Electrostatics & Capacitors · JEE Main previous year questions

A parallel plate capacitor consisting of two circular plates of radius $10\mathrm{cm}$ is being charged by a constant current of $0.15\mathrm{A}$ . If the rate of change of potential difference between the plates is $7\times 10^{8}V/s$ then the integer value of the distance between the parallel plates is ( Take, ${ϵ}_0=9\times 10^{-12}\frac{\mathrm{F}}{\mathrm{m}},\pi =\frac{22}{7}$ ) ____ $\mu \mathrm{m}$.

Consider a long thin conducting wire carrying a uniform current I. A particle having mass " $m$ " and charge " $q$ " is released at a distance " $a$ " from the wire with a speed $v_0$ along the direction of current in the wire. The particle gets attracted to the wire due to magnetic force. The particle turns round when it is at distance $x$ from the wire. The value of $x$ is [ ${\mu }_0$ is vacuum permeability]

1. A.$a\left[1-\frac{m{v}_0}{2q{\mu }_0\mathrm{I}}\right]$
2. B.$a{\mathrm{e}}^{-\frac{4\pi m{v}_0}{q{\mu }_0\mathrm{I}}}$
3. C.$a\left[1-\frac{mv}{q}{\mu }_0\mathrm{I}\right]$
4. D.$\frac{a}{2}$

Due to presence of an em-wave whose electric component is given by $E=100\sin (\omega t-kx)\mathrm{N}{\mathrm{C}}^{-1}$ a cylinder of length $200\mathrm{cm}$ holds certain amount of em-energy inside it. If another cylinder of same length but half diameter than previous one holds same amount of em-energy, the magnitude of the electric field of the corresponding em-wave should be modified as

1. A.$50\sin (\omega t-\mathrm{kx})\mathrm{N}{\mathrm{C}}^{-1}$
2. B.$400\sin (\omega t-\mathrm{kx})\mathrm{N}{\mathrm{C}}^{-1}$
3. C.$200\sin (\omega t-kx)\mathrm{N}{\mathrm{C}}^{-1}$
4. D.$25\sin (\omega t-\mathrm{kx})\mathrm{N}{\mathrm{C}}^{-1}$

Three infinitely long wires with linear charge density $\lambda$ are placed along the $x$-axis, $y$-axis and $z$-axis respectively. Which of the following denotes an equipotential surface?

1. A.$\left(x^2+y^2\right)\left(y^2+z^2\right)\left(z^2+x^2\right)=\text{constant}$
2. B.$xyz=\text{constant}$
3. C.$xy+yz+zx=\text{constant}$
4. D.$(x+y)(y+z)(z+x)=\text{constant}$

1. A.${\mathrm{C}}_2>{\mathrm{C}}_1,{\mathrm{U}}_2>{\mathrm{U}}_1$
2. B.${\mathrm{C}}_1>{\mathrm{C}}_2,{\mathrm{U}}_1>{\mathrm{U}}_2$
3. C.${\mathrm{C}}_2>{\mathrm{C}}_1,{\mathrm{U}}_2<{\mathrm{U}}_1$
4. D.${\mathrm{C}}_1>{\mathrm{C}}_2,{\mathrm{U}}_1<{\mathrm{U}}_2$

1. A.$\sqrt{\frac{qEl}{2m}}$
2. B.$\sqrt{\frac{qEl}{4m}}$
3. C.$\sqrt{\frac{qEl}{m}}$
4. D.$\sqrt{\frac{2qEl}{m}}$

Consider a parallel plate capacitor of area A (of each plate) and separation ' $d$ ' between the plates. If $E$ is the electric field and ${\epsilon }_0$ is the permittivity of free space between the plates, then potential energy stored in the capacitor is

1. A.${\epsilon }_0{\mathrm{E}}^2\mathrm{Ad}$
2. B.$\frac{3}{4}{\epsilon }_0{\mathrm{E}}^2\mathrm{Ad}$
3. C.$\frac{1}{4}{\epsilon }_0{\mathrm{E}}^2\mathrm{Ad}$
4. D.$\frac{1}{2}{\epsilon }_0{\mathrm{E}}^2\mathrm{Ad}$

A parallel plate capacitor was made with two rectangular plates, each with a length of $l=3\mathrm{cm}$ and breadth of $\mathrm{b}=1\mathrm{cm}$. The distance between the plates is $3\mu \mathrm{m}$. Out of the following, which are the ways to increase the capacitance by a factor of 10 ? A. $l=30\mathrm{cm}$, $\mathrm{b}=1\mathrm{cm}$, $\mathrm{d}=1\mu \mathrm{m}$ B. $l=3\mathrm{cm}$, $\mathrm{b}=1\mathrm{cm}$, $\mathrm{d}=30\mu \mathrm{m}$ C. $l=6\mathrm{cm}$, $\mathrm{b}=5\mathrm{cm}$, $\mathrm{d}=3\mu \mathrm{m}$ D. $l=1\mathrm{cm}$, $\mathrm{b}=1\mathrm{cm}$, $\mathrm{d}=10\mu \mathrm{m}$ E. $l=5\mathrm{cm}$, $\mathrm{b}=2\mathrm{cm}$, $\mathrm{d}=1\mu \mathrm{m}$ Choose the correct answer from the options given below:

1. A.C only
2. B.A only
3. C.B and D only
4. D.C and E only

An electron of mass ' m ' with an initial velocity $\vec{v}=v_0\hat{i}\left(v_0>0\right)$ enters an electric field $\vec{\mathrm{E}}=-{\mathrm{E}}_0\hat{\mathrm{k}}$. If the initial de Broglie wavelength is ${\lambda }_0$, the value after time t would be

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

A small uncharged conducting sphere is placed in contact with an identical sphere but having $4\times 10^{-8}\text{ C}$ charge and then removed to a distance such that the force of repulsion between them is $9\times 10^{-3}\text{ N}$. The distance between them is (Take $\frac{1}{4\pi {ϵ}_0}$ as $9\times 10^9$ in SI units)

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

1. A.$\frac{K{q}_0^2}{a}(4-2\sqrt{2})$
2. B.$\frac{K{q}_0^2}{a}(4\sqrt{2}-2)$
3. C.$\frac{K{q}_0^2}{a}(3-\sqrt{2})$
4. D.$\frac{K{q}_0^2}{a}(3\sqrt{2}-2)$

A plane electromagnetic wave of frequency $20\text{ MHz}$ travels in free space along the $+x$ direction. At a particular point in space and time, the electric field vector of the wave is ${\mathrm{E}}_y=9.3{\mathrm{Vm}}^{-1}$. Then, the magnetic field vector of the wave at that point is

1. A.${\mathrm{B}}_z=1.55\times 10^{-8}\text{ T}$
2. B.${\mathrm{B}}_z=6.2\times 10^{-8}\text{ T}$
3. C.${\mathrm{B}}_z=3.1\times 10^{-8}\text{ T}$
4. D.${\mathrm{B}}_z=9.3\times 10^{-8}\text{ T}$

Two point charges $-4\mu \text{c}$ and $4\mu \text{c}$, constituting an electric dipole, are placed at $(-9,0,0)\text{cm}$ and $(9,0,0)\text{cm}$ in a uniform electric field of strength $10^4{\mathrm{NC}}^{-1}$. The work done on the dipole in rotating it from the equilibrium through $180^{\circ }$ is :

1. A.$18.4\text{ mJ}$
2. B.$12.4\text{ mJ}$
3. C.$16.4\text{ mJ}$
4. D.$14.4\text{ mJ}$

Two charges $7\mu \mathrm{C}$ and $-4\mu \mathrm{C}$ are placed at $(-7\text{ cm},0,0)$ and $(7\text{ cm},0,0)$ respectively. Given, ${ϵ}_0=8.85\times 10^{-12}{\mathrm{C}}^2{\mathrm{N}}^{-1}{\mathrm{m}}^{-2}$, the electrostatic potential energy of the charge configuration is :

1. A.$-2.0\text{ J}$
2. B.$-1.5\text{ J}$
3. C.$-1.2\text{ J}$
4. D.$-1.8\text{ J}$

A time varying potential difference is applied between the plates of a parallel plate capacitor of capacitance $2.5\mu \text{ F}$. The dielectric constant of the medium between the capacitor plates is $1$ . It produces an instantaneous displacement current of $0.25\text{ mA}$ in the intervening space between the capacitor plates, the magnitude of the rate of change of the potential difference will be _________ ${\mathrm{Vs}}^{-1}$.

1. A.$\frac{\lambda a}{2{ϵ}_0}$
2. B.$\frac{\lambda a}{8{ϵ}_0}$
3. C.$\frac{\lambda a}{4{ϵ}_0}$
4. D.$\frac{\lambda a}{16{ϵ}_0}$

An electron is made to enter symmetrically between two parallel and equally but oppositely charged metal plates, each of $10\text{ cm}$ length. The electron emerges out of the electric field region with a horizontal component of velocity $10^6\text{ m/s}$. If the magnitude of the electric field between the plates is $9.1\text{ V/cm}$, then the vertical component of velocity of electron is (mass of electron $=9.1\times 10^{-31}\text{ kg}$ and charge of electron $=1.6\times 10^{-19}\text{ C}$ )

1. A.$1\times 10^6\text{ m/s}$
2. B.$16\times 10^6\text{ m/s}$
3. C.$16\times 10^4\text{ m/s}$
4. D.0

A parallel-plate capacitor of capacitance $40\mu \text{ F}$ is connected to a $100\text{ V}$ power supply. Now the intermediate space between the plates is filled with a dielectric material of dielectric constant $K=2$. Due to the introduction of dielectric material, the extra charge and the change in the electrostatic energy in the capacitor, respectively, are

1. A.$8\text{ mC and }2.0\text{ J}$
2. B.$4\text{ mC and }0.2\text{ J}$
3. C.$2\text{ mC and }0.2\text{ J}$
4. D.$2\text{ mC and }0.4\text{ J}$

A plane EM wave is propagating along $x$ direction. It has a wavelength of $4\text{ mm}$. If electric field is in $y$ direction with the maximum magnitude of $60{\text{ Vm}}^{-1}$, the equation for magnetic field is :

1. A.${\mathrm{B}}_z=2\times 10^{-7}\sin \left[\frac{\pi }{2}(x-3\times 10^8\mathrm{t})\right]\hat{\mathrm{k}}\mathrm{T}$
2. B.${\mathrm{B}}_z=2\times 10^{-7}\sin \left[\frac{\pi }{2}\times 10^3(x-3\times 10^8\mathrm{t})\right]\hat{\mathrm{k}}\mathrm{T}$
3. C.${\mathrm{B}}_z=60\sin \left[\frac{\pi }{2}(x-3\times 10^8\mathrm{t})\right]\hat{\mathrm{k}}\mathrm{T}$
4. D.${\mathrm{B}}_x=60\sin \left[\frac{\pi }{2}(x-3\times 10^8\mathrm{t})\right]\hat{\mathrm{i}}\mathrm{T}$

1. A.$\frac{11{ϵ}_o\mathrm{A}}{20\mathrm{d}}$
2. B.$\frac{13{ϵ}_o\mathrm{A}}{17\mathrm{d}}$
3. C.$\frac{18{ϵ}_o\mathrm{A}}{11\mathrm{d}}$
4. D.$\frac{11{ϵ}_o\mathrm{A}}{18\mathrm{d}}$

A bulb and a capacitor are connected in series across an ac supply. A dielectric is then placed between the plates of the capacitor. The glow of the bulb :

1. A.becomes zero
2. B.remains same
3. C.increases
4. D.decreases

At the centre of a half ring of radius $\mathrm{R}=10\text{ cm}$ and linear charge density $4{\text{ nC m}}^{-1}$, the potential is $x\pi \text{V}$. The value of $x$ is ______.

The magnetic field in a plane electromagnetic wave is $B_y=(3.5\times 10^{-7})\sin (1.5\times 10^{3}x+0.5\times 10^{11}t)\mathrm{T}$. The corresponding electric field will be :

1. A.$E_z=105\sin (1.5\times 10^{3}x+0.5\times 10^{11}t)\mathrm{V}{\mathrm{m}}^{-1}$
2. B.$E_y=10.5\sin (1.5\times 10^{3}x+0.5\times 10^{11}t)\mathrm{V}{\mathrm{m}}^{-1}$
3. C.$E_y=1.17\sin (1.5\times 10^{3}x+0.5\times 10^{11}t)\mathrm{V}{\mathrm{m}}^{-1}$
4. D.$E_z=1.17\sin (1.5\times 10^{3}x+0.5\times 10^{11}t)\mathrm{V}{\mathrm{m}}^{-1}$