Electric Field

A tiny spherical oil drop carrying a net charge $q$ is balanced in still air with a vertical uniform electric field of strength $\frac{81\mathrm{\pi }}{7}\times {10}^5 \mathrm{V} {\mathrm{m}}^{-1}$. When the field is switched off, the drop is observed to fall with terminal velocity $2 \times {10}^{–3} \mathrm{m} {\mathrm{s}}^{–1}$. Given $g= 9.8 \mathrm{m} {\mathrm{s}}^{–2}$, viscosity of the air $=1.8 \times {10}^{–5} \mathrm{N} \mathrm{s} {\mathrm{m}}^{–2}$ and the density of oil $= 900 \mathrm{kg} {\mathrm{m}}^{–3}$, the magnitude of $q$ is :

A continuous beam of electrons emitted by a heating filament is accelerated in free space by an electric field as shown in the figure. The two stops at the left ensure that the electron beam has a uniform cross-section. Which of the following is/are correct:

Four point charges $+ 8 \mathrm{\mu C} , - 1 \mathrm{\mu C} , - 1 \mathrm{\mu C}$ and $+ 8 \mathrm{\mu C}$ are fixed at the points, $-\sqrt{\frac{27}{2}}\mathrm{m} ,$$-\sqrt{\frac{3}{2}}\mathrm{m} ,$$+\sqrt{\frac{3}{2}}\mathrm{m}$ and $+\sqrt{\frac{27}{2}}\mathrm{m}$ respectively on the $\mathrm{y} - \mathrm{axis}.$ A particle of mass $6 \times 10{ }^{-4} \mathrm{kg}$ and of charge$+ 0.1 \mathrm{\mu C}$ moves along the $-x$ direction. Its speed at $x=+\infty$ is $v_0.$ Find the least value of $v_0$ for which the particle will cross the origin. Find also the kinetic energy of the particle at the origin. Assume that space is gravity-free. Given:$\frac{1}{\left(4{\mathrm{\pi \epsilon }}_0\right) }= 9\times {10}^9 \mathrm{N} {\mathrm{m}}^{-2} {\mathrm{C}}^{-2}$

The figure shows two infinitely large conducting plates $A$ and $B$. If the electric field at $C$ due to charge densities ${\mathrm{\sigma }}_1, {\mathrm{\sigma }}_2, {\mathrm{\sigma }}_3$ and ${\sigma }_4$ is $E\hat{\mathrm{i}}$, find ${\mathrm{\sigma }}_2$ and ${\mathrm{\sigma }}_{3 }$in terms of $E$. State whether this much information is sufficient to find ${\mathrm{\sigma }}_1$ and ${\mathrm{\sigma }}_4$ in terms of $E$. Derive a relation between${\mathrm{\sigma }}_1$ and ${\mathrm{\sigma }}_4.$ 

The potential difference between two large parallel plates is varied as $v=at$, a is a positive constant and $t$ is time. An electron starts from rest at$t=0$ from the plate which is at lower potential. If the distance between the plates is $L$, mass of electron $m$ and charge on electron$-e$ then find the velocity of the electron when it reaches the other plate.

Find the electric field potential and strength at the centre of a hemisphere of radius $R$ charged uniformly with the surface density $\sigma$.

A triangle is made from thin insulating rods of different lengths, and the rods are uniformly charged, i.e., the linear charge density on each rod is uniform and the same for all three rods. Find a particular point in the plane of the triangle at which the electric field strength is zero.

A thin non-conducting ring of radius $R$ has a linear charge density $\mathrm{\lambda }={\mathrm{\lambda }}_{0 }\mathrm{cos\phi }$, where ${\mathrm{\lambda }}_0$ is a constant, $\phi$ is the azimuthal angle. Find the magnitude of the electric field strength,

(a) at the centre of the ring,

(b) on the axis of the ring as a function of the distance $x$ from its centre. Investigate the obtained function at $x\gg R$.

A system consists of a thin charged wire ring of radius $R$ and a very long uniformly charged thread oriented along the axis of the ring with one of its ends coinciding with the centre of the ring. The total charge of the ring is equal to $q$. The charge of the thread (per unit length) is equal to $\lambda$. Find the interaction force between the ring and the thread.

A thin semi-circular ring of the radius $r$ has a positive charge $q$ distributed uniformly over it. The net field $E$ at the centre $O$ is

Charges are placed on the vertices of a square as shown. Let $E$ be the electric field and $V$ the potential at the centre. If the charges on $A$ and $B$ are interchanged with those on $D$ and $C$, respectively, then

A point charge $q$ is within an electrically neutral conducting shell whose outer surface is a sphere of radius $R$. The centre of outer surface is at $O$. Consider a point $P$ outside the conductor as shown in the figure. The magnitude of electric field at $P$

Which of the following statement(s) is/are correct? 

A wooden block performs SHM on a frictionless surface with frequency, ${\nu }_0$. The block carries a charge $+Q$ on its surface. If now a uniform electric field $E$ is switched-on as shown, then the SHM of the block will be 

Consider a system of three charges $\frac{q}{3}, \frac{q}{3}$ and $-\frac{2q}{3}$ placed at points $A, B$ and $C$, respectively, as shown in the figure. Take $O$ to be the centre of the circle of radius $R$ and angle $CAB= {60}^{°}$.

The electric field produced by a positively charged particle, placed in an $\mathrm{xy}$-plane is $7.2 (4\mathrm{i} + 3\mathrm{j}) \mathrm{N} {\mathrm{C}}^{-1}$ at the point $(3 \mathrm{cm}, 3\mathrm{cm})$ and $100\hat{\mathrm{i}} \mathrm{N} {\mathrm{C}}^{-1}$ at the point $(2 \mathrm{cm}, 0)$.

An oil drop has a charge $– 9.6 \times {10}^{–19} \mathrm{C}$ and mass $1.6 \times {10}^{–15} \mathrm{gm}$. When allowed to fall, due to air resistance force it attains a constant velocity. Then if a uniform electric field is to be applied vertically to make the oil drop ascend up with the same constant speed, which of the following are correct. $(g=10 {\mathrm{ms}}^{–2})$ (Assume that the magnitude of resistance force is same in both the cases)

A solid metallic sphere has a charge $+3Q$. Concentric with this sphere is a conducting spherical shell having charge $-Q$. The radius of the sphere is $a$ and that of the spherical shell is $b(b>a)$. What is the electric field at a distance $R(a<R<b)$ from the centre?

The electric field above a uniformly charged nonconducting sheet is $E$. If the nonconducting sheet is now replaced by a conducting sheet with the charge same as before, the new electric field at the same point is:

For an infinite line of charge having charge density $\lambda$ lying along $x$-axis, the work required in moving charge $q$ from $C$ to $A$ along arc $CA$ is: