Why Gauss's Law

Q. A long cylindrical volume contains a uniformly distributed charge of density $\rho .$ The radius of cylindrical volume is $R.$ A charge particle $\left(q\right)$ revolves around the cylinder in a circular path. The kinetic energy of the particle is:

1. $\frac{\rho q{R}^2}{4{\epsilon }_0}$
2. $\frac{q\rho }{4{\epsilon }_0{R}^2}$
3. $\frac{\rho q{R}^2}{2{\epsilon }_0}$
4. $\frac{4{\epsilon }_0{R}^2}{q\rho }$

Q. A non conducting disc of radius $a$ and uniform positive surface charge density $\sigma$ is placed on the ground with its axis vertical. A particle of mass $m$ and charge $q$ is dropped, along the axis of the disc from a height $H$ with zero initial velocity. It was observed that its velocity becomes zero just when it reaches the disc. The value of $\frac{3H}{a}$ is (​​​​​​​Given $\frac{q}{m}=\frac{4{ϵ}_{o}g}{\sigma }$)

Q. The graph $\frac{1}{\lambda }$ and stopping potential $\left(V\right)$ of three metals having work function ${\varphi }_1$, ${\varphi }_2$ and ${\varphi }_3$ in an experiment of photoelectric effect is plotted as shown in the figure. Which one of the following statement is/are correct? [Here $\lambda$ is the wavelength of the incident ray]
(i) Ratio of work functions ${\varphi }_1:{\varphi }_2:{\varphi }_3=1:2:4$
(ii) Ratio of work functions ${\varphi }_1:{\varphi }_2:{\varphi }_3=4:2:1$
(iii) $\tan \theta \propto \frac{hc}{e}$, where $h=$ Planck's constant, $c=$ speed of light
(iv) The violet colour-light can eject photoelectrons from metals 2 and 3.

1. (i), (iv)
2. (i), (iii)
3. (ii), (iii)
4. (i), (ii) and (iv)

Q. The PV graph of a process is a shown. Here $P_0={10}^{5}Pa$ and $V_0={10}^{-3}{m}^3$. The working substance is 2 moles of a mono – atomic gas. The change in internal energy of the gas and heat supplied in the process is (in joule)

1. 100, 100
2. 750, 1050
3. 700, 1400
4. 1000, 2000

Q.

Electric field given by the vector $\overrightarrow{E}=x\hat{i}+y\hat{j}$ is present in the xy plane. A small ring carrying charge +Q, which can freely slide on a smooth nonconducting 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. $(Q{L}^2/m{)}^{1/2}$
2. $2(Q{L}^2/m{)}^{1/2}$
3. $4(Q{L}^2/m{)}^{1/2}$
4. zero

Q. The plastic rod of length $L$ has uniform charge density $\lambda$. Find the electric potential at a distance $r(>>L)$ far on the perpendicular bisector of the rod, to the right of it on the x-axis.

1. $\frac{\lambda }{2\pi {ϵ}_0}\ln \left(x/L\right)$
2. $\frac{\lambda }{4\pi {ϵ}_0}ln\left(L/x\right)$
3. $\frac{\lambda }{4\pi {ϵ}_0}\ln \left(L/2x\right)$
4. $0$