2022
JEE Mains
MCQ
JEE Main 2022 (Online) 24th June Morning Shift
A vertical electric field of magnitude 4.9 $\times$ 105 N/C just prevents a water droplet of a mass 0.1 g from falling. The value of charge on the droplet will be :
(Given : g = 9.8 m/s2)
A.
1.6 $\times$ 10$-$9 C
B.
2.0 $\times$ 10$-$9 C
C.
3.2 $\times$ 10$-$9 C
D.
0.5 $\times$ 10$-$9 C
2022
JEE Mains
Numerical
JEE Main 2022 (Online) 28th July Morning Shift
Two electric dipoles of dipole moments $1.2 \times 10^{-30} \,\mathrm{Cm}$ and $2.4 \times 10^{-30} \,\mathrm{Cm}$ are placed in two different uniform electric fields of strengths $5 \times 10^{4} \,\mathrm{NC}^{-1}$ and $15 \times 10^{4} \,\mathrm{NC}^{-1}$ respectively. The ratio of maximum torque experienced by the electric dipoles will be $\frac{1}{x}$. The value of $x$ is __________.
Correct Answer: 6
Explanation:
${{{\rho _1}} \over {{\rho _2}}} = {{{\mu _1}{B_1}\sin 90} \over {{\mu _2}{B_2}\sin 90}}$
$ = {{1.2 \times {{10}^{ - 30}} \times 5 \times {{10}^4}} \over {2.4 \times {{10}^{ - 30}} \times 15 \times {{10}^4}}}$
$ = {1 \over 6}$
2022
JEE Mains
Numerical
JEE Main 2022 (Online) 27th July Morning Shift
A long cylindrical volume contains a uniformly distributed charge of density $\rho \,\mathrm{Cm}^{-3}$. The electric field inside the cylindrical volume at a distance $x=\frac{2 \varepsilon_{0}}{\rho} \mathrm{m}$ from its axis is ________ $\mathrm{Vm}^{-1}$.
Correct Answer: 1
Explanation:
$E = {{\rho r} \over {2{\varepsilon _0}}}$
at $r = {{2{\varepsilon _0}} \over \rho }$
$E = {\rho \over {2{\varepsilon _0}}}\left( {{{2{\varepsilon _0}} \over \rho }} \right)$
$ = 1$
2022
JEE Mains
Numerical
JEE Main 2022 (Online) 26th July Evening Shift
Three point charges of magnitude $5 \mu \mathrm{C}, 0.16 \mu \mathrm{C}$ and $0.3 \mu \mathrm{C}$ are located at the vertices $A, B, C$ of a right angled triangle whose sides are $A B=3 \mathrm{~cm}, B C=3 \sqrt{2} \mathrm{~cm}$ and $C A=3 \mathrm{~cm}$ and point $A$ is the right angle corner. Charge at point $\mathrm{A}$ experiences ____________ $\mathrm{N}$ of electrostatic force due to the other two charges.
Correct Answer: 17
Explanation:
${F_{AC}} = {{k(5 \times 0.3) \times {{10}^{ - 12}}} \over {9 \times {{10}^{ - 4}}}}$
${F_{AB}} = {{k(5 \times 0.16) \times {{10}^{ - 12}}} \over {9 \times {{10}^{ - 4}}}}$
${F_{net}} = {{k \times {{10}^{ - 12}}} \over {9 \times {{10}^{ - 4}}}}\sqrt {{{1.5}^2} + {{(0.8)}^2}} $
$ = {{{{10}^9} \times {{10}^{ - 12}}} \over {{{10}^{ - 4}}}} \times 1.7 = 17$
2022
JEE Mains
Numerical
JEE Main 2022 (Online) 25th July Morning Shift
The volume charge density of a sphere of radius $6 \mathrm{~m}$ is $2 \,\mu \mathrm{C} \,\mathrm{cm}^{-3}$. The number of lines of force per unit surface area coming out from the surface of the sphere is _______________ $\times 10^{10} \,\mathrm{NC}^{-1}$.
[Given : Permittivity of vacuum $\epsilon_{0}=8.85 \times 10^{-12} \,\mathrm{C}^{2}\, \mathrm{~N}^{-1}-\mathrm{m}^{-2}$ )
Correct Answer: 45
Explanation:
$\rho$ = 2 $\mu$c/cm3
R = 6 m
Number of lines of force per unit area = Electric field at surface.
$ = {{KQ} \over {{R^2}}}$
$ = {1 \over {4\pi {\varepsilon _0}}}{{\rho {4 \over 3}\pi {R^3}} \over {{R^2}}}$
$ = {{\rho R} \over {3{ \in _0}}}$
$ = {{2 \times {{10}^{ - 6}} \times {{10}^6} \times 6} \over {3 \times 8.85 \times {{10}^{ - 12}}}}$
$ = 0.45197 \times {10^{12}}$
$ = 45.19 \times {10^{10}}$ N/C
$ \simeq 45 \times {10^{10}}$
2022
JEE Mains
Numerical
JEE Main 2022 (Online) 30th June Morning Shift
Eight similar drops of mercury are maintained at 12 V each. All these spherical drops combine into a single big drop. The potential energy of bigger drop will be ____________ E. Where E is the potential energy of a single smaller drop.
Correct Answer: 32
Explanation:
From law of conservation of charge
$ \begin{aligned} &q_{i}=\mathrm{q}_{\mathrm{f}} \Rightarrow 8 \times\left(4 \pi \mathrm{E}_{0} \mathrm{R}\right) \times 12=\left(4 \pi E R^{1}\right) \times \mathrm{V}_{f} \\\\ &\Rightarrow 96 \mathrm{R}=\mathrm{V}_{\mathrm{f}} \mathrm{R}^{1} \\\\ &\text { And, } 8 \times \frac{4}{3} \pi R^{3}=\frac{4}{3} \pi R^{1} 3 \\\\ &8=\left(\frac{R^{1}}{R}\right)^{3} \end{aligned} $
$ \begin{aligned} & \mathrm{R}^{1}=2 \mathrm{R} \end{aligned} $
From (i) & (ii), we get
So, $96 \mathrm{R}=\mathrm{V}_{\mathrm{f}} \times 2 \mathrm{R} \Rightarrow \mathrm{V}_{\mathrm{f}}=48$ Volt
$ \begin{aligned} &V_{f}=\frac{1}{2} C_{f} V_{f}^{2}=\frac{1}{2} \times\left(4 \pi \varepsilon_{0} \mathrm{R}^{1}\right) \mathrm{V}_{f}^{2} \\\\ &=\frac{1}{2} \times\left(4 \pi \varepsilon_{0} \times 2 \mathrm{R}\right) \times 48^{2} \\\\ &=\left(\frac{1}{2} \times 4 \pi \varepsilon_{0} R \times 12^{2}\right) \times \frac{48^{2} \times 2}{12^{2}}=32 \,E \end{aligned} $
$ \begin{aligned} &q_{i}=\mathrm{q}_{\mathrm{f}} \Rightarrow 8 \times\left(4 \pi \mathrm{E}_{0} \mathrm{R}\right) \times 12=\left(4 \pi E R^{1}\right) \times \mathrm{V}_{f} \\\\ &\Rightarrow 96 \mathrm{R}=\mathrm{V}_{\mathrm{f}} \mathrm{R}^{1} \\\\ &\text { And, } 8 \times \frac{4}{3} \pi R^{3}=\frac{4}{3} \pi R^{1} 3 \\\\ &8=\left(\frac{R^{1}}{R}\right)^{3} \end{aligned} $
$ \begin{aligned} & \mathrm{R}^{1}=2 \mathrm{R} \end{aligned} $
From (i) & (ii), we get
So, $96 \mathrm{R}=\mathrm{V}_{\mathrm{f}} \times 2 \mathrm{R} \Rightarrow \mathrm{V}_{\mathrm{f}}=48$ Volt
$ \begin{aligned} &V_{f}=\frac{1}{2} C_{f} V_{f}^{2}=\frac{1}{2} \times\left(4 \pi \varepsilon_{0} \mathrm{R}^{1}\right) \mathrm{V}_{f}^{2} \\\\ &=\frac{1}{2} \times\left(4 \pi \varepsilon_{0} \times 2 \mathrm{R}\right) \times 48^{2} \\\\ &=\left(\frac{1}{2} \times 4 \pi \varepsilon_{0} R \times 12^{2}\right) \times \frac{48^{2} \times 2}{12^{2}}=32 \,E \end{aligned} $
2022
JEE Mains
Numerical
JEE Main 2022 (Online) 25th June Evening Shift
27 identical drops are charged at 22V each. They combine to form a bigger drop. The potential of the bigger drop will be _____________ V.
Correct Answer: 198
Explanation:
Let the charge on one drop is q and its radius is r.
So for one drop $V = {{kq} \over r}$
For 27 drops merged new charge will be Q = 27 q and new radius is R = 3r
So new potential is
$V' = {{kQ} \over R} = 9{{kq} \over r} = 9 \times 22$ V
= 198 V
2021
JEE Mains
MCQ
JEE Main 2021 (Online) 1st September Evening Shift
A cube is placed inside an electric field, $\overrightarrow E = 150{y^2}\widehat j$. The side of the cube is 0.5 m and is placed in the field as shown in the given figure. The charge inside the cube is :
A.
3.8 $\times$ 10$-$11 C
B.
8.3 $\times$ 10$-$11 C
C.
3.8 $\times$ 10$-$12 C
D.
8.3 $\times$ 10$-$12 C
2021
JEE Mains
MCQ
JEE Main 2021 (Online) 31st August Evening Shift
Choose the incorrect statement :
(1) The electric lines of force entering into a Gaussian surface provide negative flux.
(2) A charge 'q' is placed at the centre of a cube. The flux through all the faces will be the same.
(3) In a uniform electric field net flux through a closed Gaussian surface containing no net charge, is zero.
(4) When electric field is parallel to a Gaussian surface, it provides a finite non-zero flux.
Choose the most appropriate answer from the options given below
(1) The electric lines of force entering into a Gaussian surface provide negative flux.
(2) A charge 'q' is placed at the centre of a cube. The flux through all the faces will be the same.
(3) In a uniform electric field net flux through a closed Gaussian surface containing no net charge, is zero.
(4) When electric field is parallel to a Gaussian surface, it provides a finite non-zero flux.
Choose the most appropriate answer from the options given below
A.
(3) and (4) only
B.
(2) and (4) only
C.
(4) only
D.
(1) and (3) only
2021
JEE Mains
MCQ
JEE Main 2021 (Online) 31st August Morning Shift
Two particles A and B having charges 20$\mu$C and $-$5$\mu$C respectively are held fixed with a separation of 5 cm. At what position a third charged particle should be placed so that it does not experience a net electric force?
A.
At 5 cm from 20 $\mu$C on the left side of system
B.
At 5 cm from $-$5 $\mu$C on the right side
C.
At 1.25 cm from $-$5 $\mu$C between two charges
D.
At midpoint between two charges
2021
JEE Mains
MCQ
JEE Main 2021 (Online) 27th August Evening Shift
Figure shows a rod AB, which is bent in a 120$^\circ$ circular arc of radius R. A charge ($-$Q) is uniformly distributed over rod AB. What is the electric field $\overrightarrow E $ at the centre of curvature O ?
A.
${{3\sqrt 3 Q} \over {8\pi {\varepsilon _0}{R^2}}}(\widehat i)$
B.
${{3\sqrt 3 Q} \over {8{\pi ^2}{\varepsilon _0}{R^2}}}(\widehat i)$
C.
${{3\sqrt 3 Q} \over {16{\pi ^2}{\varepsilon _0}{R^2}}}(\widehat i)$
D.
${{3\sqrt 3 Q} \over {8{\pi ^2}{\varepsilon _0}{R^2}}}( - \widehat i)$
2021
JEE Mains
MCQ
JEE Main 2021 (Online) 27th August Morning Shift
A uniformly charged disc of radius R having surface charge density $\sigma$ is placed in the xy plane with its center at the origin. Find the electric field intensity along the z-axis at a distance Z from origin :-
A.
$E = {\sigma \over {2{\varepsilon _0}}}\left( {1 - {Z \over {{{({Z^2} + {R^2})}^{1/2}}}}} \right)$
B.
$E = {\sigma \over {2{\varepsilon _0}}}\left( {1 + {Z \over {{{({Z^2} + {R^2})}^{1/2}}}}} \right)$
C.
$E = {{2{\varepsilon _0}} \over \sigma }\left( {{1 \over {{{({Z^2} + {R^2})}^{1/2}}}} + Z} \right)$
D.
$E = {\sigma \over {2{\varepsilon _0}}}\left( {{1 \over {({Z^2} + {R^2})}} + {1 \over {{Z^2}}}} \right)$
2021
JEE Mains
MCQ
JEE Main 2021 (Online) 26th August Evening Shift
The two thin coaxial rings, each of radius 'a' and having charges +Q and $-$Q respectively are separated by a distance of 's'. The potential difference between the centres of the two rings is :
A.
${Q \over {2\pi {\varepsilon _0}}}\left[ {{1 \over a} + {1 \over {\sqrt {{s^2} + {a^2}} }}} \right]$
B.
${Q \over {4\pi {\varepsilon _0}}}\left[ {{1 \over a} + {1 \over {\sqrt {{s^2} + {a^2}} }}} \right]$
C.
${Q \over {4\pi {\varepsilon _0}}}\left[ {{1 \over a} - {1 \over {\sqrt {{s^2} + {a^2}} }}} \right]$
D.
${Q \over {2\pi {\varepsilon _0}}}\left[ {{1 \over a} - {1 \over {\sqrt {{s^2} + {a^2}} }}} \right]$
2021
JEE Mains
MCQ
JEE Main 2021 (Online) 26th August Morning Shift
A solid metal sphere of radius R having charge q is enclosed inside the concentric spherical shell of inner radius a and outer radius b as shown in the figure. The approximate variation electric field $\overrightarrow E $ as a function of distance r from centre O is given by
A.
B.
C.
D.
2021
JEE Mains
MCQ
JEE Main 2021 (Online) 27th July Evening Shift
What will be the magnitude of electric field at point O as shown in the figure? Each side of the figure is l and perpendicular to each other?
A.
${1 \over {4\pi {\varepsilon _0}}}{q \over {{l^2}}}$
B.
${1 \over {4\pi {\varepsilon _0}}}{q \over {(2{l^2})}}\left( {2\sqrt 2 - 1} \right)$
C.
${q \over {4\pi {\varepsilon _0}{{(2l)}^2}}}$
D.
${1 \over {4\pi {\varepsilon _0}}}{{2q} \over {2{l^2}}}\left( {\sqrt 2 } \right)$
2021
JEE Mains
MCQ
JEE Main 2021 (Online) 27th July Morning Shift
Two identical tennis balls each having mass 'm' and charge 'q' are suspended from a fixed point by threads of length 'l'. What is the equilibrium separation when each thread makes a small angle '$\theta$' with the vertical?
A.
$x = {\left( {{{{q^2}l} \over {2\pi {\varepsilon _0}mg}}} \right)^{{1 \over 2}}}$
B.
$x = {\left( {{{{q^2}l} \over {2\pi {\varepsilon _0}mg}}} \right)^{{1 \over 3}}}$
C.
$x = {\left( {{{{q^2}{l^2}} \over {2\pi {\varepsilon _0}{m^2}g}}} \right)^{{1 \over 3}}}$
D.
$x = {\left( {{{{q^2}{l^2}} \over {2\pi {\varepsilon _0}{m^2}{g^2}}}} \right)^{{1 \over 3}}}$
2021
JEE Mains
MCQ
JEE Main 2021 (Online) 25th July Evening Shift
Two ideal electric dipoles A and B, having their dipole moment p1 and p2 respectively are placed on a plane with their centres at O as shown in the figure. At point C on the axis of dipole A, the resultant electric field is making an angle of 37$^\circ$ with the axis. The ratio of the dipole moment of A and B, ${{{p_1}} \over {{p_2}}}$ is : (take $\sin 37^\circ = {3 \over 5}$)
A.
${3 \over 8}$
B.
${3 \over 2}$
C.
${2 \over 3}$
D.
${4 \over 3}$
2021
JEE Mains
MCQ
JEE Main 2021 (Online) 22th July Evening Shift
An electric dipole is placed on x-axis in proximity to a line charge of linear charge density 3.0 $\times$ 10$-$6 C/m. Line charge is placed on z-axis and positive and negative charge of dipole is at a distance of 10 mm and 12 mm from the origin respectively. If total force of 4N is exerted on the dipole, find out the amount of positive or negative charge of the dipole.
A.
0.485 mC
B.
815.1 nC
C.
8.8 $\mu$C
D.
4.44 $\mu$C
2021
JEE Mains
MCQ
JEE Main 2021 (Online) 20th July Morning Shift
A certain charge Q is divided into two parts q and (Q $-$ q). How should the charges Q and q be divided so that q and (Q $-$ q) placed at a certain distance apart experience maximum electrostatic repulsion?
A.
Q = 2q
B.
Q = 4q
C.
Q = 3q
D.
Q = ${q \over 2}$
2021
JEE Mains
MCQ
JEE Main 2021 (Online) 18th March Morning Shift
An oil drop of radius 2 mm with a density 3g cm$-$3 is held stationary under a constant electric field 3.55 $\times$ 105 V m$-$1 in the Millikan's oil drop experiment. What is the number of excess electrons that the oil drop will possess? (consider g = 9.81 m/s2)
A.
48.8 $\times$ 1011
B.
1.73 $\times$ 1010
C.
17.3 $\times$ 1010
D.
1.73 $\times$ 1012
2021
JEE Mains
MCQ
JEE Main 2021 (Online) 16th March Evening Shift
Find out the surface charge density at the intersection of point x = 3 m plane and x-axis, in the region of uniform line charge of 8 nC/m lying along the z-axis in free space.
A.
0.424 nC m$-$2
B.
4.0 nC m$-$2
C.
47.88 C/m
D.
0.07 nC m$-$2
2021
JEE Mains
MCQ
JEE Main 2021 (Online) 26th February Evening Shift
Given below are two statements:
Statement I : An electric dipole is placed at the center of a hollow sphere. The flux of the electric field through the sphere is zero but the electric field is not zero anywhere in the sphere.
Statement II : If R is the radius of a solid metallic sphere and Q be the total charge on it. The electric field at any point on the spherical surface of radius r (< R) is zero but the electric flux passing through this closed spherical surface of radius r is not zero..
In the light of the above statements, choose the correct answer from the options given below :
Statement I : An electric dipole is placed at the center of a hollow sphere. The flux of the electric field through the sphere is zero but the electric field is not zero anywhere in the sphere.
Statement II : If R is the radius of a solid metallic sphere and Q be the total charge on it. The electric field at any point on the spherical surface of radius r (< R) is zero but the electric flux passing through this closed spherical surface of radius r is not zero..
In the light of the above statements, choose the correct answer from the options given below :
A.
Both Statement I and Statement II are true
B.
Statement I is false but Statement II is true
C.
Statement I is true but Statement II is false
D.
Both Statement I and Statement II are false
2021
JEE Mains
MCQ
JEE Main 2021 (Online) 26th February Evening Shift
An inclined plane making an angle of 30$^\circ$ with the horizontal is placed in a uniform horizontal electric field $200{N \over C}$ as shown in the figure. A body of mass 1 kg and charge 5 mC is allowed to slide down from rest at a height of 1 m. If the coefficient of friction is 0.2, find the time taken by the body to reach the bottom.
[g = 9.8 m/s2; $\sin 30^\circ = {1 \over 2}$; $\cos 30^\circ = {{\sqrt 3 } \over 2}$]
[g = 9.8 m/s2; $\sin 30^\circ = {1 \over 2}$; $\cos 30^\circ = {{\sqrt 3 } \over 2}$]
A.
0.46 s
B.
0.92 s
C.
1.3 s
D.
2.3 s
2021
JEE Mains
MCQ
JEE Main 2021 (Online) 26th February Morning Shift
Find the electric field at point P (as shown in figure) on the perpendicular bisector of a uniformly charged thin wire of length L carrying a charge Q. The distance of the point P from the centre of the rod is a = ${{\sqrt 3 } \over 2}L$.
A.
${Q \over {4\pi {\varepsilon _0}{L^2}}}$
B.
${Q \over {3\pi {\varepsilon _0}{L^2}}}$
C.
${Q \over {2\sqrt 3 \pi {\varepsilon _0}{L^2}}}$
D.
${{\sqrt 3 Q} \over {4\pi {\varepsilon _0}{L^2}}}$
2021
JEE Mains
MCQ
JEE Main 2021 (Online) 25th February Evening Shift
A charge 'q' is placed at one corner of a cube as shown in figure. The flux of electrostatic field $\overrightarrow E $ through the shaded area is :
A.
${q \over {24{\varepsilon _0}}}$
B.
${q \over {48{\varepsilon _0}}}$
C.
${q \over {4{\varepsilon _0}}}$
D.
${q \over {8{\varepsilon _0}}}$
2021
JEE Mains
MCQ
JEE Main 2021 (Online) 24th February Evening Shift
Two electrons each are fixed at a distance '2d'. A third charge proton placed at the midpoint is displaced slightly by a distance x (x << d) perpendicular to the line joining the two fixed charges. Proton will execute simple harmonic motion having angular frequency : (m = mass of charged particle)
A.
${\left( {{{2{q^2}} \over {\pi {\varepsilon _0}m{d^3}}}} \right)^{{1 \over 2}}}$
B.
${\left( {{{{q^2}} \over {2\pi {\varepsilon _0}m{d^3}}}} \right)^{{1 \over 2}}}$
C.
${\left( {{{2\pi {\varepsilon _0}m{d^3}} \over {{q^2}}}} \right)^{{1 \over 2}}}$
D.
${\left( {{{\pi {\varepsilon _0}m{d^3}} \over {2{q^2}}}} \right)^{{1 \over 2}}}$
2021
JEE Mains
MCQ
JEE Main 2021 (Online) 24th February Morning Shift
A cube of side 'a' has point charges +Q located at each of its vertices except at the origin where the charge is $-$Q. The electric field at the centre of cube is :
A.
${{2Q} \over {3\sqrt 3 \pi {\varepsilon _0}{a^2}}}\left( {\widehat x + \widehat y + \widehat z} \right)$
B.
${{ - Q} \over {3\sqrt 3 \pi {\varepsilon _0}{a^2}}}\left( {\widehat x + \widehat y + \widehat z} \right)$
C.
${Q \over {3\sqrt 3 \pi {\varepsilon _0}{a^2}}}\left( {\widehat x + \widehat y + \widehat z} \right)$
D.
${{ - 2Q} \over {3\sqrt 3 \pi {\varepsilon _0}{a^2}}}\left( {\widehat x + \widehat y + \widehat z} \right)$
2021
JEE Mains
Numerical
JEE Main 2021 (Online) 25th July Morning Shift
A particle of mass 1 mg and charge q is lying at the mid-point of two stationary particles kept at a distance '2 m' when each is carrying same charge 'q'. If the free charged particle is displaced from its equilibrium position through distance 'x' (x < < 1 m). The particle executes SHM. Its angular frequency of oscillation will be ____________ $\times$ 105 rad/s if q2 = 10 C2.
Correct Answer: 6000
Explanation:
Net force on free charged particle
$F = {{k{q^2}} \over {{{(d + x)}^2}}} - {{k{q^2}} \over {{{(d - x)}^2}}}$
$F = - k{q^2}\left[ {{{4dx} \over {{{({d^2} - {x^2})}^2}}}} \right]$
$a = - {{4k{q^2}d} \over m}\left( {{x \over {{d^4}}}} \right)$
$a = - \left( {{{4k{q^2}} \over {m{d^3}}}} \right)x$
So, angular frequency
$\omega = \sqrt {{{4k{q^2}} \over {m{d^3}}}} $
$\omega = \sqrt {{{4 \times 9 \times {{10}^9} \times 10} \over {1 \times {{10}^{ - 6}} \times {1^3}}}} $
$\omega = 6 \times {10^8}$ rad/sec
$\omega = 6000 \times {10^5}$ rad/sec
2021
JEE Mains
Numerical
JEE Main 2021 (Online) 22th July Evening Shift
The total charge enclosed in an incremental volume of 2 $\times$ 10$-$9 m3 located at the origin is ___________ nC, if electric flux density of its field is found as
D = e$-$x sin y $\widehat i$ $-$ e$-$x cos y $\widehat j$ + 2z $\widehat k$ C/m2
D = e$-$x sin y $\widehat i$ $-$ e$-$x cos y $\widehat j$ + 2z $\widehat k$ C/m2
Correct Answer: 4
Explanation:
$\overline D = {\varepsilon _0}\overline E $
$Div.\,\overline E = {\rho \over {{\varepsilon _0}}}$
$ \Rightarrow div.\,\overline D = \rho $
$ \Rightarrow {\partial \over {\partial x}}\left( {{e^{ - x}}\sin y} \right) + {\partial \over {\partial y}}\left( { - {e^{ - x}}\cos y} \right) + {\partial \over {\partial z}}(2z) = \rho $
$\Rightarrow$ $\rho$ = 2 (a constant)
V = 2 $\times$ 10$-$9 m3
q = 2 $\times$ 2 $\times$ 10$-$9 = 4 nC
$Div.\,\overline E = {\rho \over {{\varepsilon _0}}}$
$ \Rightarrow div.\,\overline D = \rho $
$ \Rightarrow {\partial \over {\partial x}}\left( {{e^{ - x}}\sin y} \right) + {\partial \over {\partial y}}\left( { - {e^{ - x}}\cos y} \right) + {\partial \over {\partial z}}(2z) = \rho $
$\Rightarrow$ $\rho$ = 2 (a constant)
V = 2 $\times$ 10$-$9 m3
q = 2 $\times$ 2 $\times$ 10$-$9 = 4 nC
2021
JEE Mains
Numerical
JEE Main 2021 (Online) 20th July Morning Shift
A body having specific charge 8 $\mu$C/g is resting on a frictionless plane at a distance 10 cm from the wall (as shown in the figure). It starts moving towards the wall when a uniform electric field of 100 V/m is applied horizontally towards the wall. If the collision of the body with the wall is perfectly elastic, then the time period of the motion will be _______________ s.
Correct Answer: 1
Explanation:
Given,
q = 8$\mu$C/g = 8 $\times$ 10$-$6 C/g = 8 $\times$ 10$-$3 C/kg
s = 10 cm = 0.1 m $\Rightarrow$ E = 100 V/m
We know that, acceleration, a = ${{force(F)} \over {mass(m)}}$
$\Rightarrow$ a = ${{qE} \over m}$ [$\because$ F = qE]
= ${{8 \times {{10}^{ - 6}} \times 100} \over {{{10}^{ - 3}}}}$ = 0.8 ms$-$2
As per question, when electric field is switched on, the body strikes to the wall and then returns back.
For one oscillation,
s = ut + ${1 \over 2}$ at2
$\Rightarrow$ 0.1 = ${1 \over 2}$ $\times$ 0.8 t2 [$\because$ u = 0]
$\Rightarrow$ 0.2 = 0.8 t2
$\Rightarrow$ ${2 \over 8}$ = t2 $\Rightarrow$ t2 = ${1 \over 4}$ $\Rightarrow$ t = ${1 \over 2}$
$\therefore$ Time period = 2 $\times$ ${1 \over 2}$ = 1 s
Therefore, if the collision of the body is perfectly elastic, the time period of motion will be 1s.
q = 8$\mu$C/g = 8 $\times$ 10$-$6 C/g = 8 $\times$ 10$-$3 C/kg
s = 10 cm = 0.1 m $\Rightarrow$ E = 100 V/m
We know that, acceleration, a = ${{force(F)} \over {mass(m)}}$
$\Rightarrow$ a = ${{qE} \over m}$ [$\because$ F = qE]
= ${{8 \times {{10}^{ - 6}} \times 100} \over {{{10}^{ - 3}}}}$ = 0.8 ms$-$2
As per question, when electric field is switched on, the body strikes to the wall and then returns back.
For one oscillation,
s = ut + ${1 \over 2}$ at2
$\Rightarrow$ 0.1 = ${1 \over 2}$ $\times$ 0.8 t2 [$\because$ u = 0]
$\Rightarrow$ 0.2 = 0.8 t2
$\Rightarrow$ ${2 \over 8}$ = t2 $\Rightarrow$ t2 = ${1 \over 4}$ $\Rightarrow$ t = ${1 \over 2}$
$\therefore$ Time period = 2 $\times$ ${1 \over 2}$ = 1 s
Therefore, if the collision of the body is perfectly elastic, the time period of motion will be 1s.
2021
JEE Mains
Numerical
JEE Main 2021 (Online) 18th March Evening Shift
An infinite number of point charges, each carrying 1 $\mu$C charge, are placed along the y-axis at y = 1 m, 2 m, 4 m, 8 m ...............
The total force on a 1C point charge, placed at the origin, is x $\times$ 103 N.
The value of x, to the nearest integer, is __________. [Take ${1 \over {4\pi {\varepsilon _0}}} = 9 \times {10^9}$ Nm2/C2]
The total force on a 1C point charge, placed at the origin, is x $\times$ 103 N.
The value of x, to the nearest integer, is __________. [Take ${1 \over {4\pi {\varepsilon _0}}} = 9 \times {10^9}$ Nm2/C2]
Correct Answer: 12
Explanation:
${F_{total}} = {{k{q_1}{q_2}} \over {r_1^2}} + {{k{q_1}{q_3}} \over {r_2^2}} + {{k{q_1}{q_4}} \over {r_3^2}} + .....$
$ = 9 \times {10^9} \times {10^{ - 6}}\left[ {1 + {{\left( {{1 \over 2}} \right)}^2} + {{\left( {{1 \over {{2^2}}}} \right)}^2} + {{\left( {{1 \over {{2^3}}}} \right)}^2} + {{\left( {{1 \over {{2^\infty }}}} \right)}^2}} \right]$
$ = 9 \times {10^9} \times {10^{ - 6}}\left[ {{1 \over {1 - {1 \over 4}}}} \right]$
$ \because $ $\left[ {{S_\infty } = {a \over {1 - r}}} \right]$ for G.P.
$ = 9 \times {10^3} \times {4 \over 3}$ = 12 $\times$ 103 N
2021
JEE Mains
Numerical
JEE Main 2021 (Online) 17th March Evening Shift
The electric field in a region is given by $\overrightarrow E = {2 \over 5}{E_0}\widehat i + {3 \over 5}{E_0}\widehat j$ with ${E_0} = 4.0 \times {10^3}{N \over C}$. The flux of this field through a rectangular surface area 0.4 m2 parallel to the Y-Z plane is __________ Nm2C$-$1.
Correct Answer: 640
Explanation:
$\phi = \overrightarrow E \,.\,\overrightarrow A $
$ = {{{E_0}} \over 5}\left( {2\widehat i + 3\widehat j} \right)\,.\,\left( {0.4\widehat i} \right)$
$ = {{4000} \over 5}\left( {2 \times 0.4} \right)$
$ = 640$ Nm2 C$-$1
$ = {{{E_0}} \over 5}\left( {2\widehat i + 3\widehat j} \right)\,.\,\left( {0.4\widehat i} \right)$
$ = {{4000} \over 5}\left( {2 \times 0.4} \right)$
$ = 640$ Nm2 C$-$1
2021
JEE Mains
Numerical
JEE Main 2021 (Online) 26th February Evening Shift
27 similar drops of mercury are maintained at 10V each. All these spherical drops combine into a single big drop. The potential energy of the bigger drop is __________ times that of a smaller drop.
Correct Answer: 243
Explanation:
$(27)\left( {{4 \over 3}\pi {r^3}} \right) = {4 \over 3}\pi {R^3}$
R = 3r
Potential energy of smaller drop :
${U_1} = {3 \over 5}{{k{q^2}} \over r}$
Potential energy of bigger drop :
$U = {3 \over 5}{{k{Q^2}} \over R}$
$U = {3 \over 5}{{k{{(27q)}^2}} \over R}$
$U = {3 \over 5}k{{(27)(27){q^2}} \over {3r}}$
$U = {{(27)(27)} \over 3}\left( {{3 \over 5}{{k{q^2}} \over r}} \right)$
$U = 243\,{U_1}$
R = 3r
Potential energy of smaller drop :
${U_1} = {3 \over 5}{{k{q^2}} \over r}$
Potential energy of bigger drop :
$U = {3 \over 5}{{k{Q^2}} \over R}$
$U = {3 \over 5}{{k{{(27q)}^2}} \over R}$
$U = {3 \over 5}k{{(27)(27){q^2}} \over {3r}}$
$U = {{(27)(27)} \over 3}\left( {{3 \over 5}{{k{q^2}} \over r}} \right)$
$U = 243\,{U_1}$
2021
JEE Mains
Numerical
JEE Main 2021 (Online) 25th February Evening Shift
Two identical conducting spheres with negligible volume have 2.1 nC and $-$0.1 nC charges, respectively. They are brought into contact and then separated by a distance of 0.5 m. The electrostatic force acting between the spheres is __________ $\times$ 10$-$9 N.
[Given : $4\pi {\varepsilon _0} = {1 \over {9 \times {{10}^9}}}$ SI unit]
[Given : $4\pi {\varepsilon _0} = {1 \over {9 \times {{10}^9}}}$ SI unit]
Correct Answer: 36
Explanation:
When they brought into contact & then separated by a distance = 0.5 m
Then charge distribution will be
The electrostatic force acting b/w the sphere is
${F_e} = {{k{q_1}{q_2}} \over {{r^2}}}$
$ = {{9 \times {{10}^9} \times 1 \times {{10}^{ - 9}} \times 1 \times {{10}^{ - 9}}} \over {{{(0.5)}^2}}}$
$ = {{900} \over {25}} \times {10^{ - 9}}$
$ \Rightarrow $ ${F_e} = 36 \times {10^{ - 9}}N$
2021
JEE Mains
Numerical
JEE Main 2021 (Online) 25th February Evening Shift
Two small spheres each of mass 10 mg are suspended from a point by threads 0.5 m long. They are equally charged and repel each other to a distance of 0.20 m. The charge on each of the sphere is ${a \over {21}} \times {10^{ - 8}}$C. The value of 'a' will be ___________. [Given g = 10 ms$-$2]
Correct Answer: 20
Explanation:
T sin$\theta$ = ${{k{q^2}} \over {{r^2}}}$
T cos$\theta$ = mg
tan$\theta$ = ${{k{q^2}} \over {mg{r^2}}}$
q2 = ${{\tan \theta mg{r^2}} \over k}$
$ \because $ tan$\theta$ = ${{0.1} \over {0.5}} = {1 \over 5}$
${q^2} = {1 \over 5} \times {{10 \times {{10}^{ - 6}} \times 10 \times 0.2 \times 0.2} \over {9 \times {{10}^9}}}$
$q = {{2\sqrt 2 } \over 3} \times {10^{ - 8}}$
after comparison from the given equation a = 20
2021
JEE Mains
Numerical
JEE Main 2021 (Online) 25th February Morning Shift
The electric field in a region is given by $\overrightarrow E = \left( {{3 \over 5}{E_0}\widehat i + {4 \over 5}{E_0}\widehat j} \right){N \over C}$. The ratio of flux of reported field through the rectangular surface of area 0.2 m2 (parallel to y $-$ z plane) to that of the surface of area 0.3 m2 (parallel to x $-$ z plane) is a : b, where a = __________ [Here ${\widehat i}$, ${\widehat j}$ and ${\widehat k}$ are unit vectors along x, y and z-axes respectively.]
Correct Answer: 1
Explanation:
$\phi = \overrightarrow E \,.\,\overrightarrow A $
${\overrightarrow A _a} = 0.2\widehat i$
${\overrightarrow A _b} = 0.3\widehat j$
${\phi _a} = \left( {{3 \over 5}{E_0}\widehat i + {4 \over 5}{E_0}\widehat j} \right).\,0.2\widehat i$
$ \Rightarrow $ ${\phi _a} = {3 \over 5}{E_0} \times 0.2$
${\phi _b} = \left( {{3 \over 5}{E_0}\widehat i + {4 \over 5}{E_0}\widehat j} \right).\,0.3\widehat j$
$ \Rightarrow $ ${\phi _b} = {4 \over 5}{E_0} \times 0.3$
${a \over b} = {{{\phi _a}} \over {{\phi _b}}} = {{{3 \over 5}{E_0} \times 0.2} \over {{4 \over 5}{E_0} \times 0.3}} = {6 \over {12}} = 0.5$
${\overrightarrow A _a} = 0.2\widehat i$
${\overrightarrow A _b} = 0.3\widehat j$
${\phi _a} = \left( {{3 \over 5}{E_0}\widehat i + {4 \over 5}{E_0}\widehat j} \right).\,0.2\widehat i$
$ \Rightarrow $ ${\phi _a} = {3 \over 5}{E_0} \times 0.2$
${\phi _b} = \left( {{3 \over 5}{E_0}\widehat i + {4 \over 5}{E_0}\widehat j} \right).\,0.3\widehat j$
$ \Rightarrow $ ${\phi _b} = {4 \over 5}{E_0} \times 0.3$
${a \over b} = {{{\phi _a}} \over {{\phi _b}}} = {{{3 \over 5}{E_0} \times 0.2} \over {{4 \over 5}{E_0} \times 0.3}} = {6 \over {12}} = 0.5$
2021
JEE Mains
Numerical
JEE Main 2021 (Online) 25th February Morning Shift
512 identical drops of mercury are charged to a potential of 2V each. The drops are joined to form a single drop. The potential of this drop is ________ V.
Correct Answer: 128
Explanation:
Let charge on each drop = q
radius = r
$v = {{kq} \over r}$
$ \Rightarrow $ $2 = {{kq} \over r}$
radius of bigger
${4 \over 3}\pi {R^3} = 512 \times {4 \over 3}\pi {r^3}$
$R = 8r$
$ \therefore $ $v = {{k(512)q} \over R} = {{512} \over 8}{{kq} \over r} = {{512} \over 8} \times 2$
$ = 128V$
radius = r
$v = {{kq} \over r}$
$ \Rightarrow $ $2 = {{kq} \over r}$
radius of bigger
${4 \over 3}\pi {R^3} = 512 \times {4 \over 3}\pi {r^3}$
$R = 8r$
$ \therefore $ $v = {{k(512)q} \over R} = {{512} \over 8}{{kq} \over r} = {{512} \over 8} \times 2$
$ = 128V$
2021
JEE Mains
Numerical
JEE Main 2021 (Online) 24th February Evening Shift
A point charge of +12$\mu$C is at a distance 6 cm vertically above the centre of a square of side 12 cm as shown in figure. The magnitude of the electric flux through the square will be __________ $\times$ 103 Nm2/C.
Correct Answer: 226
Explanation:
Given, charge, q = 12 $\mu$C = 12 $\times$ 10$-$6C
Height of charge from surface, h = 6 cm = 6 $\times$ 10$-$2 m and side of square, a = 12 cm = 12 $\times$ 10$-$2 m
Using Gauss law, it is a part of cube of side 12 cm and charge at centre so;
$\phi = {Q \over {6{\varepsilon _0}}} = {{12\mu c} \over {6{\varepsilon _0}}} = 2 \times 4\pi \times 9 \times {10^9} \times {10^{ - 6}}$
$ = 226 \times {10^3}$ Nm2 / C
2020
JEE Mains
MCQ
JEE Main 2020 (Online) 6th September Evening Slot
Two identical electric point dipoles have dipole moments ${\overrightarrow p _1} = p\widehat i$ and ${\overrightarrow p _2} = - p\widehat i$ and are held on the x
axis at distance '$a$' from each other. When released, they move along the x-axis with the direction
of their dipole moments remaining unchanged. If the mass of each dipole is 'm', their speed when
they are infinitely far apart is :
A.
${p \over a}\sqrt {{3 \over {2\pi { \in _0}ma}}} $
B.
${p \over a}\sqrt {{1 \over {\pi { \in _0}ma}}} $
C.
${p \over a}\sqrt {{1 \over {2\pi { \in _0}ma}}} $
D.
${p \over a}\sqrt {{2 \over {\pi { \in _0}ma}}} $
2020
JEE Mains
MCQ
JEE Main 2020 (Online) 6th September Evening Slot
Consider the force F on a charge 'q' due to a uniformly charged spherical shell of radius R carrying
charge Q distributed uniformly over it. Which one of the following statements is true for F, if 'q' is
placed at distance r from the centre of the shell?
A.
${1 \over {4\pi {\varepsilon _0}}}{{qQ} \over {{R^2}}} > F > 0$ for r < R
B.
$F = {1 \over {4\pi {\varepsilon _0}}}{{qQ} \over {{r^2}}}$ for r > R
C.
$F = {1 \over {4\pi {\varepsilon _0}}}{{qQ} \over {{r^2}}}$ for all r
D.
$F = {1 \over {4\pi {\varepsilon _0}}}{{qQ} \over {{R^2}}}$ for r < R
2020
JEE Mains
MCQ
JEE Main 2020 (Online) 6th September Morning Slot
Charges Q1
and Q2
are at points A and B of a right angle triangle OAB (see figure). The resultant
electric field at point O is perpendicular to the hypotenuse, then
${{{Q_1}} \over {{Q_2}}}$ is proportional to :
${{{Q_1}} \over {{Q_2}}}$ is proportional to :
A.
${{x_1^3} \over {x_2^3}}$
B.
${{x_2^2} \over {x_1^2}}$
C.
${{{x_1}} \over {{x_2}}}$
D.
${{{x_2}} \over {{x_1}}}$
2020
JEE Mains
MCQ
JEE Main 2020 (Online) 5th September Evening Slot
Ten charges are placed on the circumference
of a circle of radius R with constant angular
separation between successive charges.
Alternate charges 1, 3, 5, 7, 9 have charge (+q)
each, while 2, 4, 6, 8, 10 have charge (–q) each.
The potential V and the electric field E at the
centre of the circle are respectively.
(Take V = 0 at infinity)
(Take V = 0 at infinity)
A.
V = 0; E = 0
B.
$V = {{10q} \over {4\pi {\varepsilon _0}R}}$; $E = {{10q} \over {4\pi {\varepsilon _0}{R^2}}}$
C.
$V = {{10q} \over {4\pi {\varepsilon _0}R}}$; E = 0
D.
V = 0; $E = {{10q} \over {4\pi {\varepsilon _0}{R^2}}}$
2020
JEE Mains
MCQ
JEE Main 2020 (Online) 5th September Morning Slot
A solid sphere of radius R carries a charge
Q + q distributed uniformly over its volume. A
very small point like piece of it of mass m gets
detached from the bottom of the sphere and
falls down vertically under gravity. This piece
carries charge q. If it acquires a speed v when
it has fallen through a vertical height y (see
figure), then :
(assume the remaining portion to be spherical).
(assume the remaining portion to be spherical).
A.
v2 = $y\left[ {{{qQ} \over {4\pi {\varepsilon _0}R\left( {R + y} \right)m}} + g} \right]$
B.
v2 = $2y\left[ {{{qQR} \over {4\pi {\varepsilon _0}{{\left( {R + y} \right)}^3}m}} + g} \right]$
C.
v2 = $2y\left[ {{{qQ} \over {4\pi {\varepsilon _0}R\left( {R + y} \right)m}} + g} \right]$
D.
v2 = $y\left[ {{{qQ} \over {4\pi {\varepsilon _0}{R^2}ym}} + g} \right]$
2020
JEE Mains
MCQ
JEE Main 2020 (Online) 4th September Evening Slot
A particle of charge q and mass m is subjected to an electric field
E = E0 (1 – $a$x2) in the x-direction, where $a$ and E0 are constants. Initially the particle was at rest at x = 0. Other than the initial position the kinetic energy of the particle becomes zero when the distance of the particle from the origin is :
E = E0 (1 – $a$x2) in the x-direction, where $a$ and E0 are constants. Initially the particle was at rest at x = 0. Other than the initial position the kinetic energy of the particle becomes zero when the distance of the particle from the origin is :
A.
$a$
B.
$\sqrt {{2 \over a}} $
C.
$\sqrt {{3 \over a}} $
D.
$\sqrt {{1 \over a}} $
2020
JEE Mains
MCQ
JEE Main 2020 (Online) 4th September Morning Slot
Two charged thin infinite plane sheets of
uniform surface charge density ${\sigma _ + }$ and ${\sigma _ - }$,
where |${\sigma _ + }$| > |${\sigma _ - }$|, intersect at right angle.
Which of the following best represents the
electric field lines for this system :
uniform surface charge density ${\sigma _ + }$ and ${\sigma _ - }$,
where |${\sigma _ + }$| > |${\sigma _ - }$|, intersect at right angle.
Which of the following best represents the
electric field lines for this system :
A.
B.
C.
D.
Given, |${\sigma _ + }$| > |${\sigma _ - }$|
2020
JEE Mains
MCQ
JEE Main 2020 (Online) 4th September Morning Slot
A two point charges 4q and -q are fixed
on the x-axis at x = $ - {d \over 2}$
and x = ${d \over 2}$
respectively. If a third point charge 'q' is taken from the origin to x = d along the semicircle as shown in the figure, the energy of the charge will :
A.
increase by ${{3{q^2}} \over {4\pi {\varepsilon _0}d}}$
B.
increase by ${{2{q^2}} \over {3\pi {\varepsilon _0}d}}$
C.
decrease by ${{{q^2}} \over {4\pi {\varepsilon _0}d}}$
D.
decrease by ${{4{q^2}} \over {3\pi {\varepsilon _0}d}}$
2020
JEE Mains
MCQ
JEE Main 2020 (Online) 3rd September Evening Slot
Concentric metallic hollow spheres of radii R and 4R hold charges Q1
and Q2
respectively. Given that
surface charge densities of the concentric spheres are equal, the potential difference V(R) – V(4R) is :
A.
${{3{Q_2}} \over {4\pi {\varepsilon _0}R}}$
B.
${{3{Q_1}} \over {4\pi {\varepsilon _0}R}}$
C.
${{3{Q_1}} \over {16\pi {\varepsilon _0}R}}$
D.
${{{Q_2}} \over {4\pi {\varepsilon _0}R}}$
2020
JEE Mains
MCQ
JEE Main 2020 (Online) 3rd September Morning Slot
Two isolated conducting spheres S1 and S2 of
radius ${2 \over 3}R$ and ${1 \over 3}R$ have 12 $\mu $C and –3 $\mu $C
charges, respectively, and are at a large
distance from each other. They are now
connected by a conducting wire. A long time
after this is done the charges on S1 and S2 are
respectively :
A.
4.5 $\mu $C on both
B.
+4.5 $\mu $C and –4.5 $\mu $C
C.
6 $\mu $C and 3 $\mu $C
D.
3 $\mu $C and 6 $\mu $C
2020
JEE Mains
MCQ
JEE Main 2020 (Online) 2nd September Evening Slot
A small point mass carrying some positive
charge on it, is released from the edge of a
table. There is a uniform electric field in this
region in the horizontal direction. Which of the
following options then correctly describe the
trajectory of the mass?
(Curves are drawn schematically and are not to scale).
(Curves are drawn schematically and are not to scale).
A.
B.
C.
D.
2020
JEE Mains
MCQ
JEE Main 2020 (Online) 2nd September Evening Slot
A charge Q is distributed over two concentric
conducting thin spherical shells radii r and
R (R > r). If the surface charge densities on the
two shells are equal, the electric potential at
the common centre is :
A.
${1 \over {4\pi {\varepsilon _0}}}{{\left( {R + r} \right)} \over {\left( {{R^2} + {r^2}} \right)}}$Q
B.
${1 \over {4\pi {\varepsilon _0}}}{{\left( {R + r} \right)} \over {2\left( {{R^2} + {r^2}} \right)}}Q$
C.
${1 \over {4\pi {\varepsilon _0}}}{{\left( {R + 2r} \right)Q} \over {2\left( {{R^2} + {r^2}} \right)}}$
D.
${1 \over {4\pi {\varepsilon _0}}}{{\left( {2R + r} \right)} \over {\left( {{R^2} + {r^2}} \right)}}Q$