Electrostatic Potential And Capacitance (SAQs)

Physics-2 | 5. Electrostatic Potential and Capacitance – SAQs:
Welcome to SAQs in Chapter 5: Electrostatic Potential and Capacitance. This page features the key FAQs for Short Answer Questions. Answers are provided in simple English, with a Telugu explanation, and formatted according to the exam style. This will aid in understanding the material and achieving top marks in your final exams.

---

SAQ-1 : Derive an expression for the electric potential due to a point charge.

Introduction

The electric potential at a point in an electric field represents the work done in moving a unit positive charge from infinity to that point. This section provides a derivation of the electric potential due to a point charge.

Key Formula for Electric Potential

The change in potential (VBA​) from point A to point B is expressed as:

$$V_{BA} = V_B – V_A = -\int_{BA} \vec{E} \cdot d\vec{s}$$

where E is the electric field.

Detailed Steps for Derivation

1. Electric Field of a Point Charge: For a point charge q, as we move from point A to point B, the change in electric potential (V_BA​) is given by $$V_{BA} = V_B – V_A = -\int_{BA} \vec{E} \cdot d\vec{s}$$
2. Substituting Electric Field (E): The electric field is defined as kq/r², where k is Coulomb’s constant. Substituting this into the equation, we get $$E \cdot ds = kq/r^2 \cdot dr$$
3. Integral Calculation: Inserting the expression for E into the original equation: $$V_{BA} = V_B – V_A = -\int_{r_B}^{r_A} \frac{kq}{r^2} dr$$
4. Integral Solution: Solving the integral, we obtain: $$V_{BA} = kq \left[\frac{1}{r_B} – \frac{1}{r_A}\right]$$
5. Reference Point at Infinity: Setting r_A​ = ∞ for a reference point at infinity, the equation simplifies to: $$V = \frac{kq}{r}$$ ​This is the desired expression for the electric potential due to a point charge.

Additional Information

1. Potential (V): The work done per unit charge, measured in volts (joules per coulomb).
2. Electric Field (E): The force per unit charge, pointing radially outward for a positive charge and inward for a negative charge. It is expressed in Newton per coulomb.
3. Electric Field and Potential Relationship: The electric field is the negative gradient of potential (E = −gradV). On the surface of a charged conductor, where the potential is constant, the electrostatic field is normal to the surface.

Summary

The electric potential due to a point charge is quantified by the expression $$V = \frac{kq}{r}$$ signifying the work done per unit positive charge in moving from infinity to a point a distance ‘r’ away from the charge. This formulation is crucial in understanding electrostatics and the behavior of charges in electric fields.

---

SAQ-2 : Derive an expression for the capacitance of a parallel plate capacitor.

Introduction

Capacitance is a measure of an electrical component’s ability to store electrical charge. A capacitor is a key electrical device that stores energy in an electric field formed between two conductive plates in close proximity, separated by a dielectric material. This article focuses on deriving the capacitance of a parallel plate capacitor using concepts of electric field and potential difference.

Key Formula in Capacitance Derivation

The fundamental formula for this derivation is the definition of capacitance, given by $$C = \frac{Q}{V}$$ where C represents capacitance, Q is the charge stored, and V is the potential difference between the plates.

Detailed Derivation

1. Configuration of Capacitor: A parallel plate capacitor consists of two plates with surface charge densities of +σ and −σ respectively. Let A represent the area of the plates, and d the distance between them.
2. Electric Field Calculation: The electric field (E) generated by a single charged plate is $$E = \frac{\sigma}{2\varepsilon_0}$$ directed normally outward.
3. Total Electric Field: The total field between the plates is the sum of the fields due to each plate, resulting in $$E = \frac{\sigma}{\varepsilon_0} = \frac{Q}{A\varepsilon_0}$$ considering the field from the negatively charged plate as opposite in direction.
4. Potential Difference: The potential difference (V) between the plates is $$V = Ed = \frac{Qd}{A\varepsilon_0}$$
5. Capacitance Formula: Substituting V into the capacitance formula, we get $$C = \frac{Q}{V} = \frac{A\varepsilon_0}{d}$$

Additional Insights

1. Dielectric Medium: Introducing a dielectric medium between the plates increases the capacitance by a factor of k, the relative permittivity or dielectric constant of the medium.
2. Applications: Capacitors are crucial in various fields, including digital circuits, power supply in particle accelerators and fusion research, and in oscillators for generating electrical signals.

Summary

The article elucidates the principles of a parallel plate capacitor and derives its capacitance formula, $$C = \frac{A\varepsilon_0}{d}$$ from fundamental electrostatic concepts. This understanding is vital in electrical engineering and physics for efficient manipulation and storage of electric energy.

---

SAQ-3 : Derive the formula for equivalent capacitance when the capacitors are in series.

Introduction

A capacitor is an electrical component designed to store energy in an electric field. When capacitors are connected in series, they exhibit unique properties and relationships. This article focuses on understanding the series connection of capacitors and deriving the formula for the equivalent capacitance of such an arrangement.

Key Formula for Series Capacitance

In a series connection of capacitors, the total or equivalent capacitance (C) is calculated by the inverse of the sum of the inverses of each individual capacitance:

$$\frac{1}{C} = \frac{1}{C_1} + \frac{1}{C_2} + \frac{1}{C_3} + \ldots$$

Detailed Derivation

1. Circuit Configuration: Consider a circuit with capacitors C₁​, C₂​, and C₃​ connected in series, having potential differences V₁​, V₂​, and V_3​, respectively.
2. Charge in Series Connection: In series, the total charge (Q) is the same across each capacitor due to the uniform current flow in a series circuit.
3. Total Potential Difference: The total potential difference (V) across the series connection is the sum of individual potentials: $$V = V_1 + V_2 + V_3$$
4. Potential Difference Relation: The potential difference across each capacitor is related to charge and capacitance as $$V_1 = \frac{Q}{C_1}$$ $$V_2 = \frac{Q}{C_2}$$ $$V_3 = \frac{Q}{C_3}$$
5. Effective Capacitance: Expressing the total potential difference as $$V = \frac{Q}{C}$$​ and substituting V₁​, V₂​, and V₃​, we get: $$\frac{Q}{C} = \frac{Q}{C_1} + \frac{Q}{C_2} + \frac{Q}{C_3}​$$
6. Equivalent Capacitance Formula: Simplifying the above, the relationship for the effective capacitance in a series combination is derived: $$\frac{1}{C} = \frac{1}{C_1} + \frac{1}{C_2} + \frac{1}{C_3}$$

Key Takeaways

1. In a series connection, the effective capacitance is lower than the smallest individual capacitance.
2. Series combinations are used to reduce the total capacitance in a circuit.

Summary

Understanding series connections of capacitors is crucial in electronics and electrical engineering, as it allows for the manipulation of total capacitance in electrical circuits. The derived formula for equivalent capacitance in series helps in calculating the combined effect of multiple capacitors.

---

SAQ-4 : Derive the formula for equivalent capacitance when the capacitors are in parallel.

Introduction

Capacitors are essential components in various electronic and electrical circuits. When capacitors are connected in parallel, they create a combined effect on the circuit’s overall capacitance. This article provides a detailed derivation of the formula for the equivalent capacitance in a parallel setup.

Key Formula for Parallel Capacitance

The formula for equivalent capacitance (C) of capacitors in a parallel circuit is the sum of individual capacitances:

$$C = C_1 + C_2 + C_3 + \ldots$$

Derivation Process

1. Capacitor Configuration: Consider capacitors C₁​, C₂​, and C₃​ connected in parallel, each storing charges Q₁​, Q₂​, and Q_3​, respectively.
2. Constant Potential Difference: In a parallel arrangement, the potential difference (V) across each capacitor is the same, as they are all directly connected to the source voltage.
3. Total Charge Calculation: The total charge (Q) in a parallel setup is the sum of individual charges: $$Q = Q_1 + Q_2 + Q_3$$
4. Charge-Capacitance Relation: The charge on each capacitor is expressed as $$Q_1 = C_1 \cdot V$$ $$Q_2 = C_2 \cdot V$$ and $$Q_3 = C_3 \cdot V$$
5. Equivalent Capacitance Expression: The total charge for the parallel combination is Q = C ⋅ V, where C is the equivalent capacitance. Substituting the individual charge expressions, we get: $$C \cdot V = C_1 \cdot V + C_2 \cdot V + C_3 \cdot V$$
6. Simplifying the Equation: Simplifying the equation, the formula for equivalent capacitance of the parallel combination is derived: $$C = C_1 + C_2 + C_3$$

Key Takeaways

1. The equivalent capacitance in a parallel circuit is the sum of the individual capacitances.
2. Parallel connections are typically used to increase the overall capacitance in a circuit.

Summary

Understanding the concept of parallel connections of capacitors and the method to calculate the equivalent capacitance is crucial in electronics and electrical engineering. This knowledge is key to effectively designing and implementing circuits based on specific capacitance requirements.

---

SAQ-5 : Explain the behavior of dielectrics in an external field.

Introduction

Dielectrics are insulating materials that do not conduct electric current but can support an electrostatic field. They are classified into Non-polar and Polar dielectrics, based on their behavior under an external electric field. This distinction is crucial in understanding how dielectrics interact with electric fields.

Behavior of Non-Polar Dielectrics

Non-Polar Dielectrics consist of molecules like oxygen (O2), characterized by coinciding centers of positive and negative charges, resulting in a net dipole moment of zero. In an external electric field, these molecules exhibit displacement of charges in opposite directions, leading to an induced dipole moment. The phenomenon of developing these induced moments is termed as polarization of the dielectric.

Behavior of Polar Dielectrics

Polar Dielectrics, in contrast, have molecules with separate centers of positive and negative charges, creating a permanent dipole moment. Without an external field, these dipoles are randomly oriented, rendering a net dipole moment of zero. However, in an external field, the dipoles align orderly, leading to polarization. The dipole moment per unit volume (P), also known as polarization, is given by P = χE, where χ is the electric susceptibility of the medium and E is the electric field.

Summary

The behavior of non-polar and polar dielectrics under external electric fields is fundamental in physics and engineering. This understanding is instrumental in designing various applications, including capacitors, transistors, and other electronic devices, where dielectrics play a pivotal role.