6 Most SAQ’s of Electric Charges and Fields Chapter in Inter 2nd Year Physics (TS/AP)

4 Marks

SAQ-1 : State and explain Coulomb’s inverse square law in electricity.

Introduction

Coulomb’s Law, named after the French physicist Charles-Augustin de Coulomb, is a cornerstone of electromagnetism. It describes the force between two stationary, electrically charged particles, taking into account the distance between them and the magnitude of their charges.

1. Fundamental Concept: The elementary charge (e), the charge of a proton or electron, serves as the basic unit of electric charge. All other charges are integer multiples of this fundamental charge, highlighting the quantized nature of electric charge.
2. Coulomb’s Inverse Square Law: Coulomb’s Law asserts that the force (F) between two charges is directly proportional to the product of their magnitudes (q₁​ and q₂​) and inversely proportional to the square of the distance (r) between them. The law is mathematically formulated as: $$F = k\frac{q_1q_2}{r^2}$$
   Here, k represents the Coulomb’s constant, valued at approximately 9 ×10⁹ N(m/C)² in free space or air. The constant k is defined as 1/4πε, with ε being the permittivity of the medium surrounding the charges. In a vacuum or air, k is further detailed as 1/4πε₀​, where ε₀​ (approximately 8.854×10⁻¹² C²/N·m²) is the permittivity of free space.
3. Interpretation of Coulomb’s Law: The force derived from Coulomb’s Law can be attractive or repulsive, contingent on the charges’ nature. Charges of the same sign repel each other, while charges of opposite signs attract, illustrating the law’s ability to describe both attraction and repulsion in electrical phenomena.
4. Importance and Applications: Coulomb’s Law is fundamental across physics and engineering disciplines, underpinning theories in electrostatics, electrodynamics, and electronics, and informing our understanding of atomic structure and chemical bonding. It also elucidates concepts of electric field and electric potential, marking its critical role in scientific advancements.

Summary

The concept of permittivity is pivotal in comprehending Coulomb’s Law, indicating a medium’s capacity to convey electric fields. This property measures the medium’s opposition to the field, integrating a fundamental aspect of how electric forces operate across different mediums.

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SAQ-2 : Derive an expression for the intensity of the electric field at a point on the axial line of an electric dipole.

Introduction

An electric dipole consists of two equal and opposite charges (q and −q), separated by a small distance 2a. It is a fundamental concept in electrostatics, representing a simple system that produces an electric field in the space around it.

Electric Dipole Moment

The electric dipole moment (p​) is a vector quantity defined as the product of the charge (q) and the separation distance (2a). It is directed from the negative to the positive charge. Mathematically, it is expressed as:

$$\vec{p} = q \cdot (2a) \vec{d}$$

where d is the direction from the negative to the positive charge.

Derivation of Electric Field Intensity on Axial Line

Consider a point P on the axial line of the dipole at a distance r from the center of the dipole, where r ≫ 2a to ensure the point is far from the dipole relative to its size.

1. Contributions from Both Charges: The electric field due to a single charge is given by Coulomb’s law as $$E = \frac{kq}{r^2}$$ For the dipole, the electric fields due to the positive and negative charges at point P need to be considered separately and then summed vectorially.
2. Electric Field Due to Positive Charge (E₊​): At a distance (r+a) from the positive charge, $$E_+ = \frac{kq}{(r+a)^2}$$
3. Electric Field Due to Negative Charge (E₋​): At a distance (r−a) from the negative charge, $$E_- = \frac{kq}{(r-a)^2}$$
4. Net Electric Field (E) on the Axial Line: The net electric field is the difference between E₊​ and E₋​, as they are in opposite directions along the same line, $$E = E_+ – E_- = kq\left[\frac{1}{(r-a)^2} – \frac{1}{(r+a)^2}\right]$$
5. Approximation for Long Distances: Assuming r ≫ a, we can use the binomial approximation $$\frac{1}{(r \pm a)^2} \approx \frac{1}{r^2} \mp \frac{2a}{r^3}$$ to simplify the expression, $$E \approx kq\left[\frac{2a}{r^3}\right]$$
6. Expression for Electric Field Intensity: Substituting p = q(2a) into the simplified expression, the intensity of the electric field at a point on the axial line of an electric dipole is given by: $$E = \frac{2kp}{r^3}$$
   where k is the Coulomb constant, and p is the magnitude of the electric dipole moment.

Summary

The intensity of the electric field at a point on the axial line of an electric dipole is inversely proportional to the cube of the distance from the center of the dipole. This relationship highlights the effect of the dipole moment and the distance on the electric field’s strength, underscoring the foundational principles of electrostatic interactions in dipoles.

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SAQ-3 : Derive the equation for the couple acting on an electric dipole in a uniform electric field.

Introduction

An electric dipole is characterized by two equal and opposite charges separated by a small distance. When placed in a uniform electric field, the dipole experiences a torque that aims to align it with the field direction. Despite the forces on the dipole being equal in magnitude but opposite in direction, resulting in a net force of zero, the dipole undergoes rotation due to this torque.

Derivation of Torque on an Electric Dipole

1. Positioning of the Dipole: An electric dipole with a moment p in a uniform electric field E is oriented at an angle θ to the field. This angle is crucial for calculating the torque experienced by the dipole.
2. Forces on the Charges: The positive charge experiences a force qE in the field direction, while the negative charge experiences an equal force in the opposite direction −qE. These forces create a couple that generates a torque.
3. Calculation of Torque (τ): The torque’s magnitude is found by multiplying the force by the perpendicular distance between the force lines, resulting in: $$\tau = qE(2a \sin \theta) = q(2a)E \sin \theta = pE \sin \theta$$ This equation highlights the dependence of torque on the electric field (E), dipole moment (p), and the angle (θ) between them.
4. Vector Representation: The torque can also be expressed as a vector product τ = p × E, indicating that the torque’s direction is perpendicular to the plane containing p and E.
5. Maximum Torque (τ_max​): The maximum torque is pE, occurring when θ=90^∘, illustrating that the torque is maximal when the dipole is perpendicular to the electric field.

Summary

The derivation elucidates the relationship between the torque on an electric dipole and the electric field, dipole moment, and the angle between the dipole moment and the field. This relationship is pivotal in understanding how molecules behave in electric fields, leading to polarization in materials as dipoles align with the electric field.

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SAQ-4 : State Gauss’s law in electrostatics and explain its importance.

Introduction

Electrostatics is a branch of physics that focuses on the study of stationary or slow-moving electric charges. Gauss’s law stands as a pivotal principle within this field, establishing a relationship between the electric flux through a closed surface and the charge enclosed by that surface. This law is crucial for understanding and calculating the electric fields generated by various charged entities.

Gauss’s Law: Mathematical Formulation

Gauss’s law is mathematically expressed as:

$$\oint \vec{E} \cdot d\vec{s} = \frac{Q_{\text{enclosed}}}{\varepsilon_0}$$

where E represents the electric field, ds denotes an infinitesimal area vector on the closed surface, Q_enclosed​ is the total charge enclosed within the surface, and ε₀​ is the permittivity of free space. This law serves as a powerful tool in determining the electric fields for various charge distributions, especially when the problem involves a high degree of symmetry.

Application and Importance

1. Symmetry in Charge Distributions: Gauss’s law proves especially useful in calculating electric fields resulting from charge distributions such as a charged sheet or an infinite straight charged wire, where symmetry simplifies the analysis.
2. Case Study: Charged Spherical Shell:
   • Outside the Shell: For a point outside a uniformly charged thin spherical shell, Gauss’s law facilitates the derivation of the electric field (E) as $$E = \frac{Q_{\text{enclosed}}}{4\pi\varepsilon_0r^2}$$ where r is the distance from the shell’s center to the point of interest. This outcome leverages the spherical symmetry, which dictates that the electric field is uniform across the Gaussian surface and points radially outward.
   • Inside the Shell: Intriguingly, the electric field within a uniformly charged thin sphere is zero. This results from selecting a Gaussian surface within the shell, which encloses no charge, leading to a null electric field inside the shell.

Summary

Gauss’s law is instrumental in determining the electric fields generated by various charged configurations, particularly highlighting the utility of symmetry in simplifying complex electrostatic problems. This principle not only elucidates the behavior of electric fields in the vicinity of charged objects, such as the zero electric field inside a uniformly charged shell, but also underscores the foundational aspects of electrostatic interactions.

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SAQ-5 : Define intensity of electric field at a point. Derive an expression for the intensity due to a point charge.

Introduction

Electric field intensity (E) is a fundamental concept in electrostatics, representing the force per unit positive charge at a specific point within an electric field. It is a vector quantity, characterized by both magnitude and direction, essential for understanding the effects of electric fields on charges.

Derivation of Electric Field Intensity from a Point Charge

1. Electric Field Concept: The electric field is the area around an electric charge where its influence is observable, exerting force on other charges present.
2. Coulomb’s Law Application: Given a point charge q_1​ and a unit positive test charge q_2​ at a distance r from q₁​, the force (F) acting on q₂​ due to q₁​ is given by Coulomb’s law: $$\mathbf{F} = \frac{1}{4\pi\varepsilon_0} \frac{q_1q_2}{r^2} \hat{r}$$ where ε₀​ is the permittivity of free space and r^ is the unit vector pointing from q₁​ to q₂​.
3. Definition of Electric Field Intensity (E): The intensity at the position of q₂​ is the force experienced by it per unit charge: $$\mathbf{E} = \frac{\mathbf{F}}{q_2}​$$
4. Expression for E: By inserting the expression for F into the equation for E and simplifying, we obtain the electric field intensity due to a point charge q₁​ at a distance r: $$\mathbf{E} = \frac{q_1}{4\pi\varepsilon_0r^2} \hat{r}$$

Summary

This derivation demonstrates the direct relationship between electric field intensity and the magnitude of a charge (q₁​), as well as its distance (r) from a point of interest. It’s crucial to note that the direction of the electric field intensity due to a positive charge is outward, away from the charge, whereas for a negative charge, it is directed inward, towards the charge.

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SAQ-6 : Derive an expression for the intensity of the electric field at a point on the equatorial plane of an electric dipole.

Introduction

An electric dipole is characterized by two equal and opposite charges, +q and −q, separated by a distance 2a. The equatorial plane of an electric dipole refers to the plane perpendicular to the line connecting these charges and passing through their midpoint.

Key Formulae for Derivation

1. Dipole Moment (P): Defined as P = q⋅2a, where q is the magnitude of each charge and 2a is the distance between the charges.
2. Electric Field Intensity (E): Given by $$E = \frac{1}{4\pi\varepsilon_0} \cdot \frac{q}{r^2}$$ where ε₀​ is the permittivity of free space, q is the charge, and r is the distance from the charge.

Detailed Steps for Derivation

1. Positioning: Consider the dipole charges +q and −q, and an equatorial point P.
2. Net Electric Field Components: At point P, the net electric field intensity is the sum of the horizontal components, represented as E cosθ.
3. Geometric Relationships: In triangles AOP and POB (with A and B representing the charges and O the midpoint),
   • $$AP^2 = BP^2 = d^2 + a^2$$
   • $$\cos \theta = \frac{a}{\sqrt{d^2 + a^2}}$$
4. Electric Field Intensity at P:
   • From −q, $$E_1 = k \cdot \frac{q}{d^2 + a^2}$$​ where $$k = \frac{1}{4\pi\varepsilon_0}$$
   • From +q, $$E_2 = k \cdot \frac{q}{d^2 + a^2}$$
5. Net Electric Field (Ep​): The net field at P is $$E_p = 2E \cos \theta = 2 \cdot k \cdot \frac{q}{d^2 + a^2} \cdot \frac{a}{\sqrt{d^2 + a^2}}$$
6. Simplification for Small Dipole: For a ≪ d, the intensity simplifies to $$E_p = -\frac{1}{4\pi\varepsilon_0} \cdot \frac{P}{d^3}$$

Summary

The electric field intensity at an equatorial point of an electric dipole is inversely proportional to the cube of the distance from the dipole’s midpoint (d³). Importantly, the field’s direction is opposite to the dipole moment’s direction, illustrating the dipole’s symmetric field distribution in the equatorial plane.