Electric potential

Two charges $Q_1=q$ and $Q_2=mq$ are placed at the points $P_1(a,b)$ and $P_2(ma,mb)$, respectively, in the $XY$ plane, where $a,b\ne0$ and $m\ne0,1$. If $V_1$ is the potential at a point in the $XY$ plane due to charge $Q_1$ and $V_2$ is the potential at that point due to charge $Q_2$. Correct statement(s) for the points at which $|V_1|=|V_2|$ is/are:

Two metal plates ($A$, $B$) are kept horizontally with separation of $\left(\dfrac{12}\pi\right)$ cm, with plate $A$ on the top. An atomizer jet sprays oil (density 1.5 g/cm³) droplets of radius 1 mm horizontally. All oil droplets carry a charge 5 nC. The potentials $V_A$ and $V_B$ are required on plates A and B respectively in order to ensure the droplets do not descend. The values of $V_A$ and $V_B$ are ________.
(Neglect the air resistance to the droplets and take $g=10$ m/s²)

A three coulomb charge moves from the point $(0,-2,-5)$ to the point $(5,1,2)$ in an electric field expressed as $\vec E=2x\hat i+3y^2\hat j+4\hat k$ N/C. The work done in moving the charge is ________ J.

The electric potential as a function of $x$, $y$ is given by $V=5(x^2-y^2)$ V. The electric field at a point $(2,3)$ m is ________ V/m.

A positive point charge of $10^{-8}$ C is kept at a distance of 20 cm from the center of a neutral conducting sphere of radius 10 cm. The sphere is then grounded and the charge on the sphere is measured. The grounding is then removed and subsequently the point charge is moved by a distance of 10 cm further away from the center of the sphere along the radial direction. Taking $\dfrac{1}{4\pi\epsilon_0} = 9\times10^9\ \mathrm{Nm^2/C^2}$ (where $\epsilon_0$ is the permittivity of free space), which of the following statements is/are correct:

Six infinitely large and thin non-conducting sheets are fixed in configurations I and II. As shown in the figure, the sheets carry uniform surface charge densities which are indicated in terms of $\sigma_0$. The separation between any two consecutive sheets is 1 $\mu$m. The various regions between the sheets are denoted as 1, 2, 3, 4 and 5. If $\sigma_0 = 9\ \mu\mathrm{C/m^2}$, then which of the following statements is/are correct:

(Take permittivity of free space $\epsilon_0 = 9\times10^{-12}$ F/m)

An infinitely long thin wire, having a uniform charge density per unit length of $5\,\mathrm{nC/m}$, is passing through a spherical shell of radius $1\,\mathrm{m}$, as shown in the figure. A $10\,\mathrm{nC}$ charge is distributed uniformly over the spherical shell. If the configuration of the charges remains static, the magnitude of the potential difference between points P and R, in Volt, is _____.

[Given: In SI units $\dfrac{1}{4\pi\varepsilon_0} = 9\times10^9$, $\ln 2 = 0.7$. Ignore the area pierced by the wire.]

An electric dipole is formed by two charges $+q$ and $-q$ located in $xy$-plane at $(0,2)$ mm and $(0,-2)$ mm, respectively, as shown in the figure. The electric potential at point P $(100,100)$ mm due to the dipole is $V_0$. The charges $+q$ and $-q$ are then moved to the points $(-1,2)$ mm and $(1,-2)$ mm, respectively. What is the value of electric potential at P due to the new dipole?

Six charges are placed around a regular hexagon of side length $a$ as shown in the figure. Five of them have charge $q$, and the remaining one has charge $x$. The perpendicular from each charge to the nearest hexagon side passes through the center O of the hexagon and is bisected by the side.

Which of the following statement(s) is(are) correct in SI units?

In the figure, the inner (shaded) region $A$ represents a sphere of radius $r_A = 1$, within which the electrostatic charge density varies with the radial distance $r$ from the center as $\rho_A = kr$, where $k$ is positive. In the spherical shell $B$ of outer radius $r_B$, the electrostatic charge density varies as $\rho_B = \dfrac{2k}{r}$. Assume that dimensions are taken care of. All physical quantities are in their SI units.

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

A disk of radius $R$ with uniform positive charge density $\sigma$ is placed on the $xy$ plane with its center at the origin. The Coulomb potential along the $z$-axis is
$$V(z) = \frac{\sigma}{2\epsilon_0}\left(\sqrt{R^2 + z^2} - z\right).$$
A particle of positive charge $q$ is placed initially at rest at a point on the $z$ axis with $z = z_0$ and $z_0 > 0$. In addition to the Coulomb force, the particle experiences a vertical force $\vec F = -c\,\hat k$ with $c > 0$. Let $\beta = \dfrac{2c\epsilon_0}{q\sigma}$. Which of the following statement(s) is(are) correct?

27 identical drops are charged at 22 V each. They combine to form a bigger drop. The potential of the bigger drop will be ______ V.

If the electric potential at any point $(x,y,z)$ m in space is given by $V=3x^2$ volt. The electric field at the point $(1,0,3)$ m will be :

Two uniformly charged spherical conductors A and B of radii 5 mm and 10 mm are separated by a distance of 2 cm. If the spheres are connected by a conducting wire, then in equilibrium condition, the ratio of the magnitudes of the electric fields at the surface of the sphere A and B will be :

The value of $R$ is ____ meter.

The value of $b$ is ____ meter.

A thin spherical insulating shell of radius $R$ carries a uniformly distributed charge such that the potential at its surface is $V_0$. A hole with a small area $\alpha4\pi R^2\ (\alpha \ll 1)$ is made on the shell without affecting the rest of the shell. Which one of the following statements is correct?

A point charge $+Q$ is placed just outside an imaginary hemispherical surface of radius $R$ as shown in the figure. Which of the following statements is/are correct?

There is a uniform electrostatic field in a region. The potential at various points on a small sphere centred at P, in the region, is found to vary between the limits 589.0 V to 589.8 V. What is the potential at a point on the sphere whose radius vector makes an angle of 60° with the direction of the field ?

The potential (in volts) of a charge distribution is given by

$V(z)=30-5z^2$ for |z| ≤ 1 m

V(z) = 35 − 10 |z| for |z| ≥ 1 m.

V(z) does not depend on x and y. If this potential is generated by a constant charge per unit volume $\rho_0$ (in units of $\epsilon_0$) which is spread over a certain region, then choose the correct statement.