Conservation of Energy in Case of Electrostatics

Q. The closest distance of approach of an alpha particle travelling with a velocity ‘v’ towards $A{l}_{13}$ nucleus is ‘d’. The closest distance of approach of an alpha particle travelling with velocity ‘4v’ towards $F{e}_{26}$ nucleus is

1. $\frac{d}{8}$
   
2. $\frac{d}{4}$
   
3. $\frac{d}{8}$
   
4. $8d$

Q. A proton, a deuteron and an $\alpha$-particle is accelerated through the same potential difference from rest, enters a region of uniform magnetic field, moving at right angles to it. What is the ratio of their kinetic energies?

1. $1:1:2$
2. $2:2:1$
3. $1:3:3$
4. $1:2:3$

Q. A bullet of mass m and charge q is fired towards a solid uniformly charged sphere of radius R and total charge +q. If it strikes the surface of sphere with speed u, find the minimum speed u so that it can penetrate through the sphere. (Neglect all resistance forces or friction acting on bullet except electrostatic forces)

Q. A body of mass $1\text{g}$ and carrying a charge ${10}^{-8}\text{C}$ passes through two points $\text{P}$ and $\text{Q}$. $\text{P}$ and $\text{Q}$ are at electric potential $600\text{V}$ and $0\text{V}$ respectively. The velocity of the body at $\text{Q}$ is $20\times {10}^{-2}{\text{ms}}^{-1}$. Its velocity at $P$ is $\sqrt{x\times {10}^{-3}}{\text{ms}}^{-1}$. The value of $x$ is

Q. The magnetic field of a plane electromagnetic wave is given by:
$\overrightarrow{B}=B_0\hat{i}\left[\cos \right(kz-\omega t\left)\right]+B_1\hat{j}\cos (kz+\omega t),$ Where $B_0=3\times {10}^{-5}\text{T}\text{and}{B}_1=2\times {10}^{-6}\text{T}$
The rms value of the force experienced by a stationary charge $Q={10}^{-4}\text{C}\text{at}z=0$ is closest to

1. $0.1\text{N}$
2. $0.6\text{N}$
3. $3\times {10}^{-2}\text{N}$
4. $0.9\text{N}$

Q. A block of mass $1\text{kg}$ containing a net positive charge ${10}^{-10}\text{C}$ is placed on a smooth horizontal table which terminates in a vertical wall as shown in figure. The distance of the block from the wall is $2\text{m}$. A horizontal electric field ${10}^{10}\text{N/C}$ towards right is switched on. Assuming elastic collision (if any), find the time period of the resulting oscillatory motion.

1. $1\text{s}$
2. $2\text{s}$
3. $3\text{s}$
4. $4\text{s}$

Q. The electric field intensity due to a thin infinitely long wire of uniform linear charge density $\lambda$ at $\text{O}$ as shown in the figure is

1. $\frac{\lambda }{2\pi {ϵ}_{0}R}$
2. $\frac{\sqrt{2}\lambda }{2\pi {ϵ}_{0}R}$
3. $\frac{\lambda \sqrt{5}}{2\pi {ϵ}_{0}R}$
4. $0$

Q. Figure shows a charge $+Q$ clamped at a point in free space. From a large distance another charge particle of charge $-q$ and mass of $m$ is thrown towards $+Q$ with an impact parameter $d$ as shown with speed $v$. How many positions for minimum separation would be attained for the particle.

1. $1$
2. $2$
3. $3$
4. $4$

Q. A charged particle enters a uniform magnetic field, directed perpendicular to its velocity. The magnetic field may

1. increase the kinetic energy of the particle.
2. decrease the kinetic energy of the particle.
3. change the direction of motion of the particle.
4. Both options $\left(A\right)$ and $\left(C\right)$

Q. Energy per unit volume for a capacitor having area $A$ and separation $d$ kept at potential difference $V$ is given by

1. $\frac{1}{2}\frac{{ϵ}_0{V}^2}{d^2}$
2. $\frac{{ϵ}_0{V}^2}{d^2}$
3. $\frac{1}{2}C{V}^2$
4. $\frac{Q^2}{2C}$

Q. A charged particle q is fired towards another charged particle Q which is fixed, with a speed v. It approaches Q upto a closest distance r and then returns. If q is given speed 2v, the closest distance of approach would be: [AIEEE-2004]

1. r/4
2. r/2
3. 2r
4. r

Q. A charge $3\mu \text{C}$ is released from rest from a point $P$ in an electric field where electric potential is $20\text{V}$. When the charge reaches slowly at infinity its kinetic energy becomes:

1. $60\mu \text{J}$
2. $45\mu \text{J}$
3. $30\mu \text{J}$
4. $120\mu \text{J}$

Q. The energy of a $100\mu \text{F}$ capacitor charged to $6\text{kV}$ is used to lift a $50\text{kg}$ mass (by powering a motor by energy of capacitor) without incurring any losses. What would be the greatest vertical height through which mass could be raised?
(Take $g=10{\text{m/s}}^2$)

1. $0.6\text{mm}$
2. $3.6\text{m}$
3. $1.6\text{m}$
4. $1.2\text{mm}$

Q. A particle of mass $0.01\text{kg}$ and charge ${10}^{-6}\text{C}$ is placed at rest in a uniform electric field $2\times {10}^6\text{N/C}$ and then released. The kinetic energy attained by the particle after moving a distance of $2\text{m}$ is
[Assume gravity free space]

1. $1\text{J}$
2. $2\text{J}$
3. $3\text{J}$
4. $4\text{J}$

Q. An elementary particle of mass m and charge +e is projected with velocity at a much more massive particle of charge $Z_e$, where Z > 0. What is the closest possible approach of the incident particle

1. $\frac{Z{e}^2}{8\pi {ϵ}_{0}m{v}^2}$
2. $\frac{Ze}{4\pi {ϵ}_{0}m{v}^2}$
3. $\frac{Z{e}^2}{2\pi {ϵ}_{0}m{v}^2}$
4. $\frac{Ze}{8\pi {ϵ}_{0}m{v}^2}$

Q. A block of mass $4\text{kg}$ carrying a charge $50\mu \text{C}$ is connected to a spring for which spring constant is $100\text{N/m}$. The block lies on a frictionless horizontal track and the system is immersed in a uniform electric field of magnitude $5\times {10}^5\text{V/m}$, directed as shown in figure. If the block is released suddenly from rest when the spring is unstretched then the maximum extension in the spring is

1. $20\text{cm}$
2. $30\text{cm}$
3. $40\text{cm}$
4. $50\text{cm}$

Q. An automobile spring extends $0.2\text{m}$ for $5000\text{N}$ load. The ratio of potential energy stored in the spring to the potential energy stored in a $10\mu \text{F}$ capacitor at a potential difference of $10,000\text{V}$ will be

1. $\frac{1}{4}$
2. $1$
3. $\frac{1}{2}$
4. $2$

Q. A particle of mass $m$ and charge $q$ is released from rest in an electric field $E$. Then the K.E. after time $t$ will be

1. $\frac{2E^2{t}^2}{mq}$
2. $\frac{E^2{q}^2{t}^2}{2m}$
3. $\frac{E{q}^{2}m}{2t^2}$
4. $\frac{Eqm}{2t}$

Q. A particle of specific charge (charge/mass) $\alpha$ starts moving from the origin from rest under the action of an electric field $\overrightarrow{E}=E_0\hat{i}$ and magnetic field $\overrightarrow{B}=B_0\hat{k}$. Its velocity at $(x_0,y_0,0)$ is $(4\hat{i}-3\hat{j})$. The value of $x_0$ is :
(All parameters are in SI unit.)

1. $\frac{13\alpha E_0}{2B_0}$
2. $\frac{16\alpha B_0}{E_0}$
3. $\frac{25}{2\alpha E_0}$
4. $\frac{5\alpha }{2B_0}$

Q. A particle of charge $q$ and mass $m$ moving under the influence of uniform electric field $E\hat{i}$ and a uniform magnetic field $-B\hat{i}$ moves from point $\text{P}(0,a)$ to $\text{Q}(2a,a)$ where $\text{O}(0,0)$ is origin. The velocity of the particle at Point $\text{P}$ is $v\hat{i}$ and at $\text{Q}$ is $2v\hat{i}$. The rate of work done by the electric field & magnetic field on particle at point $\text{P}$ will be respectively:

1. $\frac{3}{4}\frac{m{v}^3}{a},0$
2. $\frac{3}{4}\frac{m{v}^3}{a},\frac{3}{2}\frac{m{v}^3}{a}$
3. $\frac{3}{2a}m{v}^3,0$
4. $0,\frac{m{v}^3}{2a}$

Q. Two unchanged spheres are rubbed together and then separated by a distance of 2cm in air. If the force of attraction between them is 22.5N. Then the number of electrons lost by positively charged sphere is

1. $625$
2. $6.25\times {10}^{18}$
3. $6.25\times {10}^{12}$
4. $225$

Q. A test charge $q$ is made to move in the electric field of a point charge $Q$ along two different closed paths (Figure). First path has sections along and perpendicular to lines of electric field. Second path is a rectangular loop of the same area as the first loop. How does the work done compare in the two cases?

Q. The potential barrier between the $p-n$ junction is $0.8\text{V}$. If the width of the depletion region is $8\times {10}^{-7}\text{m}$ then the intensity of electric field in this region is ____$\times {10}^6\text{V/m}$.

Q. There exists a region of electric field with potential difference of 25 V as shown in figure. An electron enters this region at speed of v = 3 $\times {10}^6$ m/s at angle of $\alpha$ and emerges out of region at angle $\beta$. The $\frac{sin\alpha }{sin\beta }$ is [Electric field exist in given region only]
$e=1.6\times {10}^{-}19,m=9.1\times {10}^{-}31$ (2 decimal places)

Q. For the situation described in the figure, the magnetic field changes with time according to, $B=(2.00t^3-4.00t^2+0.8)\text{T}$. What is the magnitude of electric field at point ${\text{P}}_1$ when $t=0.02\text{s}$ and $r_1=0.02\text{m}$.

1. $1\text{N/C}$
2. $2.5\text{N/C}$
3. $3\text{N/C}$
4. $0.3\text{N/C}$

Q. A particle of charge $-1\text{C}$ and mass $0.2\text{g}$ moves in a circle of radius $1\text{m}$ around an infinite long line charge of linear charge density $+1\text{nC/m}$. The time period of motion is

1. $\frac{\pi }{150}\text{s}$
2. $\frac{\pi }{75}\text{s}$
3. $\frac{\pi }{100}\text{s}$
4. $\frac{\pi }{60}\text{s}$

Q. A particle of specific charge $\alpha$ starts moving from the origin under the action of an electric field $\overrightarrow{E}=E_0\hat{i}$ and magnetic field $\overrightarrow{B}=B_0\hat{k}$. Its velocity at $(x_0,y_0,0)$ is $(4\hat{i}+3\hat{j})$. The value of $x_0$ is:

1. $\frac{5\alpha }{2B_0}$
2. $\frac{13\alpha E_0}{2B_0}$
3. $\frac{16\alpha E_0}{E_0}$
4. $\frac{25}{2\alpha B_0}$

Q. An electron with a kinetic energy of $100\text{eV}$ enters the space between the plates of a parallel plate capacitor made of two dense metal grids at an angle of ${30}^{\circ }$ with the plates of capacitor and leaves this space at an angle of ${45}^{\circ }$ with the plates. What is the potential difference of the capacitor ?

1. $100\text{V}$
2. $50\text{V}$
3. $150\text{V}$
4. $200\text{V}$

Q. For the shown system of fixed charges, where should a third charge of $3nC$ be placed, so that the system remains in equilibrium.

1. $0.37$ $\text{m}$ from $4nC$ charge toward left
2. $0.47$ $\text{m}$ from $4nC$ charge toward left
3. $0.67$ $\text{m}$ from $4nC$ charge toward left
4. $0.57$ $\text{m}$ from $4nC$ charge toward left

Q.

Two electric charges of $+{10}^{-9}C$ and $-{10}^{-9}C$ are placed at the corners A and B of an equilateral triangle ABC side 5cm.The electric intensity at C is

1. $1800N/C$
2. $3600N/C$
3. $900N/C$
4. $2700N/C$