Q.1. For two charges $\displaystyle {q_1}$ and $\displaystyle {q_2}$, if force between them for some separation in air is $\displaystyle {F}$, then force between them in a medium of permittivity $\displaystyle {\varepsilon}$ will be:

Q.2. Two point charges placed at a certain distance $\displaystyle {r}$ in air exert a force $\displaystyle {F}$ on each other. Then, distance $\displaystyle {r'}$ at which these charges will exert the same force in a medium of dielectric constant $\displaystyle {K}$ is given by:

Q.3. Charge $\displaystyle {q_2}$ of mass $\displaystyle {m}$ revolves around a stationary charge $\displaystyle {q_1}$ in a circular orbit of radius $\displaystyle {r}$. The orbital periodic time of $\displaystyle {q_2}$ would be:

Q.4. Three identical positive point charges are placed at the vertices of an isosceles right angled triangle as shown. Which of the vectors numbered coincides in direction with the electric field at the mid-point $\displaystyle {M}$ of the hypotenuse due to the system of charges?

Q.5. A charged cork ball of mass $\displaystyle {2\text{ kg}}$ is suspended by light insulated string in electric field $\displaystyle {\vec{E} = \hat{i} + \hat{j}}$ making angle $\displaystyle {45^\circ}$ with vertical. Find charge on the ball if it is in equilibrium:

[Suspended Cork Ball in Electric Field E = i + j ] Diagram showing string at $45^\circ$ angle in electric field $\vec{E}$.

Q.6. Two small spheres each having charge $\displaystyle {+Q}$ are suspended by insulating threads of length $\displaystyle {L}$ from a hook in space with no gravitational effect. The angle between suspensions and tension in each thread will be:

Q.7. The bob of a simple pendulum has mass $\displaystyle {2\text{ g}}$ and charge $\displaystyle {5.0\,\mu\text{C}}$. It is at rest in a uniform horizontal electric field of intensity $\displaystyle {2000\text{ V/m}}$. At equilibrium, the angle that the pendulum makes with vertical is ($\displaystyle {g = 10\text{ m/s}^2}$):

Directions for Q.13 to Q.16: Choose correct option:
(a) Both Assertion and Reason are correct and Reason is correct explanation of Assertion.
(b) Both Assertion and Reason are correct but Reason is not correct explanation.
(c) Assertion is correct but Reason is incorrect.
(d) Assertion is incorrect but Reason is correct.

Q.13. Assertion: When we produce charge $\displaystyle {q_1}$ on a body by rubbing against another body which gets charge $\displaystyle {q_2}$, then $\displaystyle {q_1 + q_2 = 0}$.
Reason: Charge on an isolated system remains constant.

Q.14. Assertion: At macroscopic level, quantisation of charge has no practical consequence and can be ignored.
Reason: $\displaystyle {1\,\mu\text{C}}$ charge contains $\displaystyle {10^{13}}$ times electronic charge $\displaystyle {e}$ approximately.

Q.15. Assertion: Charge on a body can be increased or decreased in terms of electronic charge $\displaystyle {e}$.
Reason: Charge on a body is integral multiple of electronic charge $\displaystyle {e}$, this is called quantisation of charge.

Q.16. Assertion: When bodies are charged through friction, there is transfer of charge from one body to another, but no creation or destruction of charge.
Reason: This follows from conservation of electric charges.

Q.17. Write Coulomb’s law in vector form. What is the importance of expressing it in vector form?

Write any two limitations of Coulomb’s law.

Q.18. (i) Depict electric field lines due to two positive charges kept a distance apart.
(ii) Depict electric field lines due to an electric dipole (two opposite charges).

Q.19. Consider two hollow concentric spheres, $\displaystyle {S_1}$ & $\displaystyle {S_2}$, enclosing charges $\displaystyle {2Q}$ & $\displaystyle {4Q}$ respectively.

[Concentric Spheres S1 and S2 Enclosing Charges ] Diagram showing inner sphere $S_1$ with charge $2Q$ and outer sphere $S_2$ with charge $4Q$.

(i) Find out the ratio of the electric flux through them.
(ii) How will electric flux through $\displaystyle {S_1}$ change if medium of dielectric constant $\displaystyle {\varepsilon_r}$ is introduced inside $\displaystyle {S_1}$?

Q.20. Two point charges $\displaystyle {+4\,\mu\text{C}}$ and $\displaystyle {+1\,\mu\text{C}}$ are separated by $\displaystyle {2\text{ m}}$ in air. Find point on line-joining charges where net electric field is zero.

An electric dipole is held in a uniform electric field. Write expression for torque acting on it in vector form. Specify direction and identify two pairs of perpendicular vectors.

Q.21. Given uniform electric field $\displaystyle {\vec{E} = 6 \times 10^3\hat{i}\text{ N/C}}$. Find flux through a square of $\displaystyle {10\text{ cm}}$ side whose plane is parallel to Y-Z plane. What is flux if plane makes $\displaystyle {30^\circ}$ angle with x-axis?

Q.22. (i) An electric dipole is held in a uniform electric field. Show with diagram that it undergoes no translatory motion. Derive expression for torque.
(ii) What happens if field is non-uniform?
(iii) What happens if external field $\displaystyle {\vec{E}}$ is increasing (a) parallel to $\displaystyle {\vec{p}}$, (b) anti-parallel to $\displaystyle {\vec{p}}$?

Q.23. (a) Derive expression for electric field $\displaystyle {E}$ due to dipole of length $\displaystyle {2a}$ at distance $\displaystyle {r}$ on axial line.
(b) Draw graph of $\displaystyle {E}$ versus $\displaystyle {r}$ for $\displaystyle {r \gg a}$.

Q.24. Obtain expression for electric field intensity at a point on equatorial line of electric dipole of moment $\displaystyle {\vec{p}}$ and length $\displaystyle {2a}$. Direction of field?

Q.25. Using Gauss’s law, derive expression for electric field intensity due to infinitely long straight wire of linear charge density $\displaystyle {\lambda\text{ C/m}}$.

Q.26. Using Gauss’s law, obtain expression for electric field intensity due to infinitely large plane sheet of charge density $\displaystyle {\sigma\text{ C/m}^2}$. Field direction for (i) positive sheet, (ii) negative sheet?

Q.27. Using Gauss’s law, deduce expression for electric field due to uniformly charged spherical shell of radius $\displaystyle {R}$ at point (i) outside ($\displaystyle {r > R}$), (ii) inside ($\displaystyle {r < R}$). Plot graph of $\displaystyle {E}$ vs $\displaystyle {r}$.

Q.28. Given electric field components $\displaystyle {E_x = \alpha x, E_y = 0, E_z = 0}$. Calculate flux through cube of side $\displaystyle {a}$ and effective charge inside.

[Cube in Non-Uniform Electric Field Ex = alpha * x ] Diagram showing cube of side $a$ placed in coordinate system with electric field along x-axis.

An electric dipole at $\displaystyle {30^\circ}$ with field $\displaystyle {2 \times 10^5\text{ N/C}}$ experiences torque $\displaystyle {4\text{ N}\cdot\text{m}}$. Calculate charge magnitude if dipole length is $\displaystyle {2\text{ cm}}$.

Q.29. Case Study 1: Read passage on charging by friction and charge conservation, then answer:

(i) Cause of charging is: (a) transfer of protons   (b) transfer of electrons   (c) transfer of neutrons
(ii) When glass rod is rubbed with silk: (a) negative on silk only   (b) equal opposite charges on both   (c) positive on glass only
(iii) Positively charged object mass: (a) remains same   (b) increases   (c) decreases slightly by factor of electron mass
(iv) Cause of quantisation: (a) transfer of integral number of electrons   (b) transfer of protons

Q.30. Case Study 2: Read passage on Coulomb's law $\displaystyle {F = {\frac{{1}}{{4{\pi}{{\varepsilon}_{0}}}}}{\frac{{{q}_{1}}{{q}_{2}}}{{{r}^{2}}}}}$, then answer:

(i) Proportionality constant $\displaystyle {k}$ depends on: (a) force   (b) nature of medium   (c) magnitude of charges
(ii) Dimensional formula for $\displaystyle {{\varepsilon}_{0}}$: (a) $\displaystyle {[M^{-1}L^{-3}T^4 A]}$   (b) $\displaystyle {[M^{-1}L^{-3}T^4 A^2]}$
(iii) Repulsion force between two $\displaystyle {1\text{ C}}$ charges $\displaystyle {1\text{ m}}$ apart in vacuum: (a) $\displaystyle {9 \times 10^9\text{ N}}$   (b) $\displaystyle {9 \times 10^{11}\text{ N}}$
(iv) Two identical charges repel with $\displaystyle {10\text{ mg wt}}$ at $\displaystyle {0.6\text{ m}}$ apart. Charge value: (a) $\displaystyle {2\,\mu\text{C}}$   (b) $\displaystyle {2\text{ nC}}$

1. When a body is connected to the earth, then electrons from the earth flow into the body. It means that the body is:

2. Three charges, each equal to $\displaystyle {+q}$, are placed at the corners of an equilateral triangle. If the force between any two charges is $\displaystyle {F}$, then net force on either charge will be:

3. A metallic spherical shell has an inner radius $\displaystyle {R_1}$ and outer radius $\displaystyle {R_2}$. A charge $\displaystyle {q}$ is placed at the centre of the spherical cavity. The surface charge density on the inner surface is:

4. If a charge $\displaystyle {q}$ is placed at the centre of the line joining two equal charges $\displaystyle {Q}$ such that the system is in equilibrium, then the value of $\displaystyle {q}$ is:

5. A cylinder of radius $\displaystyle {R}$ and length $\displaystyle {L}$ is placed in a uniform electric field $\displaystyle {E}$ parallel to the axis of the cylinder. The total flux over the curved surface of the cylinder is:

6. At the centre of a cubical box $\displaystyle {+Q}$ charge is placed. The value of total flux coming out of a single wall is:

7. Two point charges A and B, having charges $\displaystyle {+q}$ and $\displaystyle {-q}$ respectively, are placed at a certain distance apart and force acting between them is $\displaystyle {F}$. If $25\%$ charge of A is transferred to B, then force between the charges becomes:

8. An electron falls from rest through a vertical distance $\displaystyle {h}$ in a uniform and vertically upward directed electric field $\displaystyle {E}$. The direction of electric field is now reversed, keeping its magnitude the same. A proton is allowed to fall from rest in it through the same vertical distance $\displaystyle {h}$. The time of fall of the electron, in comparison to the time of fall of the proton is:

9. Suppose a closed square loop whose area vector is $\displaystyle {5\hat{i} - 6\hat{j}}$ is placed in an electric field of $\displaystyle {2\hat{i} - 4\hat{j}}$, then what will be electric flux?

10. A dipole with charges $\displaystyle {+4\,\mu\text{C}}$ and $\displaystyle {-4\,\mu\text{C}}$ separated by $\displaystyle {2\text{ mm}}$ is in a field $\displaystyle {7 \times 10^4\text{ N/C}}$ at $\displaystyle {30^\circ}$. What is the torque?

11. What explains why the electric field due to a dipole can be zero at certain points along its equatorial plane?

12. Consider three point objects P, Q and R. P and Q repel each other, while P and R attract. What is the nature of force between Q and R?

13. The net electric field inside a hollow charged conducting sphere is:

14. Two positive point charges of magnitude $\displaystyle {+Q}$ are placed on the x-axis at distances $\displaystyle {a}$ and $\displaystyle {3a}$ from the origin, respectively. What is the electric field at the point $\displaystyle {(2a, 0)}$?

15. Two equal negative charges $\displaystyle {-q}$ are fixed at points $\displaystyle {(0, a)}$ and $\displaystyle {(0, -a)}$ on the Y-axis. A positive charge $\displaystyle {Q}$ is released from rest at a point on the X-axis. The charge $\displaystyle {Q}$ will:

16. Two point charges $\displaystyle {+8q}$ and $\displaystyle {-2q}$ are located at $\displaystyle {x = 0}$ and $\displaystyle {x = L}$ respectively. The location of a point on the x-axis at which the net electric field due to these two point charges is zero is:

17. Two charged spheres separated at a distance exert a force $\displaystyle {F}$ on each other. If they are immersed in a liquid of dielectric constant 2, then what is the net force on each sphere?

18. What is the SI unit of permittivity of free space?

19. How many electrons must be removed from a neutral metal plate to give it a charge of $\displaystyle {+1.0\text{ C}}$?

20. Two identical charged spheres exert an initial electrostatic force $\displaystyle {F}$ on each other. They are touched together momentarily and then placed back to their original distance apart. The new force is $\displaystyle {F'}$. Which of the following is true?

Q.31. (i) $\displaystyle {ABC}$ is equilateral triangle of side $\displaystyle {l}$. Charges $\displaystyle {+2\,\mu\text{C}}$ at $\displaystyle {B}$ and $\displaystyle {C}$. Find magnitude and sign of charge $\displaystyle {q}$ at midpoint $\displaystyle {M}$ of $\displaystyle {BC}$ so net field at $\displaystyle {A}$ is zero.
(ii) Electric dipole of charges $\displaystyle {-1.0\,\mu\text{C}}$ and $\displaystyle {+1.0\,\mu\text{C}}$ at $\displaystyle {(0,0)}$ and $\displaystyle {(3\text{ mm}, 4\text{ mm})}$ in field $\displaystyle {\vec{E} = 1000\hat{i}\text{ V/m}}$. Find torque.

(i) Write Coulomb’s law in vector form for system of point charges.
(ii) Two long straight parallel wires 1 and 2 with charge densities $\displaystyle {\lambda_1 = 10\text{ C/m}}$ and $\displaystyle {\lambda_2 = 20\text{ C/m}}$ separated by $\displaystyle {30\text{ cm}}$. Find net force on electron held at point $\displaystyle {P}$ distance $\displaystyle {10\text{ cm}}$ from wire 1.

Q.32. (i) Define electric flux, SI unit.
(ii) Explain how Gauss's law is based on inverse-square dependence in Coulomb's law.
(iii) Charges $\displaystyle {A(q)}$ and $\displaystyle {B(2q)}$ at $\displaystyle {(0,0)}$ and $\displaystyle {(a,a)}$. Find force exerted by $\displaystyle {A}$ on $\displaystyle {B}$ in unit vectors $\displaystyle {\hat{i}, \hat{j}}$.

(i) Three parallel infinite line charges $\displaystyle {+\lambda, +2\lambda, -\lambda}$ placed in plane. Find electric field at point $\displaystyle {P}$.
(ii) Charge $\displaystyle {q}$ enclosed by sphere. Flux change if (a) radius doubled, (b) shape changed to cube?

Q.33. (i) Two identical electric dipoles along diagonals of square $\displaystyle {ABCD}$ of side $\displaystyle {\sqrt{2}\text{ m}}$. Find net field magnitude and direction at center.
(ii) Three metal spherical shells $\displaystyle {A, B, C}$ radius $\displaystyle {R}$ with inner balls $\displaystyle {R/10}$. Shell charges $\displaystyle {+6q, -4q, 14q}$ and inner ball charges $\displaystyle {-2q, +8q, -10q}$. Compare electric fields at distance $\displaystyle {3R}$.

1. Assertion: In a uniform electric field, an electron moves in direction opposite to electric field.
Reason: This is because of negative charge on electron.

2. Assertion: If dipole of moment $\displaystyle {30 \times 10^{-5}\text{ C}\cdot\text{m}}$ is enclosed by closed surface, net flux is zero.
Reason: Dipole consists of two equal and opposite charges.

3. Assertion: Metallic shield in form of hollow shell blocks external electric field.
Reason: Electric field inside hollow conductor is zero at every point.

4. Assertion: Flux through a cube enclosing charge $\displaystyle {q}$ is independent of cube side length.
Reason: Gauss’s law flux is independent of size and shape of Gaussian surface.

5. Assertion: When bodies are charged through friction, charge is transferred but no creation or destruction occurs.
Reason: Follows from conservation of electric charges.

Case Study 1: Parallel Charged Metal Plates

Two large thin metal plates A and B are placed parallel and close to each other with opposite surface charge densities of magnitude $\displaystyle {\sigma = 17.0 \times 10^{-22}\text{ C/m}^2}$.

Q1. Electric field $\displaystyle {E}$ in outer region of plate A: (a) $\displaystyle {\sigma/\varepsilon_0}$   (b) Zero   (c) $\displaystyle {2\sigma/\varepsilon_0}$
Q2. Electric field $\displaystyle {E}$ in outer region of plate B: (a) $\displaystyle {\sigma/\varepsilon_0}$   (b) Zero   (c) $\displaystyle {2\sigma/\varepsilon_0}$
Q3. Electric field $\displaystyle {E}$ between the plates: (a) Zero   (b) $\displaystyle {1.9 \times 10^{-10}\text{ N/C}}$   (c) $\displaystyle {3.8 \times 10^{-10}\text{ N/C}}$
Q4. Ratio of field at distances $2\text{ cm}$ and $4\text{ cm}$ from plate A: (a) $\displaystyle {1:2}$   (b) $\displaystyle {1:1}$   (c) $\displaystyle {2:1}$

Case Study 2: Electric Dipole in Uniform Field

An electric dipole consists of charges $\displaystyle {\pm 1.0\,\mu\text{C}}$ separated by $\displaystyle {2.0\text{ cm}}$ in field $\displaystyle {10^5\text{ N/C}}$.

Q1. Torque expression in vector form: (a) $\displaystyle {\vec{\tau} = \vec{p} \times \vec{E}}$   (b) $\displaystyle {\vec{\tau} = \vec{p} \cdot \vec{E}}$
Q2. Maximum torque on dipole: (a) $\displaystyle {2 \times 10^{-3}\text{ N}\cdot\text{m}}$   (b) $\displaystyle {1 \times 10^{-3}\text{ N}\cdot\text{m}}$
Q3. Torque is minimum when angle $\displaystyle {\theta}$ is: (a) $\displaystyle {90^\circ}$   (b) $\displaystyle {0^\circ\text{ or }180^\circ}$
Q4. Net force on dipole in uniform field: (a) Zero   (b) $\displaystyle {qE}$

1. (a) Name two basic properties of electric charge.
(b) What does $\displaystyle {q_1 + q_2 = 0}$ signify in electrostatics?

2. Define dielectric constant of a medium in terms of force between charges. What is its S.I. unit?

3. Two identical balls with charge $\displaystyle {q}$ are suspended by insulating strings. What is the effect on repulsive force when a plastic sheet is inserted between them?

4. Define electric field intensity. Write its S.I. unit. Is it scalar or vector?

5. (i) What is the physical significance of electric field?
(ii) Write expression for force acting on test charge $\displaystyle {q_0}$ in field $\displaystyle {\vec{E}}$.

6. State any two properties of electric field lines.

7. What is the importance of electric field lines?

8. (i) Trace field lines due to charge $\displaystyle {+Q}$ near a conducting surface.
(ii) Draw electric field lines due to thin spherical shell when charge is (a) positive, (b) negative.

9. Why do electric field lines never cross each other?

10. Why are electric field lines always normal to the surface of a conductor?

1. (a) Derive an expression for electric field $\displaystyle {E}$ due to dipole of length $\displaystyle {2a}$ at distance $\displaystyle {r}$ from centre on axial line.
(b) Draw graph of $\displaystyle {E}$ versus $\displaystyle {r}$ for $\displaystyle {r \gg a}$.

2. Derive expression for electric field intensity at a point on equatorial line of electric dipole of moment $\displaystyle {\vec{p}}$ and length $\displaystyle {2a}$. Direction of field?

3. Using Gauss’s law, deduce expression for electric field due to uniformly charged spherical conducting shell of radius $\displaystyle {R}$ at point (i) outside ($\displaystyle {r > R}$), (ii) inside ($\displaystyle {r < R}$). Plot graph of $\displaystyle {E}$ vs $\displaystyle {r}$.