m$$ . The energy stored in the dielectric in Joules is
GATE EE 2013
13
A capacitor is made with a polymeric dielectric having an $$\varepsilon_0$$ of 2.26 and a
dielectric breakdown strength of 50 kV/cm. The permittivity of free space is
8.85 pF/m. If the rectangular plates of the capacitor have a width of 20 cm and a
length of 40 cm, then the maximum electric charge in the capacitor is
GATE EE 2011
14
$$$F\left(x,y\right)=\left(x^2\;+\;xy\right)\;{\widehat a}_x\;+\;\left(y^2\;+\;xy\right)\;{\widehat a}_y$$$. Its line integral over the straight line from (x, y)=(0,2) to (2,0) evaluates to
GATE EE 2009
15
A capacitor consists of two metal plates each $$500 \times 500\,\,m{m^2}$$ and spaced $$6$$ $$mm$$ apart. The space between the metal plates is filled with a glass plate of $$4$$ $$mm$$ thickness and a layer of paper of $$2$$ $$mm$$ thickness. The relative permittivity of the glass and paper are $$8$$ and $$2$$ respectively. Neglecting the fringing effect, the capacitance will be (Given that $${\varepsilon _0} = 8.85 \times {10^{ - 12}}\,F/m$$ )
GATE EE 2008
16
Two point charges $${Q_1} = 10\,\,\mu C$$ and $${Q_2} = 20\,\,\mu C$$ are placed at coordinates $$(1,1,0)$$ and $$\left( { - 1, - 1,0} \right)$$ respectively. The total electric flux passing through a plane $$z=20$$ will be
GATE EE 2008
17
A solid sphere made of insulating material has a radius $$R$$ and has a total charge $$Q$$ distributed uniformly in its volume. What is the magnitude of the electric field intensity, $$E,$$ at distribution $$r\left( {0 < r < R} \right)$$ inside the sphere?
GATE EE 2007
18
A composite parallel plate capacitor is made up of two different dielectric materials with different thickness ($${t_1}$$ and $${t_2}$$) as shown in Fig. The two different dielectric materials are separated by a conducting foil $$F.$$ The voltage of the conducting foil is
GATE EE 2003
19
A point charge of $$+1$$ $$nC$$ is placed in a space with a permittivity of $$8.85 \times {10^{ - 12}}\,F/m$$ as shown in Fig. The potential difference $${V_{PQ}}$$ between two points $$P$$ and $$Q$$ at distance of $$40mm$$ and $$20$$ $$mm$$ respectively from the point charge is
GATE EE 2003
20
A parallel plate capacitor has an electrode area of $$100$$ $$m{m^2}$$, with a spacing of $$0.1$$ $$mm$$ between the electrodes. The dielectric between the plates is air with a permittivity of $$8.85 \times {10^{ - 12}}\,\,F/m.$$ The charge on the capacitor is $$100$$ $$V$$. The stored energy in the capacitor is
GATE EE 2003
21
Given the potential function in free space to be $$V\left( x \right) = \left( {50{x^2} + 50{y^2} + 50{z^2}} \right)$$ volts, the magnitude (in volts /metre) and the direction of the electric field at a point $$\left( {1,\,\, - 1,\,\,1} \right),$$ where the dimensions are in metres, are
GATE EE 2001
22
The electric field $$\overrightarrow E $$ (in volts/metre) at the point $$(1, 1, 0)$$ due to a point charge of $$+1$$ $$\mu C$$ located at $$\left( { - 1,\,1,\,1} \right)$$ (coordinates in metres) is
GATE EE 2001
23
The capacitance of an isolated sphere of radius 10 cm in air is equal to _____ pF.
GATE EE 1997
24
The capacitance of the arrangement shown in Fig. is ................... $$pF$$
GATE EE 1997
Marks 5
1
Show via the construction of a suitable Gaussian surface, that the capacitance of a spherical capacitor consisting of two concentric shells of radii $$a$$ and $$b$$ is given by $$C = 4\pi {\varepsilon _0}{{ab} \over {\left( {b - a} \right)}}$$ where $${\varepsilon _0}$$ is the free space permittivity.
GATE EE 2000
2
A charge $$+Q$$ is uniformly distributed throughout the volume of a dielectric sphere of radius $$R$$ and dielectric constant $$'{\varepsilon _R}'.$$ Based on Gauss law, determine the expressions for the electric field $$E$$ as a function of distance $$'r'$$ from the center of the sphere, within the ranges $$0 < r < R$$ and $$R \le r.$$ Indicate expression $$(s)$$ for the
critical point $$(s)$$ on the sketch.
GATE EE 1999
3
An infinite number of charges, each equal to $$'q'$$ are placed along the $$x=1,x=2, x=4, x=8, x=16$$ and so on, Find the potential and electric field at point $$x=0,$$ due to these system of charges.
GATE EE 1998
4
Determine the electric field intensity at the point $$P$$ for the arrangement shown in Fig.
GATE EE 1997