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

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

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)

1. $\frac{q}{\sqrt{2\pi {\epsilon }_{0}mR}}$
2. $\frac{q}{\sqrt{4\pi {\epsilon }_{0}mR}}$
3. $\frac{q}{\sqrt{8\pi {\epsilon }_{0}mR}}$
4. $\frac{\sqrt{3}q}{\sqrt{4\pi {\epsilon }_{0}mR}}$

Q. Electric field given by the vector $x\hat{i}+y\hat{i}$ is present in the XY plane. A small ring carrying charge +Q, which can freely slide on a smooth non conducting rod, is projected along the rod from the point (0, L) such that it can reach the other end of the rod. What minimum velocity should be given to the ring?(Assume zero gravity)

1. ${\left(\frac{Q{L}^2}{m}\right)}^{\frac{1}{2}}$
2. $2{\left(\frac{Q{L}^2}{m}\right)}^{\frac{1}{2}}$
3. $4{\left(\frac{Q{L}^2}{m}\right)}^{\frac{1}{2}}$
4. ${\left(\frac{Q{L}^2}{2m}\right)}^{\frac{1}{2}}$

Q. Four charges, two $+8\mu C$ and two $-1\mu C$ are fixed at $(0,\pm \sqrt{\frac{27}{2}m})$ and $(0,\pm \sqrt{\frac{3}{2}m})$ respectively. A third charge of $\frac{1}{3}\mu C$ and mass 2g is at a large distance along the x-axis. Find the minimum velocity to be given to the $\frac{1}{3}\mu C$ charge such that it just reaches the origin (in $m{s}^{-1}$).

1. 2
2. 3
3. 5
4. 10

Q. Two identical particles of mass m carry a charge Q each. Initially one is at rest on a smooth horizontal plane and the other is projected along the plane directly towards first particle from a large distance with speed $v_0$. The closed distance of approach be

1. $\frac{k{Q}^2}{mv}$
2. $\frac{4k{Q}^2}{m{v}^2}$
3. $\frac{2k{Q}^2}{m{v}^2}$
4. $\frac{k{Q}^2}{4m{v}^2}$

Q.
A solid sphere and sperical shell of same mass m and same radius R are given charges +Q and -Q. Both spheres are kept on rough surface where friction is sufficient to avoid sliding. Initial seperation between spheres is 10 R. If m = 126 kg, $Q^2=23\times {10}^{-9}{C}^2$ and R = 1m.

(A)	Speed of solid sphere just before collision is	1) $\frac{3}{5}$ m/s
(B)	Speed of spherical shell just before collision is	2) 0 m/$s^2$
(C)	Speed of center of mass just before collision is	3) $\frac{5}{7}$ m/s
(D)	Acceleration of center of mass	4) Towards left
		5) Towards right

1. A $\to$ 3, B $\to$ 1, 4, C $\to$ 5, D $\to$ 5
2. A $\to$ 1, B $\to$ 3, 4, C $\to$ 4, D $\to$ 5
3. A $\to$ 1, B $\to$ 1, 4, C $\to$ 5, D $\to$ 4
4. A $\to$ 3, B $\to$ 3, 4, C $\to$ 4, D $\to$ 4

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. In the figure below, $10\text{mC}$ charge is fixed. A $50\mu \text{C}$ charge having mass $10\text{g}$ is projected from infinity towards the $10\text{mC}$ charge. Calculate initial velocity of projection if minimum distance between the charges is $10\text{cm}$.

1. $4000{\text{ms}}^{-1}$
2. $2000{\text{ms}}^{-1}$
3. $5000{\text{ms}}^{-1}$
4. $3000{\text{ms}}^{-1}$