Applications of Gauss's Law

Applying Gauss theorem, the expression for the electric field intensity at a point due to an infinitely long, thin, uniformly charged straight wire is

A thin infinite sheet charge and an infinite line charge of respective charge densities $+\sigma$ and $+\lambda$ are placed parallel at $5 \mathrm{m}$ distance from each other. Points $P$ and $Q$ are at $\frac{3}{\pi } \mathrm{m}$ and $\frac{4}{\mathrm{\pi }} \mathrm{m}$ perpendicular distances from line charge towards sheet charge, respectively. $E_p$ and $E_q$ are the magnitudes of resultant electric field intensities at point $P$ and $Q$ respectively. If $\frac{E_P}{E_Q}=\frac{4}{a}$ for $2\left|\sigma \right|=\left|\lambda \right|,$ then the value of $a$ is _____.

An electron revolves around an infinite cylindrical wire having uniform linear charge density $2\times {10}^{-8}$ $\mathrm{C} {\mathrm{m}}^{-1}$ in circular path under the influence of attractive electrostatic field as shown in the figure. The velocity of electron with which it is revolving is ______________ $\times {10}^6 \mathrm{m} {\mathrm{s}}^{-1}$. Given mass of electron $=9\times {10}^{-31} \mathrm{kg}$

Two identical infinite negative line charges are held along the lines $y=\pm 4$ in the $xy$ plane. A charge-$Q$ is placed at origin & is restricted to move along $x$ axis.Its equilibrium is 

A positive charge q having mass $m$, is released from rest in front of an infinite non-conducting sheet with surface charge density equal to $\sigma$. Speed of the charge particle after time $\text{'}t \text{'}$is

Consider an isolated, thin conducting spherical shell $A$ of radius $20 \mathrm{cm}$. The shell is given a charge $Q_A$ that distributes uniformly and due to $Q_A$ only the maximum strength of the resulting electric field has a value $11250 \mathrm{N} {\mathrm{C}}^{-1}$ due to $Q_A$ only. $B$ is another isolated thin conducting spherical shell of radius $40 \mathrm{cm}$. It is given a charge $Q_B$ that distributes uniformly and in this case, maximum strength of the resulting field has a value $4500 \mathrm{N} {\mathrm{C}}^{-1}$ due to $Q_B$ only. The two shells $A$ and $B$ are now kept in a concentric manner as shown in figure. $x,y$ and $z$ are points as shown in figure, at distance $10 \mathrm{cm}, 30 \mathrm{cm}, 50 \mathrm{cm}$ from the centre, respectively. [Take $V=0$ at $r=\infty$]

Electric field at distance $L$ and $2 L$ from uniformly charged large non conducting sheet of surface charge density $\sigma$ will be

A thin square planar lamina is given a charge $Q_1$. A straight thin cylindrical conductor is kept parallel to the plane and given a charge $Q_2$. The length of the cylinder is also parallel to one of sides of the square lamina. The straight line joining the center of the plane $C_1$ and the center of the cylinder $C_2$ is perpendicular to the plane and also perpendicular to the geometric length of the cylinder. The distance $C_1{C}_2$ is equal to the side length of the square lamina. It is observed that at the midpoint of the line $C_1{C}_2$ the net electric field is zero. Assuming that the side length of the square is huge, calculate the ratio of charges $\frac{Q_1}{Q_2}$. (Given that length of the cylinder is equal to the side length of the square plate)

The charge density of uniformly charged infinite plane is $\sigma$. A simple pendulum is suspended vertically downward near it. Charge $q_0$ is placed on metallic bob. If the angle made by the string is $\theta$ with vertical direction then _____

A hollow charged metal sphere has radius $r$. If the potential difference between its surface and a point at a distance $3r$ from the centre is$V$, then electric field intensity at a distance $3r$ is:

A parallel plate capacitor has two square plates with equal and opposite charges. The surface charge densities on the plates are $+\mathrm{\sigma }$ and $–\mathrm{\sigma }$ respectively. In the region between the plates the magnitude of the electric field is

The ratio of electric field intensity at $\mathrm{P}$ & $\mathrm{Q}$ in the shown arrangement is

In the figure shown, there is a large sheet of charge of uniform surface charge density $\sigma$. A charge particle of charge $-q$ and mass $m$ is projected from a point $A$ on the sheet with a speed $u$ with angle of projection such that it lands at maximum distance from $A$ on the sheet. Neglecting gravity, find the time of flight.

A conducting spherical shell of radius $R$ has a charge $+q$units. The electric field due to the shell at a point

A conducting spherical shell of radius $R$ has a charge $+q$units. The electric field due to the shell at a point

An electron is moving around an infinite linear charge in a circular path of diameter $0.3 \mathrm{m}$. If linear charge density is ${10}^{-6} \mathrm{C}/\mathrm{m}$ and the speed of the electron is written as$n\times {10}^7 \mathrm{m}/\mathrm{s}$, then find $n$ accurate up to two digits after the decimal point. ($m_e=9\times {10}^{-31} \mathrm{kg}$, $e=1.6\times {10}^{-19} \mathrm{C}$)

$\mathrm{A}$ solid conducting sphere of radius $20 \mathrm{cm}$ is enclosed by a thin metallic shell of radius $40 \mathrm{cm}.$ $\mathrm{A}$ charge of $40 \mathrm{\mu C}$ is given to the inner sphere. If the metallic shell is earthed, then the heat generated in the process is

A large sheet carries uniform surface charge density $\sigma$. A rod of length $2\ell$ has a linear charge density $\lambda$ on one half and $-\lambda$ on the second half. The rod is hinged at mid-point $\mathrm{O}$ and makes an angle $\theta$ with the normal to the sheet. The electric force experienced by the rod is,

An electron of ${10}^{3 }\mathrm{eV}$ energy is fired from a distance of $5 \mathrm{mm}$ perpendicularly towards an infinite charged conducting plate. What should be the minimum charge density on plate so that electron fails to strike the plate?

A particle of mass $9\times {10}^{-5} \mathrm{gm}$ is placed at some height above a uniformly charged horizontal infinite non conducting plate having a surface charge density $5\times {10}^{-5} \mathrm{C}/{\mathrm{m}}^2$. What should be the charge on the particle so that on releasing it will not fall down. Take, $g=9.8 \mathrm{m}/{\mathrm{s}}^2$