Electrostatics & Capacitors · JEE Main previous year questions

In LC circuit the inductance $\mathrm{L}=40\text{ mH}$ and capacitance $\mathrm{C}=100\mu \mathrm{F}$. If a voltage $\mathrm{V}(t)=10\sin (314t)$ is applied to the circuit, the current in the circuit is given as :

1. A.$0.52\cos 314t$
2. B.$5.2\cos 314t$
3. C.$0.52\sin 314t$
4. D.$10\cos 314t$

1. A.${\mathrm{Q}}_{\mathrm{A}}=\frac{CV}{e}$, ${\mathrm{Q}}_{\mathrm{B}}=\frac{VC}{2}$
2. B.${\mathrm{Q}}_{\mathrm{A}}=\frac{CV}{2}$, ${\mathrm{Q}}_{\mathrm{B}}=\frac{VC}{e}$
3. C.${\mathrm{Q}}_{\mathrm{A}}=VC$, ${\mathrm{Q}}_{\mathrm{B}}=\frac{VC}{e}$
4. D.${\mathrm{Q}}_{\mathrm{A}}=VC$, ${\mathrm{Q}}_{\mathrm{B}}=CV$

An electron of mass m and magnitude of charge |e| initially at rest gets accelerated by a constant electric field E. The rate of change of de-Broglie wavelength of this electron at time t ignoring relativistic effects is :

1. A.$\frac{-h}{|e|Et}$
2. B.$\frac{-h}{|e|E\sqrt{t}}$
3. C.$\frac{-h}{|e|E{t}^2}$
4. D.$\frac{|e|Et}{h}$

In finding the electric field using Gauss Law the formula $\left|\vec{E}\right|=\frac{q_{\text{enc}}}{{\epsilon }_0|A|}$ is applicable. In the formula ${\epsilon }_0$ is permittivity of free space, A is the area of Gaussian surface and ${\mathrm{q}}_{\mathrm{enc}}$ is charge enclosed by the Gaussian surface. The equation can be used in which of the following situation?

1. A.Only when $\left|\vec{E}\right|=\text{constant}$ on the surface.
2. B.For any choice of Gaussian surface.
3. C.Only when the Gaussian surface is an equipotential surface.
4. D.Only when the Gaussian surface is an equipotential surface and $\left|\vec{E}\right|$ is constant on the surface.

The critical angle of a medium for a specific wavelength, if the medium has relative permittivity 3 and relative permeability $\frac{4}{3}$ for this wavelength, will be :

1. A.$45^{\circ }$
2. B.$15^{\circ }$
3. C.$30^{\circ }$
4. D.$60^{\circ }$

1. A.$\frac{3\sqrt{3}q}{4\pi {\epsilon }_0{d}^2}$
2. B.$\frac{\sqrt{3}q}{4\pi {\epsilon }_0{d}^2}$
3. C.$\frac{\sqrt{3}q}{\pi {\epsilon }_0{d}^2}$
4. D.$\frac{2\sqrt{3}q}{\pi {\epsilon }_0{d}^2}$

Effective capacitance of parallel combination of two capacitors $C_1$ and $C_2$ is $10\mu \mathrm{F}$. When these capacitors are individually connected to a voltage source of 1V, the energy stored in the capacitor $C_2$ is 4 times that of $C_1$. If these capacitors are connected in series, their effective capacitance will be :

1. A.$4.2\mu \mathrm{F}$
2. B.$8.4\mu \mathrm{F}$
3. C.$1.6\mu \mathrm{F}$
4. D.$3.2\mu \mathrm{F}$

A plane electromagnetic wave of frequency 25 GHz is propagating in vacuum along the z-direction. At a particular point in space and time, the magnetic field is given by $\vec{B}=5\times 10^{-8}\hat{j}\text{T}$. The corresponding electric field $\vec{E}$ is (speed of light $\text{c}=3\times 10^8{\text{ ms}}^{-1}$)

1. A.$15\hat{i}\text{ V / m}$
2. B.$-15\hat{i}\text{ V / m}$
3. C.$1.66\times 10^{-16}\hat{i}\text{ V / m}$
4. D.$-1.66\times 10^{-16}\hat{i}\text{ V / m}$

1. A.
2. B.
3. C.
4. D.

1. A.$\frac{{\epsilon }_0{a}^2}{d}\left(1+\frac{\alpha a}{d}\right)$
2. B.$\frac{{\epsilon }_0{a}^2}{d}\left(1-\frac{\alpha a}{4d}\right)$
3. C.$\frac{{\epsilon }_0{a}^2}{d}\left(1-\frac{\alpha a}{2d}\right)$
4. D.$\frac{{\epsilon }_0{a}^2}{d}\left(1-\frac{3\alpha a}{2d}\right)$

An electron (mass m) with initial velocity $\vec{v}=v_0\hat{i}+v_0\hat{j}$ is in an electric field $\vec{E}=-E_0\hat{k}$. If ${\lambda }_0$ is initial de-Broglie wavelength of electron, its de-Broglie wave length at time t is given by :

1. A.$\frac{{\lambda }_0}{\sqrt{1+\frac{e^2{E}_0^2{t}^2}{m^2{v}_0^2}}}$
2. B.$\frac{{\lambda }_0\sqrt{2}}{\sqrt{1+\frac{e^2{E}_0^2{t}^2}{m^2{v}_0^2}}}$
3. C.$\frac{{\lambda }_0}{\sqrt{1+\frac{e^2{E}_0^2{t}^2}{2m^2{v}_0^2}}}$
4. D.$\frac{{\lambda }_0}{\sqrt{2+\frac{e^2{E}_0^2{t}^2}{m^2{v}_0^2}}}$

Consider two charged metallic spheres ${\mathrm{S}}_1$ and ${\mathrm{S}}_2$ of radii ${\mathrm{R}}_1$ and ${\mathrm{R}}_2$, respectively. The electric fields ${\mathrm{E}}_1$ (on ${\mathrm{S}}_1$) and ${\mathrm{E}}_2$ (on ${\mathrm{S}}_2$) on their surfaces are such that ${\mathrm{E}}_1/{\mathrm{E}}_2={\mathrm{R}}_1/{\mathrm{R}}_2$. Then the ratio ${\mathrm{V}}_1$ (on ${\mathrm{S}}_1$) / ${\mathrm{V}}_2$ (on ${\mathrm{S}}_2$) of the electrostatic potentials on each sphere is :

1. A.$({\mathrm{R}}_1/{\mathrm{R}}_2{)}^2$
2. B.$({\mathrm{R}}_2/{\mathrm{R}}_1)$
3. C.$({\mathrm{R}}_1/{\mathrm{R}}_2{)}^3$
4. D.${\mathrm{R}}_1/{\mathrm{R}}_2$

Consider a circular coil of wire carrying constant current I, forming a magnetic dipole. The magnetic flux through an infinite plane that contains the circular coil and excluding the circular coil area is given by ${\varphi }_i$. The magnetic flux through the area of the circular coil area is given by ${\varphi }_0$. Which of the following option is correct?

1. A.${\varphi }_i={\varphi }_0$
2. B.${\varphi }_i<{\varphi }_0$
3. C.${\varphi }_i>{\varphi }_0$
4. D.${\varphi }_i=-{\varphi }_0$

1. A.$\frac{\sigma }{{ϵ}_0}\left[\left(1+\frac{\sqrt{3}}{2}\right)\hat{y}+\frac{\hat{x}}{2}\right]$
2. B.$\frac{\sigma }{2{ϵ}_0}\left[(1+\sqrt{3})\hat{y}+\frac{\hat{x}}{2}\right]$
3. C.$\frac{\sigma }{2{ϵ}_0}\left[(1+\sqrt{3})\hat{y}-\frac{\hat{x}}{2}\right]$
4. D.$\frac{\sigma }{2{ϵ}_0}\left[\left(1-\frac{\sqrt{3}}{2}\right)\hat{y}-\frac{\hat{x}}{2}\right]$

If the magnetic field in a plane electromagnetic wave is given by $\vec{B}=3\times 10^{-8}\sin (1.6\times 10^{3}x+48\times 10^{10}t)\hat{j}\mathrm{T}$, then what will be expression for electric field ?

1. A.$\vec{E}=(9\sin (1.6\times 10^{3}x+48\times 10^{10}t)\hat{k}V/m)$
2. B.$\vec{E}=(60\sin (1.6\times 10^{3}x+48\times 10^{10}t)\hat{k}V/m)$
3. C.$\vec{E}=(3\times 10^{-8}\sin (1.6\times 10^{3}x+48\times 10^{10}t)\hat{i}V/m)$
4. D.$\vec{E}=(3\times 10^{-8}\sin (1.6\times 10^{3}x+48\times 10^{10}t)\hat{j}V/m)$

1. A.$\frac{A{ϵ}_{0}K}{d}\left(1+{\left(\frac{\alpha d}{2}\right)}^2\right)$
2. B.$\frac{A{ϵ}_{0}K}{d}\left(1+\frac{\alpha d}{2}\right)$
3. C.$\frac{A{ϵ}_{0}K}{d}\left(1+\frac{{\alpha }^2{d}^2}{2}\right)$
4. D.$\frac{A{ϵ}_{0}K}{d}(1+\alpha d)$

A particle of mass m and charge q has an initial velocity $\vec{v}=v_0\hat{j}$. If an electric field $\vec{E}=E_0\hat{i}$ and magnetic field $\vec{B}=B_0\hat{i}$ act on the particle, its speed will double after a time:

1. A.$\frac{3m{v}_0}{q{E}_0}$
2. B.$\frac{\sqrt{2}m{v}_0}{q{E}_0}$
3. C.$\frac{\sqrt{3}m{v}_0}{q{E}_0}$
4. D.$\frac{2m{v}_0}{q{E}_0}$

An elevator in a building can carry a maximum of 10 persons, with the average mass of each person being 68 kg. The mass of the elevator itself is 920 kg and it moves with a constant speed of 3 m/s. The frictional force opposing the motion is 6000 N. If the elevator is moving up with its full capacity, the power delivered by the motor to the elevator ($\text{g}=10{\text{ m/s}}^2$) must be at least :

1. A.48000 W
2. B.62360 W
3. C.56300 W
4. D.66000 W

In a building there are 15 bulbs of 45 W, 15 bulbs of 100 W, 15 small fans of 10 W and 2 heaters of 1 kW. The voltage of electric main is 220 V. The minimum fuse capacity (rated value) of the building will be :

1. A.15 A
2. B.20 A
3. C.25 A
4. D.10 A

The electric field of a plane electromagnetic wave is given by $$\vec{E}=E_0\frac{\hat{i}+\hat{j}}{\sqrt{2}}\cos (kz+\omega t)$$ At $t=0$, a positively charged particle is at the point $(x,y,z)=\left(0,0,\frac{\pi }{k}\right)$. If its instantaneous velocity at $(t=0)$ is $v_0\hat{k}$, the force acting on it due to the wave is :

1. A.parallel to $\hat{k}$
2. B.parallel to $\frac{\hat{i}+\hat{j}}{\sqrt{2}}$
3. C.antiparallel to $\frac{\hat{i}+\hat{j}}{\sqrt{2}}$
4. D.zero

A 60 pF capacitor is fully charged by a 20 V supply. It is then disconnected from the supply and is conneced to another uncharged 60 pF capacitor in parallel. The electrostatic energy that is lost in this process by the time the charge is redistributed between them is (in nJ) _____

1. A.surface charge density on the inner surface is uniform and equal to $\frac{(Q/2)}{4\pi a^2}$
2. B.surface charge density on the inner surface of the shell is zero everywhere
3. C.surface charge density on the outer surface depends on $\left|\vec{P}\right|$
4. D.electric field outside the shell is the same as that of a point charge at the centre of the shell

1. A.$\frac{E_1}{E_2}=\frac{(K_1+K_2+K_3)(K_2{K}_3+K_3{K}_1+K_1{K}_2)}{K_1{K}_2{K}_3}$
2. B.$\frac{E_1}{E_2}=\frac{K_1{K}_2{K}_3}{(K_1+K_2+K_3)(K_2{K}_3+K_3{K}_1+K_1{K}_2)}$
3. C.$\frac{E_1}{E_2}=\frac{(K_1+K_2+K_3)(K_2{K}_3+K_3{K}_1+K_1{K}_2)}{9K_1{K}_2{K}_3}$
4. D.$\frac{E_1}{E_2}=\frac{9K_1{K}_2{K}_3}{(K_1+K_2+K_3)(K_2{K}_3+K_3{K}_1+K_1{K}_2)}$

A point dipole $\vec{p}=-p_0\hat{x}$ is kept at the origin. The potential and electric field due to this dipole on the y-axis at a distance d are, respectively: (Take V= 0 at infinity)

1. A.$\frac{|\vec{p}|}{4\pi {ϵ}_0{d}^2}$, $\frac{-\vec{p}}{4\pi {ϵ}_0{d}^3}$
2. B.$0$, $\frac{\vec{p}}{4\pi {ϵ}_0{d}^3}$
3. C.$\frac{|\vec{p}|}{4\pi {ϵ}_0{d}^2}$, $\frac{\vec{p}}{4\pi {ϵ}_0{d}^3}$
4. D.$0$, $\frac{-\vec{p}}{4\pi {ϵ}_0{d}^3}$