Electric Potential and Potential Difference

Considering a group of positive charges, which of the following statements is correct?

Which of the following correctly represents the variation of electric potential $\left(V\right)$ of a charged spherical conductor of radius $\left(R\right)$ with radial distance $\left(r\right)$ from the centre?

For a charged spherical ball, electrostatic potential inside the ball varies with $r$ as $V=2a{r}^2+b$.

Here, $a$ and $b$ are constant and $r$ is the distance from the center. The volume charge density inside the ball is $-\lambda a\epsilon$. The value of $\lambda$ is ______.

$\epsilon =$ permittivity of medium.

A point charge $2\times {10}^{–2} \mathrm{C}$ is moved from $P$ to $S$ in a uniform electric field of $30 \mathrm{N} {\mathrm{C}}^{–1}$ directed along positive $x$-axis. If coordinates of $P$ and $S$ are $(1, 2, 0) \mathrm{m} \text{and} (0, 0, 0) \mathrm{m}$ respectively, the work done by electric field will be

The electric potential at the centre of two concentric half rings of radii $R_1$ and $R_2$, having same linear charge density $\lambda$ is

.

$27$ identical drops are charged at $22 \mathrm{V}$ each. They combine to form a bigger drop. The potential of the bigger drop will be _____  $\mathrm{V}$.

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

The two thin coaxial rings, each of radius $a$ and having charges $+\mathrm{Q}$ and $-\mathrm{Q}$ respectively are separated by a distance of $s$. The potential difference between the centres of the two rings is :

In the given figure, a battery of emf $E$ is connected across a conductor $PQ$ of length $l$ and different area of cross-sections having radii $r_1$ and $r_2\left(r_2<r_1\right).$

Choose the correct option as one moves from $P$ to $Q$.

$512$ identical drops of mercury are charged to a potential of $2 \mathrm{V}$ each. The drops are joined to form a single drop. The potential of this drop is $V$ in Volt.

Ten charges are placed on the circumference of a circle of radius $R$ with constant angular separation between successive charges. Alternate charges$1,3,5,7,9$ have charge $(+q)$ each, while $2,4,6,8,10$ have charge $(–q)$ each. The potential V and the electric field E at the centre of the circle are respectively : (Take $V=0$ at infinity)

Concentric metallic hollow spheres of radii $R$ and $4R$ hold charges $Q_1$ and $Q_2$ respectively. Given that surface charge density of the concentric spheres are equal, the potential difference $V\left(R\right)-V\left(4R\right)$ is:

A charge Q is distributed over two concentric conducting thin spherical shells radii $\mathrm{r}$ and $\mathrm{R} \left(\mathrm{R}>\mathrm{r}\right)$. If the surface charge densities on the two shells are equal, the electric potential at the common centre is : 

Consider two charged metallic spheres $S_1$ and $S_2$ of radii $R_1$ and $R_2,$ respectively. The electric fields $E_1$ (on $S_1$ ) and $E_2$ (on $S_2$ ) on their surfaces are such that $\frac{E_1}{E_2}=\frac{R_1}{R_2}.$ Then the ratio $V_1$ (on $S_1$ )$/$$V_2$ (on $S_2$ ) of the electrostatic potentials on each sphere is:

A uniformly charged ring of radius $3a$ and total charge $q$ is placed in $x‐y$ plane centred at origin. A point charge $q$ is moving towards the ring along the $z-$ axis and has speed $v$ at $z=4a$ . The minimum value of $v$ such that it crosses the origin is:

A solid conducting sphere, having a charge $\mathrm{Q},$ is surrounded by an uncharged conducting hollow spherical shell. Let the potential difference between the surface of the solid sphere and that of the outer surface of the hollow shell be $\mathrm{V}.$ If the shell is now given a charge of $–4\mathrm{Q},$ the new potential difference between the same two surfaces is:

The electric field in a region is given by $\overrightarrow{\mathrm{E}}=\left(\mathrm{A}\mathrm{x}+\mathrm{B}\right)\mathrm{ }\widehat{\mathrm{i}} ,$ where $\mathrm{E}$ is in $\mathrm{N}{\mathrm{C}}^{-1}$ and $x$ is in metres. The values of constants are $\mathrm{A}=20\mathrm{ }\mathrm{S}\mathrm{I}$ unit and $\mathrm{B}=10\mathrm{ }\mathrm{S}\mathrm{I}$ unit. If the potential at $x=1$ is ${\mathrm{V}}_1$ and that at $x=-5$ is ${\mathrm{V}}_2,$ then ${\mathrm{V}}_1-{\mathrm{V}}_2$ is

A positive point charge is released from rest at a distance ${\mathrm{r}}_0$ from a positive line charge with uniform charge density. The speed $\left(\mathrm{v}\right)$ of the point charge, as a function of instantaneous distance $\mathrm{r}$ from line charge, is proportional to

A charge $Q$ is distributed over three concentric spherical shells of radii $a, b, c \left(a<b<c\right)$ such that their surface charge densities are equal to one another.

The total potential at a point at distance $r$ from their common centre, where $r<a,$ would be:

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 $589.0 \mathrm{V}$ to $589.8 \mathrm{V}$. What is the potential at a point on the sphere whose radius vector makes an angle of $60°$ with the direction of the field?