Electrostatics · Physics · Class 12

Figure shows variation of Coulomb force $(F)$ acting between two point charges with $\frac{1}{r^2}, r$ being the separation between the two charges $\left(q_1, q_2\right)$ and $\left(q_2\right. q_3$ ). If $q_2$ is positive and least in magnitude, then the magnitudes of $q_1, q_2$ and $q_3$ are such that

Assertion (A): Equal amount of positive and negative charges are distributed uniformly on two halves of a thin circular ring as shown in figure. The resultant electric field at the centre O of the ring is along OC.

Reason (R): It is so because the net potential at $O$ is not zero.

The magnitude of the electric field due to a point charge object at a distance of $$4.0 \mathrm{~m}$$ is $$9 \frac{\mathrm{N}}{\mathrm{C}}$$. From the same charged object the electric field of magnitude, $$16 \frac{\mathrm{N}}{\mathrm{C}}$$ will be at a distance of

A point $$P$$ lies at a distance $$x$$ from the mid point of an electric dipole on its axis. The electric potential at point $$\mathrm{P}$$ is proportional to

Beams of electrons and protons move parallel to each other in the same direction. They

Assertion (A) : Work done in moving a charge around a closed path, in an electric field is always zero.

Reason (R) : Electrostatic force is a conservative force.

An electric dipole of length $$2 \mathrm{~cm}$$ is placed at an angle of $$30^{\circ}$$ with an electric field $$2 \times 10^5 \mathrm{~N} / \mathrm{C}$$. If the dipole experiences a torque of $$8 \times 10^{-3} \mathrm{~Nm}$$, the magnitude of either charge of the dipole, is

(a) (i) Two point charges $5 \mu \mathrm{C}$ and $-1 \mu \mathrm{C}$ are placed at points ( $-3 \mathrm{~cm}, 0,0$ ) and ( $3 \mathrm{~cm}, 0,0$ ) respectively. An external electric field $\vec{E}=\frac{A}{r^2} \hat{r}$

where $A=3 \times 10^5 \mathrm{Vm}$ is switched on in the region.

Calculate the change in electrostatic energy of the system due to the electric field.

(ii) A system of two conductors is placed in air and they have net charge of $+80 \mu \mathrm{C}$ and $-80 \mu \mathrm{C}$ which causes a potential difference of 16 V between them.

(1) Find the capacitance of the system.

(2) If the air between the capacitor is replaced by a dielectric medium of dielectric constant 3, what will be the potential difference between the two conductors?

(3) If the charges on two conductors are changed to $+160 \mu \mathrm{C}$ and $-160 \mu \mathrm{C}$, will the capacitance of the system change?

Give reason for your answer.

OR

(b) (i) Consider three metal spherical shells $A, B$ and $C$, each of radius $R$. Each shell is having a concentric metal ball of radius $R / 10$. The spherical shells $\mathrm{A}, B$ and $C$ are given charges $+6 q,-4 q$ and $14 q$ respectively. Their inner metal balls are also given charges $-2 q$, $+8 q$ and $-10 q$ respectively. Compare the magnitude of the electric fields due to shells $A, B$ and $C$ at a distance $3 R$ from their centres.

(ii) A charge $-6 \mu \mathrm{C}$ is placed at the centre $B$ of a semicircle of radius 5 cm , as shown in the figure. An equal and opposite charge is placed at point D at a distance of 10 cm from B. A charge $+5 \mu \mathrm{C}$ is moved from point ' $C$ ' to point ' $A$ ' along the circumference. Calculate the work done on the charge.

(a) Four point charges of $1 \mu \mathrm{C},-2 \mu \mathrm{C}, 1 \mu \mathrm{C}$ and $-2 \mu \mathrm{C}$ are placed at the corners $\mathrm{A}, \mathrm{B}, \mathrm{C}$ and D respectively, of a square of side 30 cm . Find the net force acting on a charge of $4 \mu \mathrm{C}$ placed at the centre of the square.

OR

(b) Three point charges, 1 pC each, are kept at the vertices of an equilateral triangle of side 10 cm . Find the net electric field at the centroid of triangle.

(a) Define the term 'electric flux' and write its dimensions.

(b) A plane surface, in shape of a square of side 1 cm is placed in an electric field $\vec{E}=\left(100 \frac{\mathrm{~N}}{\mathrm{C}}\right) \hat{i}$ such that the unit vector normal to the surface is given by $\hat{n}=0.8 \hat{i}+0.6 \hat{k}$. Find the electric flux through the surface.

(a) (i) Draw equipotential surfaces for an electric dipole.

(ii) Two point charges $q_1$ and $q_2$ are located at $\overrightarrow{r_1}$ and $\vec{r}_2$ respectively in an external electric field $\vec{E}$. Obtain an expression for the potential energy of the system.

(iii) The dipole moment of a molecule is $10^{-30} \mathrm{Cm}$. It is placed in an electric field $\vec{E}$ of $10^5 \mathrm{~V} / \mathrm{m}$ such that its axis is along the electric field. The direction of $\vec{E}$ is suddenly changed by $60^{\circ}$ at an instant. Find the change in the potential energy of the dipole, at that instant.

OR

(b) (i) A thin spherical shell of radius R has a uniform surface charge density $\sigma$. Using Gauss' law, deduce an expression for electric field (i) outside and (ii) inside the shell.

(ii) Two long straight thin wires AB and CD have linear charge densities $10 \mu \mathrm{C} / \mathrm{m}$ and $-20 \mu \mathrm{C} / \mathrm{m}$, respectively. They are kept parallel to each other at a distance 1 m . Find magnitude and direction of the net electric field at a point midway between them.

(a) (i) Use Gauss' law to obtain an expression for the electric field due to an infinitely long thin straight wire with uniform linear charge density $$\lambda$$.

(ii) An infinitely long positively charged straight wire has a linear charge density $$\lambda$$. An electron is revolving in a circle with a constant speed $$v$$ such that the wire passes through the centre, and is perpendicular to the plane, of the circle. Find the kinetic energy of the electron in terms of magnitudes of its charge and linear charge density $$\lambda$$ on the wire.

(iii) Draw a graph of kinetic energy as a function of linear charge density $$\lambda$$.

OR

(b) (i) Consider two identical point charges located at points $$(0,0)$$ and $$(a, 0)$$.

(1) Is there a point on the line joining them at which the electric field is zero?

(2) Is there a point on the line joining them at which the electric potential is zero?

Justify your answers for each case.

(ii) State the significance of negative value of electrostatic potential energy of a system of charges.

Three charges are placed at the corners of an equilateral triangle $$A B C$$ of side $$2.0 \mathrm{~m}$$ as shown in figure. Calculate the electric potential energy of the system of three charges.

Depict the orientation of an electric dipole in (a) stable and (b) unstable equilibrium in an external uniform electric field. Write the potential energy of the dipole in each case.