CBSE Class 12 Physics Electrostatics Assignment

CBSE Class 12 Physics Electrostatics Assignment. Please refer to the examination notes which you can use for preparing and revising for exams. These notes will help you to revise the concepts quickly and get good marks.

MISCELLANEOUS ASSIGNMENT

Comprehension-1

Van de Graaff generators like the one shown in figure are used to produce very high voltage. In the figure, the + signs represent positive charge and the -ve signs represent negative charge. In this common Van de Graaff generator, charge is separated by the frictional contact of the belt and the lower pulley shown. Positive charge collects on the lower pulley and an equal amount of negative charge spreads out along the inside of the belt. Electrons from the ground are attracted to the outside of the belt by the net positive charge on the lower portion of the belt-pulley system. These electrons travel up the belt and are transferred to the dome, which is a hollow metal sphere. A high negative charge density can be built up on the dome, because the electrons from the outside of the belt do not experience a repulsive force from the charge built up on the outside of the sphere.
The electric potential of the dome is V = Er where E is the electric field just outside the dome and r is the radius. The charges on the surface of the dome do not affect the electric-field inside the cavity. The potential that can build up on the dome is limited by the dielectric strength of the air, which is about 30,000 V/cm for dry air. Beyond this air molecules are ionized. This enables the air to conduct electricity.
Van de Graaff generators are routinely used in college physics laboratories. When a student gets within a few inches of a Van de Graaff generator, she may draw a spark with an instantaneous current of 10A and remain uninjured. An instantaneous current is the transfer of charge within 1 μs.

[Figure: Van de Graaff generator with an Upper pulley, a Belt, a Lower pulley, the dome on top, and the Ground connection at the bottom. Positive and negative charges are shown on the belt and the pulleys]

1. The 660 volt rails on a subway can kill a person upon contact. A 10,000 volt Van de Graaff generator, however, will only give a mild shock. Which of the following best explains this paradox?
(a) The generator provides more energy per charge, but since it has few charges it transfers a lesser amount of energy
(b) The generator provides more energy, but since there is little energy per charge the current is small
(c) Most of the energy provided by the generator is dissipated in the air because air presents a smaller resistance than the human body
(d) Most of the energy flows directly to the ground without going through the human body since the generator is grounded

Answer: (a) The generator provides more energy per charge, but since it has few charges it transfers a lesser amount of energy

1. The energy given to a body is the energy per charge multiplied by the number of charges transferred.
2. The generator has a high voltage, so its energy per charge is high, but it transfers very few charges in a short time.
3. The subway rail has a low voltage, but it can supply a large current, which is the cause of the danger.
4. Hence, the total energy transferred by the generator is small, and the correct option is (a).

Teacher's Note:
a) Shock depends on the energy transferred and the current, not on the voltage alone.
b) Option (b) is contradictory, because a higher energy per charge does not make the energy per charge small.

2. What is the maximum potential the dome, with a radius of 10 cm, can sustain in dry air?
(a) 3 kV
(b) 5 kV
(c) 300 kV
(d) 500 kV

Answer: (c) 300 kV

1. The maximum electric field in dry air is 30,000 V/cm.
2. The radius r = 10 cm, so V = E r = 30,000 V/cm × 10 cm = 300,000 V.
3. Hence, the maximum potential is 300 kV, and the correct option is (c).

Teacher's Note:
a) The units cancel: V/cm multiplied by cm gives volts.
b) Above this potential, the air molecules ionise and conduct electricity.

3. What is the work required to move a charge q from the top of the belt to the surface of the dome, if the amount of charge on the dome is Q and q is the only charge on the belt?
(a) 0
(b) \( \frac{kQq}{2r} \)
(c) \( \frac{kQq}{r} \)
(d) \( \frac{kq}{r} \)

Answer: (a) 0

1. The top of the belt is inside the hollow dome, where the electric field is zero.
2. So the potential is the same at the top of the belt and at the surface of the dome.
3. The potential difference is zero, so the work done is W = q × 0 = 0, and the correct option is (a).

Teacher's Note:
a) The passage says that the charges on the dome do not affect the field inside the cavity.
b) Work is zero whenever the two points are at the same potential.

4. A spherical conductor with a radius of 10 cm is given a charge of -1.0 C. It is then further charged by a current of 0.5 A for three seconds, discharged and recharged by an instantaneous current of 10 A. At what point does the sphere have the highest potential?
(a) When it has a charge of -1.0 C
(b) Just after being charged by the 0.5 A current
(c) Just after being discharged
(d) After being charged by the 10 A current

Answer: (b) Just after being charged by the 0.5 A current

1. Charge added by the 0.5 A current in 3 s = 0.5 A × 3 s = 1.5 C.
2. The charge just after this is -1.0 C + 1.5 C = +0.5 C, which is the largest positive charge reached in the sequence.
3. The 10 A instantaneous current transfers only 10 A × 1 μs = 0.00001 C, which is negligible, and discharging reduces the charge.
4. Since V = kQ/r, the potential is highest when Q is largest, so the correct option is (b).

Teacher's Note:
a) Potential is proportional to charge for a given sphere, so find the largest charge first.
b) Check the charge after each step, not only the first and last.

Comprehension-2

Each plate of a parallel plate capacitor is a square of side a. The plates are kept at a separation d (d << a), air being the dielectric medium between the plates. The capacitor is connected to a battery. A block of length a, width a and thickness y ≈ d is inserted a distance x into the capacitor as shown. Assuming dielectric constant of material is K Answer the following questions.

[Figure: Parallel plate capacitor with a square plate of side a and separation d. A dielectric block of length a, width a and thickness y, tending to d, is inserted a distance x into the capacitor, with y marked at the left side of the block]

5. Electric energy stored in the capacitor, as a function of x, can be expressed as
(a) \( \frac{\varepsilon_0 a v^2}{2d} \) (a + x(K + 1))
(b) \( \frac{\varepsilon_0 a v^2}{2d} \) (a + x(K - 1))
(c) \( \frac{\varepsilon_0 a v^2}{2d} \) (x + a(K - 1))
(d) \( \frac{\varepsilon_0 a v^2}{2d} \) (x + a(K + 1))

Answer: (b) \( \frac{\varepsilon_0 a v^2}{2d} \) (a + x(K - 1))

1. The air part of the capacitor has width a and length (a - x), and the dielectric part has length x.
2. So C = \( \frac{\varepsilon_0}{d} \) [a(a - x) + Kax] = \( \frac{\varepsilon_0 a}{d} \) (a + x(K - 1)).
3. With the battery connected, the energy is U = \( \frac{1}{2} \) C v^2 = \( \frac{\varepsilon_0 a v^2}{2d} \) (a + x(K - 1)), which is option (b).

Teacher's Note:
a) Add the capacitance of the air part and the dielectric part as two capacitors in parallel.
b) At x = 0 the energy is \( \frac{\varepsilon_0 a^2 v^2}{2d} \), which matches the empty capacitor.

6. Magnitude of force that acts on the block is
(a) \( \frac{\varepsilon_0 a v^2 (K + 1)}{2d} \)
(b) \( \frac{\varepsilon_0 a v^2}{2d (K - 1)} \)
(c) \( \frac{\varepsilon_0 a v^2 (K - 1)}{2d} \)
(d) \( \frac{\varepsilon_0 a v^2}{2d (K + 1)} \)

Answer: (c) \( \frac{\varepsilon_0 a v^2 (K - 1)}{2d} \)

1. The force on the block is F = \( \frac{\partial U}{\partial x} \) at constant voltage.
2. From Q5, U = \( \frac{\varepsilon_0 a v^2}{2d} \) (a + x(K - 1)), so \( \frac{\partial U}{\partial x} \) = \( \frac{\varepsilon_0 a v^2 (K - 1)}{2d} \).
3. Hence, the magnitude of the force is \( \frac{\varepsilon_0 a v^2 (K - 1)}{2d} \), and the correct option is (c).

Teacher's Note:
a) The force pulls the block into the capacitor when K is greater than 1.
b) The force does not depend on x, because the energy is linear in x.

7. Which of the following is correct?
(a) Energy density is maximum when x = 0
(b) Energy density is maximum when x = d
(c) Energy density is minimum when x = 0
(d) Energy density is independent of x

Answer: (d) Energy density is independent of x

1. The battery keeps the voltage v fixed, so the field between the plates is E = v/d at every x.
2. The energy density in the air region is \( \frac{\varepsilon_0 E^2}{2} \), and in the dielectric region it is \( \frac{K \varepsilon_0 E^2}{2} \), and neither depends on x.
3. Hence, the energy density does not change with x, and the correct option is (d).

Teacher's Note:
a) The total energy changes with x, but the energy per unit volume in each region does not.
b) Only the fraction of the volume filled by the dielectric changes with x.

8. Energy density between the capacitor plates can be expressed as
(a) \( \frac{\varepsilon_0 v^2}{2d^2} \)
(b) \( \frac{\varepsilon_0 v^2 x}{2d^3} \)
(c) \( \frac{\varepsilon_0 v^2 x^2}{2d^4} \)
(d) none of these

Answer: (a) \( \frac{\varepsilon_0 v^2}{2d^2} \)

1. In the air region, E = v/d, so the energy density is u = \( \frac{1}{2} \) ε0 E² = \( \frac{\varepsilon_0 v^2}{2d^2} \).
2. This does not contain x, so it matches option (a).
3. Hence, the correct option is (a).

Teacher's Note:
a) Energy density is energy per unit volume, and its unit is J/m³.
b) Options (b) and (c) contain x, which is not present in the energy density of the air region.

Each of the questions given below consists of two statements, an assertion (A) and reason (R). Select the number corresponding to the appropriate alternative as follows
(a) If both A and R are true and R is the correct explanation of A
(b) If both A and R are true but R is not the correct explanation of A
(c) If A is true but R is false
(d) If A is false but R is true

9. A: Coulomb force is a long range force.
R: Coulomb force acts between two charged particles.
(a) If both A and R are true and R is the correct explanation of A
(b) If both A and R are true but R is not the correct explanation of A
(c) If A is true but R is false
(d) If A is false but R is true

Answer: (b) If both A and R are true but R is not the correct explanation of A

1. A is true, because the Coulomb force falls off as 1/r² but never becomes zero at finite distance.
2. R is true, because the Coulomb force does act between two charged particles.
3. R does not explain why the force is long range, so the correct option is (b).

Teacher's Note:
a) The printed key gives (a), but a force acting between charges does not explain its long range, so (b) is correct.
b) Long range comes from the inverse square law, which the reason does not state.

10. A: The Coulomb force is dominating force in the universe.
R: The Coulomb force is weaker than the gravitational force.
(a) If both A and R are true and R is the correct explanation of A
(b) If both A and R are true but R is not the correct explanation of A
(c) If A is true but R is false
(d) If A is false but R is true

Answer: (d) If A is false but R is true

1. A is false, because on large scales the charges of matter are nearly neutral, so gravity dominates the universe.
2. R is false, because the Coulomb force between two charged particles is much stronger than gravity.
3. Neither statement is true, so none of the options is fully correct. The nearest choice is (d), since A is false.

Teacher's Note:
a) The printed reason says the Coulomb force is weaker than gravity, which is wrong, because it is stronger at atomic scales.
b) Gravity dominates the universe at large scales, which makes the assertion false.

11. A: Two conductors separated by a small distance carry the same positive charge and the potential difference between them may or may not be zero
R: Potential of a conductor depends on the charge and capacitance.
(a) If both A and R are true and R is the correct explanation of A
(b) If both A and R are true but R is not the correct explanation of A
(c) If A is true but R is false
(d) If A is false but R is true

Answer: (a) If both A and R are true and R is the correct explanation of A

1. For a conductor, V = Q/C, so with the same charge the potential depends on the capacitance.
2. If the two conductors have equal capacitances, the potential difference is zero; otherwise it is not zero.
3. Hence A is true, R is true, and R explains A, so the correct option is (a).

Teacher's Note:
a) Equal charges do not mean equal potentials when the capacitances are different.
b) The formula V = Q/C is the link between the reason and the assertion.

12. A: Guass's law is valid for stationary as well as moving charges.
R: Guass's law holds good for any closed surface of any shape.
(a) If both A and R are true and R is the correct explanation of A
(b) If both A and R are true but R is not the correct explanation of A
(c) If A is true but R is false
(d) If A is false but R is true

Answer: (b) If both A and R are true but R is not the correct explanation of A

1. A is true, because Gauss's law for the electric field holds for any charge distribution, including moving charges.
2. R is true, because Gauss's law applies to any closed surface, whatever its shape.
3. R does not explain why the law is valid for moving charges, so the correct option is (b).

Teacher's Note:
a) The shape of the surface is a separate point from the type of charge.
b) Gauss's law is a general law, so its validity does not depend on whether the charges are moving.

13. A: We can have non-zero electric potential in the space, where electric field strength is zero.
R: Both electric potential and electric field strength depend on source charge and not on test charge.
(a) If both A and R are true and R is the correct explanation of A
(b) If both A and R are true but R is not the correct explanation of A
(c) If A is true but R is false
(d) If A is false but R is true

Answer: (b) If both A and R are true but R is not the correct explanation of A

1. A is true, for example inside a charged conducting shell the field is zero but the potential is non-zero.
2. R is true, because both the field and the potential are set by the source charges and do not depend on a test charge.
3. R does not explain why the field can be zero where the potential is not, so the correct option is (b).

Teacher's Note:
a) The field is the gradient of the potential, so zero field does not force zero potential.
b) Use the shell example to test the assertion.

14. A: Vehicles carrying inflammable materials usually have metallic chains attached to them, which remain in contact with ground during motion.
R: The charge on the vehicles escapes to ground.
(a) If both A and R are true and R is the correct explanation of A
(b) If both A and R are true but R is not the correct explanation of A
(c) If A is true but R is false
(d) If A is false but R is true

Answer: (a) If both A and R are true and R is the correct explanation of A

1. A is true, because the metallic chain touches the ground and is used as a safety measure.
2. R is true, because the chain provides a path for the charge built up on the vehicle to escape to the ground.
3. R explains why the chains are used, so the correct option is (a).

Teacher's Note:
a) Charge build-up by friction can cause sparks, which may ignite inflammable materials.
b) The chain must stay in contact with the ground while the vehicle moves.

15. A: A solid conducting sphere and a hollow conducting sphere of same radius are charged to same potential, then they will carry the same charge.
R: The charge on a sphere having given potential is independent of the radius.
(a) If both A and R are true and R is the correct explanation of A
(b) If both A and R are true but R is not the correct explanation of A
(c) If A is true but R is false
(d) If A is false but R is true

Answer: (c) If A is true but R is false

1. A is true, because both spheres have the same radius and the same potential, so they have the same capacitance and the same charge Q = CV.
2. R is false, because for a sphere Q = 4πε0 r V, so for a given potential the charge depends on the radius.
3. Hence, the correct option is (c).

Teacher's Note:
a) A is true only because the radii are the same; the reason does not explain it.
b) If the radii were different, the charges would also be different at the same potential.

16. A: When electric dipole is perpendicular to the electric field, maximum torque acts on it.
R: If \( \tau \) = 90°, then
\( \tau \) = PE sin 90°
or
\( \tau \) = PE (max).
(a) If both A and R are true and R is the correct explanation of A
(b) If both A and R are true but R is not the correct explanation of A
(c) If A is true but R is false
(d) If A is false but R is true

Answer: (a) If both A and R are true and R is the correct explanation of A

1. The torque on a dipole in a uniform field is τ = PE sin θ, where θ is the angle between the dipole and the field.
2. The torque is maximum when sin θ = 1, that is, when θ = 90°. So A is true.
3. R shows that at 90°, τ = PE sin 90° = PE (max), which is the reason for A. Hence, the correct option is (a).

Teacher's Note:
a) The angle in the torque formula is between the dipole moment and the electric field.
b) When the dipole is parallel to the field, the torque is zero.