Electric Potential & Equipotentials

A charge $Q$ is placed at each corner of a cube of side $r$. The potential at the centre of the cube is ($\epsilon_0$ = permittivity of free space)

Five positive charges each having charge $q$ are placed at the vertices of a pentagon as shown in the figure. The electric potential $(V)$ and the electric field $(\vec{E})$ at the center $O$ of the pentagon due to these five positive charges are:

'$n$' small spherical drops of the same size which are charged to 'V' volt each coalesce to form a single big drop. The potential of a big drop is

Three small identical bubbles of water having same charge on each coalesce to form a bigger bubble. Then the ratio of the potentials on one initial bubble and that on the resultant bigger bubble is:

The electric potential as a function of $x, y$ is given by $V = 5\left(x^2 - y^2\right)V$. The electric field at a point $(2, 3)\text{ m}$ is $\underline{\qquad}\ \text{V/m}$.

If charge $+q$ is taken from one point to another over an equipotential surface, then

Which of the following statements is NOT TRUE?

a. Work done to move a charge on an equipotential surface is not zero.

b. Equipotential surfaces are the surfaces where the potential is constant.

c. Equipotential surfaces for a uniform electric field are parallel and equidistant from each other. d. Electric field is always perpendicular to an equipotential surface.

Three charges $2q, -q$ and $-q$ are located at the vertices of an equilateral triangle. At the centre of the triangle

Four electric charges $+q, +q, -q$, and $-q$ are placed in order at the corners of a square of side '$2r$'. The electric potential at a point midway between the two negative charges is

A semicircular arc of radius '$R$' is charged uniformly and has charge per unit length as $\lambda$. The electric potential $V$ at the centre of curvature is [$\varepsilon_0$ = permittivity of vacuum]

The electric potential at the center of two concentric half rings of radii $R_1$ and $R_2$, having same linear charge density $\lambda$ is ($\epsilon_0$ = permittivity of free space)

A unit positive point charge is taken slowly through an infinitesimally thin tube that is inside a charged dielectric sphere of radius $R$, having uniform positive charge density $\rho$, as shown in the figure. The initial and final positions of the charge are marked by $A$ and $B$ at distance $2R$ and $3R$ respectively, from the centre of the sphere. In this process, the magnitude of the total work done on the point charge is $\frac{\rho R^2}{n\varepsilon_0}$. The value of $n$ is: ( $\varepsilon_0$ is the permittivity of vacuum)

Two spheres A and B of radius $a$ and $b$ respectively are at the same electric potential. The ratio of the surface charge densities of A to that of B is

Two concentric spheres kept in air have radii $R$ and $r$. They have similar charge and equal surface charge density $\sigma$. The electric potential at their common centre is ($\epsilon_0$ = permittivity of free space)

A large insulated sphere of radius $r$, charged with $Q$ C is placed in contact with a small insulated uncharged sphere of radius $r_1$ and is then separated. The charge on the smaller sphere will now be

Two isolated metallic spheres of radii $R$ and $2R$ are charged such that both have the same charge density $\sigma$. The spheres are then connected by a thin conducting wire. If the new charge density of the larger sphere is $\sigma_1$. The ratio $\sigma_1$ to $\sigma$ is

Two metal sphere of radius $R$ and $3R$ have same surface charge density $\sigma$. If they are brought in contact and then separated, the surface charge density on smaller and bigger sphere becomes $\sigma_1$ and $\sigma_2$, respectively. The ratio $\frac{\sigma_1}{\sigma_2}$ is.

Two charged conducting spheres $S_1$ and $S_2$ of radii 8 cm and 18 cm are connected to each other by a wire. After equilibrium is established, the ratio of electric fields on $S_1$ and $S_2$ spheres are $E_{S1}$ and $E_{S2}$ respectively. The value of $\frac{E_{S1}}{E_{S2}}$ is ________.

Following figure indicates the electric potential V as a function through 4 regions on x-axis. Which of the following is correct for the electric field E in these regions?

Electric field in a region is given by $\vec{E} = Ax\hat{i} + By\hat{j}$, where $A = 10\text{ V/m}^2$ and $B = 5\text{ V/m}^2$. If the electric potential at a point $(10, 20)$ is 500 V, then the electric potential at origin is ______ V.

Three concentric metallic spherical shells of radii $R, 3R$ and $9R$ are given charges $Q_1, Q_2$ and $Q_3$, respectively. It is found that surface charge densities on the outer surfaces of the shells are equal. Then the ratio of charges given to shells $Q_1 : Q_2 : Q_3$, is (Neglect the induction effects)

There are three concentric conducting spherical shells $A, B$ and $C$ of radii $a, b$ and $c$ respectively ( $c > b > a$) and they are charged with charge $q_1, q_2$ and $q_3$ respectively. The potentials of the spheres $A, B$ and $C$ respectively, are:

The electrostatic potential in a charged spherical region of radius $r$ varies as $V = ar^3 + b$, where $a$ and $b$ are constants. The total charge in the sphere of unit radius is $\alpha \times \pi a\epsilon_0$. The value of $\alpha$ is ______. (permittivity of vacuum is $\epsilon_0$)

A regular hexagon of side 6 cm has a charge of $2\,\mu\text{C}$ at each of its vertices, what is the potential at the centre of the hexagon?

$\Bigg[\frac{1}{4\pi\varepsilon_0} = 9 \times 10^{9}$ SI unit $\Bigg]$

Charges of $2\,\mu\text{C}$ and $-3\,\mu\text{C}$ are placed at two points A and B separated by distance of 1 m. The distance of the point from $A$ where net potential is zero is

A conducting sphere of radius ' R ' is given a charge ' $Q$ ' uniformly. The electric field and the electric potential at the centre of the sphere are respectively [ $\varepsilon_0 =$ permittivity of free space]

Given below are two statements: one is labelled as Assertion $\mathbf{A}$ and the other is labelled as Reason $\mathbf{R}$

Assertion A: Work done in moving a test charge between two points inside a uniformly charged spherical shell is zero, no matter which path is chosen.

Reason R: Electrostatic potential inside a uniformly charged spherical shell is constant and is same as that on the surface of the shell.

In the light of the above statements, choose the correct$$ answer from the options given below.

The electrostatic potential on the surface of uniformly charged spherical shell of radius $R = 10\text{ cm}$ is 120 V. The potential at the centre of shell, at a distance $r = 5\text{ cm}$ from centre, and at a distance $r = 15$ cm from the centre of the shell respectively, are:

Assuming the drops to be spherical, 27 identical drops of mercury are charged simultaneously to the same potential of 20 volt. If all the charged drops are made to combine to form one big drop, then potential of big drop will be

' $n$ ' identical small spherical drops of water, each of radius ' $r$ ' and charged to the same potential ' v ' are combined to form a big drop. The potential of a big drop is

' n ' small spherical drops of same size which are charged to ' $V$ ' volt each coalesce to form a single big drop. The potential of the big drop is

The electric potential ' V ' is given as a function of distance ' $x$ ' (metre) by $V = \left(4x^2 + 8x - 3\right)V$. The value of electric field at $x = 0.5\text{ m}$, in $\text{V/m}$ is

The electric charges ' $+2q$ ', ' $+2q$ ', ' $-2q$ ' and ' $-2q$ ' are placed at the corners of square of side ' 2 L ' as shown in figure. The electric potential at point 'A', midway between the two charges ' $+2q$ ' and ' $+2q$ ' is

( $\varepsilon_0 =$ permittivity of free space)

Two metal spheres of radius R and 3R have same surface charge density σ. If they are brought in contact and then separated, the surface charge density on smaller and bigger sphere becomes σ1 and σ2, respectively. The ratio $\frac{\sigma_1}{\sigma_2}$ is

Two large plane parallel conducting plates are kept 10 cm apart as shown in figure. The potential difference between them is V. The potential difference between the points A and $B$ (shown in the figure) is:

Three infinitely long wires with linear charge density $\lambda$ are placed along the $x\text{-axis}, y\text{-axis}$ and $z\text{-}$ axis respectively. Which of the following denotes an equipotential surface?

Three concentric charged metallic spherical sheets $A, B$ and $C$ have radii $a, b, c$ potentials $V_A, V_B, V_C$ and charge densities $+\sigma, -\sigma$ and $+\sigma$ respectively. The value of potential $V_A$ is ( $\varepsilon_0 =$ permittivity of free space)

A regular hexagon of side 10 cm has a charge $1\,\mu\text{C}$ at each of its vertices. The potential at the centre of hexagon is $\Bigg[\frac{1}{4\pi\varepsilon_0} = 9 \times 10^{9}$ SI unit $\Bigg]$

Four point charges each +q is placed on the circumference of a circle of diameter 2 d in such a way that they form a square. The potential at the centre is proportional to

The value of electric potential at a distance of $9\text{ cm}$ from the point charge $4 \times 10^{-7}\text{ C}$ is [Given $\frac{1}{4\pi\varepsilon_0} = 9 \times 10^{9}\text{ N m}^2\text{C}^{-2}$ ]:

An electric charge $10^{-6}\,\mu\text{C}$ is placed at origin $(0, 0)\text{ m}$ of $X - Y$ co-ordinate system. Two points $P$ and $Q$ are situated at $(\sqrt{3}, \sqrt{3})\text{ m}$ and $(\sqrt{6}, 0)\text{ m}$ respectively. The potential difference between the points $P$ and $Q$ will be:

Charges of $2\,\mu\text{C}$ and $-3\,\mu\text{C}$ are placed at two points A and B separated by 1 m. The distance of the point from A, where net potential is zero, is

A thin spherical shell is charged by some source. The potential difference between the two points $C$ and $P$ (in V) shown in the figure is:

(Take $\frac{1}{4\pi\varepsilon_0} = 9 \times 10^{9}$ SI units)

The electric potential at the surface of an atomic nucleus $(z = 50)$ of radius $9 \times 10^{-13}\text{ cm}$ is __________ $\times 10^{6}\text{ V}$.

' $n$ ' small drops of same size are charged to ' $V$ ' volt each. If they coalesce to form a single large drop, then its potential will be

Two charged conducting spheres of radii $a$ and $b$ are connected to each other by a conducting wire. The ratio of charges of the two spheres respectively is:

If a unit charge is taken from one point to another point over an equipotential surface, then

Given below are two statements: one is labelled as Assertion (A) and the other is labelled as Reason (R).

Assertion (A): Work done by electric field on moving a positive charge on an equipotential surface is always zero.

Reason (R): Electric lines of forces are always perpendicular to equipotential surfaces. In the light of the above statements, choose the most appropriate answer from the options given below:

Four electric charges $+q, +q, -q$ and -q are placed in order at the corners of a square of side 2 L. The electric potential at point midway between the two positive charges is

Three charges $2q, -q$ and $-q$ are located at the vertices of an equilateral triangle. At the centre of the triangle

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 ($\varepsilon_0 =$ permittivity of free space)

At the centre of a half ring of radius $R = 10\text{ cm}$ and linear charge density $4\text{ nC m}^{-1}$, the potential is $x\pi\text{ V}$. The value of $x$ is _________.

A metallic sphere ' A ' isolated from ground is charged to $+50\,\mu\text{C}$. This sphere is brought in contact with other isolated metallic sphere ' $B$ ' of half the radius of sphere ' $A$ '. Then the charge on the two isolated spheres A \& B are in the ratio

The electrostatic potential inside a charged spherical ball is given by $V = ar^2 + b$ where ' r ' is the distance from its centre and ' $a$ ' and ' $b$ ' are constants. The volume charge density of the ball is [ $\varepsilon_0 =$ permittivity of free space $]$

Electric potential at a point '$P$' due to a point charge of $5 \times 10^{-9}\text{ C}$ is $50\text{ V}$. The distance of '$P$' from the point charge is:

(Assume, $\frac{1}{4\pi\varepsilon_0} = 9 \times 10^{+9}\text{ N m}^2\text{C}^{-2}$)

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

If a conducting sphere of radius $R$ is charged. Then the electric field at a distance $r\,(r > R)$ from the centre of the sphere would be, $(V =$ potential on the surface of the sphere)

For a uniformly charged thin spherical shell, the electric potential (V) radially away from the centre (O) of shell can be graphically represented as -

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

64 identical drops each charged upto potential of $10\text{ mV}$ are combined to form a bigger drop. The potential of the bigger drop will be __________ $\text{mV}$.

When a charge of $3\text{ C}$ is placed in uniform electric field, it experiences a force of $3000\text{ N}$. Within this field, potential difference between two points separated by a distance of $1\text{ cm}$ is

The charges $2q, -q, -q$ are located at the vertices of an equilateral triangle. At the circumcentre of the triangle

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:

For a charged spherical ball, electrostatic potential inside the ball varies with $r$ as $V = 2ar^2 + b$. Here, $a$ and $b$ are constant and r is the distance from the center. The volume charge density inside the ball is $-\lambda a\varepsilon$. The value of $\lambda$ is ____________.

$\varepsilon$ = permittivity of the medium

A charged spherical conductor of radius '$R$' is connected momentarily to another uncharged spherical conductor of radius '$r$' by means of a thin conducting wire, then the ratio of the surface charge density of the first to the second conductor is

Two isolated metallic solid spheres of radii $R$ and $2R$ are charged such that both have same charge density $\sigma$. The spheres are then connected by a thin conducting wire. If the new charge density of the bigger sphere is $\sigma'$. The ratio $\frac{\sigma'}{\sigma}$ is:

Assume that an electric field $\vec{E} = 30x^{2}\hat{i}$ exists in space. If '$V_0$' is the potential at the origin and '$V_A$' is the potential at $x = 2\text{ m}$, then the potential difference $(V_A - V_0)$ is

Three concentric spherical metallic shells X, Y and Z of radius a, b and c respectively [a < b < c] have surface charge densities $\sigma, -\sigma$ and $\sigma$ respectively. The shells X and Z are at same potential. If the radii of X & Y are 2 cm and 3 cm, respectively. The radius of shell Z is _________ cm.

Two hollow conducting spheres of radii R1 and R2 (R1 >> R2) have equal charges. The potential would be

27 identical drops are charged at 22V 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 = 3x2 volt. The electric field at the point (1, 0, 3) m will be:

The angle between the electric lines of force and the equipotential surface is

Two uniformly charged spherical conductors $A$ and $B$ of radii $5\text{ mm}$ and $10\text{ mm}$ are separated by a distance of $2\text{ cm}$. 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:

Sixty four conducting drops each of radius 0.02 m and each carrying a charge of 5 $\mu$C are combined to form a bigger drop. The ratio of surface charge density of bigger drop to the smaller drop will be:

Two charged spherical conductors of radius R1 and R2 are connected by a wire. Then the ratio of surface charge densities of the spheres ($\sigma$1 / $\sigma$2) is:

Twenty seven drops of same size are charged at 220V each. They combine to form a bigger drop. Calculate the potential of the bigger drop.

512 identical drops of mercury are charged to a potential of 2V each. The drops are joined to form a single drop. The potential of this drop is ________ V.

Let A, B and C be the three points in a uniform electric field ( $\vec{E})$ as shown. The electric potential is

Four electric charges $+q, +q, -q$ and $-q$ are placed in order at the corners of a square of side $2L$. The electric potential at point midway between the two positive charges is