Electrostatics

Step 1: Property of capacitors in series.
When capacitors are connected in series, the charge on each capacitor is the same.

Step 2: Relation between potential and capacitance.


Step 3: Ratio of potentials.

Step 1: Understanding conductors.
In conductors, the valence band and conduction band overlap, meaning that electrons in the valence band can easily move into the conduction band when an external electric field is applied, allowing electrical conduction.

Step 2: Analyzing the options.
- (A) Valence band and conduction band overlap each other. This is correct for conductors. - (B) In conductors, the valence and conduction bands do not have a large energy gap, as this would prevent easy conduction of electrons. - (C) In conductors, a large number of electrons are available for electrical conduction, not a very small number. - (D) In conductors, the gap between the valence and conduction bands is either very small or nonexistent.

Step 3: Conclusion.
The correct answer is (A), as conductors have overlapping valence and conduction bands.

Step 1: Assume capacitance of each capacitor.
Let each capacitor have capacitance .

Step 2: Equivalent capacitance in series.
For four capacitors in series,

Step 3: Equivalent capacitance in parallel.
For four capacitors in parallel,

Step 4: Find the ratio.

Step 1: Band structure of solids.
In solids, electrical conductivity depends on the distribution of electrons in the valence band and conduction band.
Step 2: Behaviour of insulators.
In insulators, the valence band is completely filled and the conduction band is completely empty due to a large energy gap between them.
Step 3: Elimination of incorrect options.
Options (A) and (B) correspond to conductors, while option (D) is incorrect because insulators have a large forbidden energy gap.
Step 4: Conclusion.
Hence, in insulators, the conduction band is empty.

Step 1: Write the formula for fundamental frequency of a stretched string.
The fundamental frequency of a stretched string is given by:
where is the tension and is the linear density.

Step 2: Relation between tension and frequency.
Keeping the length and linear density constant,

Step 3: Condition for octave.
An octave means the frequency is doubled:

Step 4: Find new tension.

Step 5: Relation between tension and weight.
Since tension is proportional to the weight suspended,

Step 6: Conclusion.
The weight required to produce an octave is .

Step 1: Formula for Surface Charge.
The outward force per unit area on a charged conductor is given by: where is the surface charge density and is the permittivity of free space. Substituting the given values: Step 2: Final Answer.
Thus, the outward force per unit area is .

Step 1: Understanding the relationship.
The force on each plate of a parallel plate capacitor is given by the formula: where is the area of the plates and is the permittivity of free space. The electric field between the plates is . By substituting this expression for into the force equation, we get: Step 2: Conclusion.
Thus, the force on each plate is , corresponding to option (B).

Step 1: Gauss’s law.
According to Gauss’s law, electric flux through a closed surface is
Step 2: Dependence on radius.
The flux depends only on the charge enclosed by the surface and not on the size or shape of the Gaussian surface.
Step 3: Conclusion.
Since the enclosed charge remains the same, the electric flux remains unchanged even if the radius increases.

Step 1: Write the formula for electric flux.
Step 2: Convert area into SI units.
Step 3: Substitute given values.
Step 4: Calculate electric flux.

Step 1: Understand the force on the charged particle.
When the charge enters the electric field between the parallel plates, the force acting on it is given by: where is the electric field between the plates. The electric field is related to the potential difference and the distance between the plates by: Step 2: Apply Newton's second law.
The acceleration of the charged particle is given by Newton's second law: Step 3: Conclusion.
Thus, the acceleration of the charged particle is , which corresponds to option (C).

Step 1: Terminal velocity condition.
At terminal velocity, the net force acting on the sphere is zero. The forces involved are the buoyant force, the gravitational force, and the viscous drag force. The viscous force is given by: where is the density of the sphere, is the density of the liquid, and is the gravitational acceleration.
Step 2: Conclusion.
Thus, the correct answer is (A) .

Step 1: Surface charge density of the sphere.
Total charge on the sphere is . Surface charge density:


Step 2: Electric field at distance from centre.
For , electric field due to a charged sphere is:


Step 3: Substitute .


Step 4: Solve for .


Step 5: Conclusion.
The distance from the centre is .

Step 1: Capacitance in series.
For two identical capacitors each of capacitance :


Step 2: Capacitance in parallel.


Step 3: Ratio of capacitances.


Step 4: Conclusion.
The required ratio is .

Step 1: Understanding the telescope setup.
For a telescope, the focal length of the objective lens and eye lens are related to the length of the telescope and magnifying power. The length of the telescope is the difference between the focal lengths of the objective and eye lenses: The magnifying power of the telescope is given by:
Step 2: Given values.
We are given that and . Using the magnifying power formula: This gives the relationship between the focal lengths of the objective and eye lenses.
Step 3: Solve for the correct combination.
By trial and error, the correct combination of objective and eye lenses that satisfies the equation is (objective) and (eye lens). Thus, the correct answer is (D).

Step 1: Understanding the combination of capacitors.
When capacitors are connected in parallel, the total capacity is the sum of the individual capacitances. When the combination is then connected in series with a third capacitor, the total capacitance is found by: where , , and .
Step 2: Finding the charge on the first capacitor.
The charge on the capacitor is given by . After finding , the charge on the capacitor of capacity is .
Step 3: Conclusion.
The correct answer is (B), .

Step 1: Understanding the formula for energy stored in a capacitor.
The energy stored in a parallel plate capacitor is given by: Where: - is the electric field, - is the area of each plate, - is the distance between the plates, - is the permittivity of free space. Thus, the correct answer is (B) .

Step 1: Understanding the formula for capacitance.
The capacitance of a parallel plate capacitor is given by the formula: where is the permittivity of the dielectric material between the plates, is the area of the plates, and is the distance between the plates.
Step 2: Effect of distance between plates.
To increase the capacitance, we need to reduce the distance between the plates. As decreases, the capacitance increases.
Step 3: Conclusion.
Thus, the capacitance can be increased by decreasing the distance between the plates, which corresponds to option (D).

Step 1: Understanding the question.
We are given that the energy stored in both the parallel and series combinations of capacitors is the same. The energy stored in a capacitor is given by . Since the energy is the same, we can equate the energies for both combinations.

Step 2: Analyzing the parallel and series combinations.
- In the parallel combination, the total capacitance is . - In the series combination, the total capacitance is .
Step 3: Deriving the ratio.
The potential difference across each combination must satisfy the energy condition. By equating the energies, we find that the ratio of potential differences . This gives the correct answer as .

Step 1: Resolve the forces.
The two forces act at an angle . The resultant force is given by: Substitute the values:
Step 2: Calculate the acceleration.
Using Newton's second law, the acceleration is:
Step 3: Conclusion.
Thus, the correct answer is (C) .

Step 1: Analyzing the circuit.
The circuit consists of capacitors in series and parallel. The total potential difference of 12 volts is applied across the series combination of the capacitors. To find the potential difference across the 4.5 capacitor, we need to use the equivalent capacitance of the series combination. For two capacitors in series, the formula for the total capacitance is: where and . Solving for , we get: Step 2: Finding the potential difference.
The total voltage volts is distributed across the capacitors. The potential difference across each capacitor in series is inversely proportional to its capacitance. The potential difference across the 4.5 capacitor is 8 volts, as obtained from the voltage divider rule. Step 3: Conclusion.
Thus, the correct answer is (B) 8 volt.

Step 1: Write energy density of electric field.
Energy density in an electric field is
Step 2: Find volume between capacitor plates.

Step 3: Calculate total energy stored.

Step 4: Conclusion.
The energy stored in the capacitor is .

Step 1: Understanding the capacitor's equation.
The capacitance of a capacitor is given by: Where is the charge stored and is the potential across the capacitor. The change in charge is proportional to the change in voltage, thus: Therefore, the capacitance is given by , and the correct answer is (D).

Step 1: Concept
For an isobaric process: and .
Step 2: Analysis
Ratio .
Step 3: Calculation
We know .
.
Divide numerator and denominator by :
.
Step 4: Conclusion
Hence, the ratio is .
Final Answer: (C)

Concept:
To convert galvanometer to voltmeter: Step 1: Find galvanometer resistance.
Step 2: Required total resistance.
Step 3: Series resistance.
Step 4: Conclusion. Required resistance = (series)

Step 1: Use the formula for force in an electric field
The force experienced by a charge in an electric field is given by:
Step 2: Substitute the given values
Given: - Charge - Electric field
Answer:
Therefore, the force experienced by the charge is . So, the correct answer is option (1).