Electrostatics & Capacitors · JEE Main previous year questions

Match List I with List II $$\begin{array}{llll}& \text{List I}& & \text{List II}\\ \text{A.}& \text{Gauss’s Law in Electrostatics}& \text{I.}& \oint \vec{E}\cdot \overrightarrow{dl}=-\frac{d{\varphi }_B}{dt}\\ \text{B.}& \text{Faraday’s Law}& \text{II.}& \oint \vec{B}\cdot \overrightarrow{dA}=0\\ \text{C.}& \text{Gauss’s Law in Magnetism}& \text{III.}& \oint \vec{B}\cdot \overrightarrow{dl}={\mu }_0{i}_c+{\mu }_0{ϵ}_0\frac{d{\varphi }_E}{dt}\\ \text{D.}& \text{Ampere-Maxwell Law}& \text{IV.}& \oint \vec{E}\cdot \overrightarrow{ds}=\frac{q}{{ϵ}_0}\end{array}$$ Choose the correct answer from the options given below :

1. A.A-I, B-II, C-III, D-IV
2. B.A-III, B-IV, C-I, D-II
3. C.A-IV, B-I, C-II, D-III
4. D.A-II, B-III, C-IV, D-I

A point charge of $10\mu \mathrm{C}$ is placed at the origin. At what location on the X-axis should a point charge of $40\mu \mathrm{C}$ be placed so that the net electric field is zero at $x=2$cm on the X-axis?

1. A.$x=6$ cm
2. B.$x=8$ cm
3. C.$x=4$ cm
4. D.$x=-4$ cm

A capacitor has capacitance $5\mu \mathrm{F}$ when its parallel plates are separated by air medium of thickness d. A slab of material of dielectric constant 1.5 having area equal to that of plates but thickness $\frac{d}{2}$ is inserted between the plates. Capacitance of the capacitor in the presence of slab will be ____ $\mu \mathrm{F}$.

If two charges ${\mathrm{q}}_1$ and ${\mathrm{q}}_2$ are separated with distance 'd' and placed in a medium of dielectric constant K. What will be the equivalent distance between charges in air for the same electrostatic force?

1. A.$d\sqrt{k}$
2. B.$1.5d\sqrt{k}$
3. C.$k\sqrt{d}$
4. D.$2d\sqrt{k}$

If $\vec{E}$ and $\vec{K}$ represent electric field and propagation vectors of the EM waves in vacuum, then magnetic field vector is given by : ($\omega$ - angular frequency) :

1. A.$\frac{1}{\omega }(\vec{K}\times \vec{E})$
2. B.$\vec{K}\times \vec{E}$
3. C.$\omega (\vec{K}\times \vec{E})$
4. D.$\omega (\vec{E}\times \vec{K})$

A stream of positively charged particles having $\frac{q}{m}=2\times 10^{11}\frac{\mathrm{C}}{\mathrm{kg}}$ and velocity ${\vec{v}}_0=3\times 10^7\hat{i}m/s$ is deflected by an electric field $1.8\hat{j}kV/m$. The electric field exists in a region of $10\mathrm{cm}$ along $x$ direction. Due to the electric field, the deflection of the charged particles in the $y$ direction is ____ $\mathrm{mm}$.

The electric field and magnetic field components of an electromagnetic wave going through vacuum is described by ${\mathrm{E}}_x={\mathrm{E}}_o\sin (kz-\omega t)$ ${\mathrm{B}}_y={\mathrm{B}}_o\sin (kz-\omega t)$ Then the correct relation between ${\mathrm{E}}_0$ and ${\mathrm{B}}_0$ is given by

1. A.${\mathrm{E}}_o{\mathrm{B}}_o=\omega k$
2. B.${\mathrm{E}}_0=k{\mathrm{B}}_0$
3. C.$k{\mathrm{E}}_0=\omega {\mathrm{B}}_0$
4. D.$\omega {\mathrm{E}}_0=k{\mathrm{B}}_0$

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

A parallel plate capacitor with air between the plates has a capacitance of 15pF. The separation between the plates becomes twice and the space between them is filled with a medium of dielectric constant 3.5. Then the capacitance becomes $\frac{x}{4}\mathrm{pF}$. The value of $x$ is ___________.

1. A.$\frac{{\rho }_{0}r}{4{\epsilon }_0}\left(\frac{3}{4}-\frac{r}{R}\right)$
2. B.$\frac{{\rho }_{0}r}{3{\epsilon }_0}\left(\frac{3}{4}-\frac{r}{R}\right)$
3. C.$\frac{{\rho }_{0}r}{4{\epsilon }_0}\left(1-\frac{r}{R}\right)$
4. D.$\frac{{\rho }_{0}r}{5{\epsilon }_0}\left(1-\frac{r}{R}\right)$

Given below are two statements. Statement I : Electric potential is constant within and at the surface of each conductor. Statement II : Electric field just outside a charged conductor is perpendicular to the surface of the conductor at every point. In the light of the above statements, choose the most appropriate answer from the options given below.

1. A.Both Statement I and Statement II are correct
2. B.Both Statement I and Statement II are incorrect
3. C.Statement I is correct but Statement II is incorrect
4. D.Statement I is incorrect but Statement II is correct

If the potential barrier across a p-n junction is $0.6\mathrm{V}$. Then the electric field intensity, in the depletion region having the width of $6\times 10^{-6}\mathrm{m}$, will be ____ $\times 10^{5}N/C$.

Two identical metallic spheres A and B when placed at certain distance in air repel each other with a force of $F$. Another identical uncharged sphere C is first placed in contact with A and then in contact with B and finally placed at midpoint between spheres A and B. The force experienced by sphere C will be:

1. A.$3F/2$
2. B.$3F/4$
3. C.$F$
4. D.$2F$

Two identical thin metal plates have charges $q_1$ and $q_2$ respectively such that $q_1>q_2$. The plates were brought close to each other to form a parallel plate capacitor of capacitance C. The potential difference between them is :

1. A.$\frac{(q_1+q_2)}{C}$
2. B.$\frac{(q_1-q_2)}{C}$
3. C.$\frac{(q_1-q_2)}{2C}$
4. D.$\frac{2(q_1-q_2)}{C}$

A capacitor of capacitance $500\mu \mathrm{F}$ is charged completely using a dc supply of $100\mathrm{V}$. It is now connected to an inductor of inductance $50\mathrm{mH}$ to form an LC circuit. The maximum current in LC circuit will be ____ $\mathrm{A}$.

Two capacitors, each having capacitance $40\mu \text{F}$ are connected in series. The space between one of the capacitors is filled with dielectric material of dielectric constant $K$ such that the equivalence capacitance of the system became $24\mu \text{F}$. The value of $K$ will be :

1. A.1.5
2. B.2.5
3. C.1.2
4. D.3

Two electric dipoles of dipole moments $1.2\times 10^{-30}\text{ Cm}$ and $2.4\times 10^{-30}\text{ Cm}$ are placed in two different uniform electric fields of strengths $5\times 10^4{\text{ NC}}^{-1}$ and $15\times 10^4{\text{ NC}}^{-1}$ respectively. The ratio of maximum torque experienced by the electric dipoles will be $\frac{1}{x}$. The value of $x$ is _________.

A slab of dielectric constant $\text{K}$ has the same cross-sectional area as the plates of a parallel plate capacitor and thickness $\frac{3}{4}\text{ d}$, where $\text{d}$ is the separation of the plates. The capacitance of the capacitor when the slab is inserted between the plates will be : (Given ${\text{C}}_0=\text{capacitance of capacitor with air as medium between plates}$.)

1. A.$\frac{4K{C}_0}{3+K}$
2. B.$\frac{3K{C}_0}{3+K}$
3. C.$\frac{3+K}{4K{C}_0}$
4. D.$\frac{K}{4+K}$

1. A.${\tan }^{-1}(4)$
2. B.${\tan }^{-1}(2)$
3. C.${\tan }^{-1}\left(\frac{1}{3}\right)$
4. D.${\tan }^{-1}(3)$

A beam of light travelling along $X$-axis is described by the electric field $E_y=900\sin \omega (t-x/c)$. The ratio of electric force to magnetic force on a charge $q$ moving along $Y$-axis with a speed of $3\times 10^7{\text{ ms}}^{-1}$ will be : (Given speed of light $=3\times 10^8{\text{ ms}}^{-1}$)

1. A.$1:1$
2. B.$1:10$
3. C.$10:1$
4. D.$1:2$

An electron (mass $\mathrm{m}$) with an initial velocity $\vec{v}=v_0\hat{i}(v_0>0)$ is moving in an electric field $\vec{E}=-E_0\hat{i}(E_0>0)$ where $E_0$ is constant. If at $t=0$ de Broglie wavelength is ${\lambda }_0=\frac{h}{m{v}_0}$, then its de Broglie wavelength after time t is given by

1. A.${\lambda }_0$
2. B.${\lambda }_0\left(1+\frac{e{E}_{0}t}{m{v}_0}\right)$
3. C.${\lambda }_{0}t$
4. D.$\frac{{\lambda }_0}{\left(1+\frac{e{E}_{0}t}{m{v}_0}\right)}$

To light, a $50\text{ W},100\text{ V}$ lamp is connected, in series with a capacitor of capacitance $\frac{50}{\pi \sqrt{x}}\mu \text{F}$, with $200\text{ V},50\text{ Hz}$ AC source. The value of $x$ will be _________.

A $1\text{ m}$ long copper wire carries a current of $1\text{ A}$. If the cross section of the wire is $2.0{\text{ mm}}^2$ and the resistivity of copper is $1.7\times 10^{-8}\Omega \text{m}$, the force experienced by moving electron in the wire is _________ $\times 10^{-23}\text{ N}$. (charge on electron $=1.6\times 10^{-19}\text{ C}$)

A charge of $4\mu \mathrm{C}$ is to be divided into two. The distance between the two divided charges is constant. The magnitude of the divided charges so that the force between them is maximum, will be :

1. A.$1\mu \mathrm{C}$ and $3\mu \mathrm{C}$
2. B.$2\mu \mathrm{C}$ and $2\mu \mathrm{C}$
3. C.$0$ and $4\mu \mathrm{C}$
4. D.$1.5\mu \mathrm{C}$ and $2.5\mu \mathrm{C}$