Electric Charges and Fields Class 12 Physics Chapter 1 Notes

October 1, 2023 / By / CBSE Class 12 Physics Notes

Electric Charges and Fields Class 12 Physics Chapter 1 Notes

Electric Charge

  1. Historical Discovery
    • Thales of Miletus, Greece, around 600 BC, discovered that amber rubbed with wool or silk attracts light objects.
    • The term “electricity” is derived from the Greek word “elektron,” which means amber.
    • Many materials were known to attract light objects when rubbed.
  2. Observations of Electric Charges a. Glass rods rubbed with wool/silk repel each other. b. Plastic rods rubbed with cat’s fur repel each other. c. Glass rods attract plastic rods. d. Glass rods repel fur.
  3. Types of Electric Charges
    • There are two types of electric charges: positive and negative.
    • Like charges (positive-positive or negative-negative) repel each other.
    • Unlike charges (positive-negative) attract each other.
    • The property distinguishing these charges is called “polarity.”
  4. Charge Transfer
    • When objects are rubbed, one acquires positive charge, and the other acquires negative charge.
    • The charges neutralize when objects with opposite charges are brought into contact.
  5. Charge Nomenclature
    • Benjamin Franklin named charge on a glass rod or cat’s fur as positive and on plastic rod or silk as negative.
    • Objects with electric charge are electrified or charged, while those without charge are electrically neutral.
  6. Gold-Leaf Electroscope
    • An apparatus to detect charge.
    • Consists of a vertical metal rod with two thin gold leaves at its bottom.
    • When a charged object touches the metal knob, charge flows to the leaves, causing them to diverge. The degree of divergence indicates the charge’s magnitude.
  7. Electric Forces in Matter
    • All matter is composed of atoms and/or molecules.
    • Materials are normally electrically neutral but contain balanced charges.
    • Various forces (e.g., molecular attraction, surface tension) in matter are fundamentally electrical in nature, arising from interactions between charged particles.
  8. Electrifying a Neutral Body
    • To charge a neutral body, one kind of charge must be added or removed.
    • In solids, some electrons, less tightly bound in atoms, are transferred.
    • A body can be positively charged by losing electrons and negatively charged by gaining electrons during rubbing.
    • No new charge is created in this process, and the transferred electrons are a small fraction of the body’s total electrons.

Conductors and Insulators

  1. Definition
    • Conductors: Substances that allow the easy flow of electricity through them.
      • Contain electric charges (electrons) that can move freely within the material.
      • Examples include metals, human and animal bodies, and the Earth.
    • Insulators: Substances that offer high resistance to the passage of electricity.
      • Electric charges do not move easily within them.
      • Examples include glass, porcelain, plastic, nylon, and wood.
  2. Charge Distribution
    • Conductors: When charge is transferred to a conductor, it quickly distributes over the entire surface.
    • Insulators: When charge is added to an insulator, it remains localized at the same place.
  3. Electrification Examples
    • Nylon or plastic comb becomes electrified when used to comb dry hair or rubbed.
    • Metal objects like spoons do not become electrified.
    • The charges on metals can flow through the body to the ground due to their conductivity.
    • Metal rods with non-conductive handles can show signs of charging when rubbed without touching the metal part.

Basic Properties of Electric Charge

  1. Two Types of Charges
    • Electric charges come in two types: positive and negative.
    • Positive charges are repelled by other positive charges and attracted to negative charges.
    • Negative charges are repelled by other negative charges and attracted to positive charges.
    • Unlike charges attract, while like charges repel each other.
  2. Cancellation of Effects
    • When positive and negative charges are combined, their effects tend to cancel each other out.
    • This cancellation results in electric neutrality, where a system has an equal amount of positive and negative charges.
  3. Point Charges
    • When charged bodies are significantly smaller in size compared to the distances between them, they are treated as point charges.
    • In this model, all the charge of the body is considered to be concentrated at a single point in space.

Additivity of Charges

  1. Quantitative Definition of Charge
    • A quantitative definition of electric charge will be provided in the next section.
    • For now, it is assumed that charge can be quantitatively defined and measured.
  2. Addition of Charges
    • In a system containing two point charges, q1 and q2, the total charge of the system is obtained by algebraically adding q1 and q2.
    • Charges add up like real numbers or scalars, similar to the addition of masses.
    • If a system contains n charges, q1, q2, q3, …, qn, then the total charge of the system is q1 + q2 + q3 + … + qn.
  3. Magnitude and Sign of Charge
    • Charge possesses magnitude but no direction, similar to mass.
    • Unlike mass, charge can be either positive or negative.
    • Proper signs must be considered when adding charges in a system.
  4. Example
    • If a system contains five charges: +1, +2, -3, +4, and -5 (in some arbitrary unit), the total charge is calculated as: (+1) + (+2) + (-3) + (+4) + (-5) = -1 in the same unit.

Conservation of Charge

  1. Charge Transfer During Rubbing
    • When bodies are charged by rubbing, electrons are transferred from one body to another.
    • Importantly, no new charges are created, and none are destroyed in this process.
    • The gain in charge by one body is precisely equal to the loss in charge by the other body.
  2. Isolated System and Redistribution
    • Within an isolated system comprising many charged bodies, interactions among these bodies may lead to the redistribution of charges.
    • However, it has been experimentally confirmed that the total charge of an isolated system remains constant.
  3. Experimental Confirmation
    • Conservation of charge is an established scientific principle supported by experimental evidence.
    • It is impossible to create or destroy the net charge carried by any isolated system, even though the charge-carrying particles within the system may undergo changes.
  4. Creation of Charged Particles
    • Nature can create charged particles through processes like the transformation of a neutron into a proton and an electron.
    • These newly created proton and electron have equal and opposite charges, resulting in a net charge of zero before and after the creation.

Quantisation of Charge

  1. Basic Unit of Charge
    • Experiments have shown that all free charges exist in integral multiples of a fundamental unit of charge, denoted as “e.”
    • The charge on a body, q, is expressed as q = ne, where “n” is any integer, positive or negative.
    • Electrons and protons carry this fundamental charge, with electrons having a charge of -e and protons having a charge of +e.
  2. History of Quantisation
    • The concept of quantisation of charge was first suggested by Michael Faraday’s experimental laws of electrolysis.
    • It was experimentally demonstrated by Robert Millikan in 1912.
  3. Coulomb as the SI Unit
    • In the International System (SI) of Units, the unit of charge is the coulomb (C).
    • One coulomb is defined in terms of the unit of electric current (ampere, A).
    • Specifically, one coulomb is the charge flowing through a wire in 1 second when the current is 1 ampere.
    • In SI units, the value of the fundamental charge “e” is approximately e = 1.602192 × 10^(-19) C.
  4. Magnitude of e and Practical Usage
    • In electrostatics, charges of this magnitude (e) are rarely encountered at the macroscopic level.
    • Smaller units are used, such as microcoulombs (μC) and millicoulombs (mC), to represent charges more relevant to everyday situations.
  5. Charge Conservation
    • Considering only protons and electrons as basic charges in the universe, all observable charges must be integral multiples of “e.”
    • The charge on any body is always an integral multiple of “e” and can increase or decrease in steps of “e.”
  6. Macroscopic vs. Microscopic Levels
    • At the macroscopic level (dealing with large charges), the grainy nature of charge in units of “e” is not apparent, and charge appears continuous.
    • In contrast, at the microscopic level (with charges of the order of a few tens or hundreds of “e”), quantisation of charge is significant.
  7. Practical Consequences
    • At the macroscopic level, where charges are enormous compared to “e,” quantisation of charge has no practical consequence.
    • However, at the microscopic level, where charges can be counted, the quantisation of charge is essential and cannot be ignored.

Coulomb’s Law

  • Coulomb’s law quantitatively describes the force between two point charges.
  • Point charges are treated as if they have negligible size compared to the distance between them.
  1. Coulomb’s Experiments
    • Coulomb used a torsion balance to measure the force between two charged metallic spheres.
    • The spheres were treated as point charges, but their charges were initially unknown.
  2. Relationship from Coulomb’s Experiments
    • Coulomb discovered that the force between two point charges varied:
      • Inversely as the square of the distance between them (r).
      • Directly proportional to the product of the magnitudes of the two charges (q1 and q2).
      • Acted along the line joining the two charges.
    • Mathematically, if two point charges q1 and q2 are separated by a distance r in vacuum, the force (F) between them is given by Coulomb’s law:
      • F = (1/4πε₀) * (q1 * q2) / r²
  3. Experimental Approach to Coulomb’s Law
    • Coulomb’s experimental method involved:
      • Varying the distance while keeping charges constant.
      • Varying the charges while keeping the distance constant.
    • Comparing forces for different pairs of charges and distances allowed him to establish Coulomb’s law.
  4. Units of Charge and Coulomb’s Law
    • Coulomb’s law does not provide the explicit magnitude of charge but can be used to define a unit of charge.
    • In SI units, the value of ε₀ (permittivity of free space) is approximately 8.854 × 10⁻¹² C²/N·m².
    • The coulomb (C) is the SI unit of charge, defined based on Coulomb’s law.
    • 1 C is the charge that, when placed 1 m apart from an equal charge in a vacuum, experiences a repulsive force of 9 × 10⁹ N.
  5. Vector Formulation of Coulomb’s Law
    • Coulomb’s law can be expressed in vector notation.
    • Position vectors of charges q1 and q2 are r1 and r2, respectively.
    • Forces F12 and F21 denote the force on q1 due to q2 and vice versa.
    • The unit vectors ɵ21 and ɵ12 specify the direction.
    • Coulomb’s force law between two point charges q1 and q2 located at r1 and r2 is expressed as: F₁₂ = (1/4πε₀) * (q₁ * q₂) * (r₁₂ / |r₁₂|³)
    • The force on q1 due to q2 is equal in magnitude and opposite in direction to the force on q2 due to q1, satisfying Newton’s third law.
  6. Charge Sign and Attraction/Repulsion
    • Coulomb’s law is valid for charges of any sign (positive or negative).
    • Like charges (q₁ and q₂ of the same sign) result in repulsion.
    • Unlike charges (q₁ and q₂ of opposite signs) result in attraction.
    • Coulomb’s law covers both cases accurately.
  7. Coulomb’s Law in Matter
    • Coulomb’s law describes the force between charges in vacuum.
    • When charges are in matter or in the presence of charged constituents of matter, the situation becomes more complex, discussed in the next chapter.

Forces Between Multiple Charges

  • Coulomb’s law describes the force between two point charges.
  • When multiple charges are present, calculating the net force on a specific charge becomes essential.
  1. Principle of Superposition
    • In a system of stationary charges q₁, q₂, q₃, …, qₙ in vacuum, the force on a particular charge (e.g., q₁) due to all other charges can be calculated by vector addition.
    • The forces of electrostatic origin add up as vectors, similar to the parallelogram law for mechanical forces.
    • This concept is known as the principle of superposition.
  2. Calculation of Forces
    • Consider a system of three charges: q₁, q₂, and q₃ (Fig. 1.5(a)).
    • To find the force on q₁ due to q₂ and q₃, calculate the individual forces F₁₂ and F₁₃ using Coulomb’s law.
    • The total force F₁ on q₁ due to q₂ and q₃ is the vector sum of these forces: F₁ = F₁₂ + F₁₃
  3. Generalization to More Charges
    • The same principle applies to systems with more than three charges (Fig. 1.5(b)).
    • The total force F₁ on q₁ due to all other charges q₂, q₃, …, qₙ is obtained by summing the forces F₁₂, F₁₃, …, F₁ₙ.
    • The vector addition is expressed as: F₁ = F₁₂ + F₁₃ + … + F₁ₙ
      • Each F₁ᵢ is calculated using Coulomb’s law.
    • The total force F₁ is a vector sum of all these forces.
  4. Principle of Superposition
    • The principle of superposition states that in a system of charges q₁, q₂, …, qₙ, the force on one charge due to another charge is unaffected by the presence of other charges.
    • This principle is fundamental to electrostatics and is a consequence of Coulomb’s law.

Electric Field

  • Electric field is a concept introduced to describe the influence of a charge on its surroundings.
  • When a charge Q is placed at the origin O in vacuum, it creates an electric field throughout the surrounding space.
  • The electric field at any point r is denoted by E(r) and is defined as the force experienced by a unit positive test charge placed at that point.
  1. Mathematical Expression
    • The electric field E at a point r due to a point charge Q at the origin O is given by: E(r) = (1 / 4πε₀) * (Q / r²) * rˆ
    • Here, ε₀ (epsilon naught) is the permittivity of free space, r is the distance from the charge Q to the point r, and rˆ is the unit vector in the direction of r.
  2. Interpretation
    • The electric field at a point signifies the force that a unit positive test charge would experience if placed at that point.
    • The charge Q, which produces the electric field, is called the source charge, while the charge q, which tests the field’s effect, is the test charge.
    • As q approaches zero (becomes negligibly small), the ratio F/q remains finite and defines the electric field: E = lim (q → 0) (F / q)
  3. Key Points
    • The electric field E is independent of the magnitude of the test charge q.
    • It depends on the position r in space, and its direction is radially outward for positive charges and radially inward for negative charges.
    • The electric field due to a point charge exhibits spherical symmetry; it is the same at equal distances from the charge.

The electric field is a fundamental concept in electromagnetism, providing a way to describe how charges influence their surroundings without needing to consider the presence of other charges. It plays a crucial role in understanding the behavior of electric charges and is a key component of many fundamental principles in physics.

Electric Field Due to a System of Charges

  • The electric field at a point in space due to a system of multiple charges is defined as the force experienced by a unit positive test charge placed at that point without disturbing the original positions of the charges.
  • For a system of charges q₁, q₂, …, qₙ with position vectors r₁, r₂, …, rₙ relative to some origin O, the electric field at a point P with position vector r due to charge qᵢ is given by Coulomb’s law: Eᵢ = (1 / (4πε₀)) * (qᵢ / rᵢ²) * r̂ᵢ
  • Here, ε₀ (epsilon naught) is the permittivity of free space, r̂ᵢ is a unit vector from charge qᵢ to point P, and rᵢ is the distance between charge qᵢ and point P.
  1. Mathematical Expression
    • The electric field at point P due to the entire system of charges is given by the vector sum of the fields due to each individual charge: E(r) = E₁ + E₂ + … + Eₙ
    • Using the expressions for Eᵢ, we can write the total electric field as: E(r) = (1 / (4πε₀)) * ∑ (qᵢ / rᵢ²) * r̂ᵢ
    • This expression accounts for the contributions of all charges q₁, q₂, …, qₙ to the electric field at point P with position vector r.
  2. Key Points
    • The electric field E is a vector quantity that varies from point to point in space.
    • Each charge qᵢ in the system contributes to the electric field at point P, and the total field is found by summing up these contributions.
    • The electric field due to a system of charges is determined by the positions and magnitudes of the charges in the system.
    • The electric field can be used to calculate the force experienced by a test charge placed at any point in space without disturbing the original charges.

Physical significance of electric field

The physical significance of the electric field can be summarized as follows:

  1. Characterization of Electrical Environment: The electric field at a point in space around a system of charges characterizes the electrical environment created by those charges. It describes how the charges affect the space around them. For example, the electric field at a point tells you the force a unit positive test charge would experience if placed at that point without disturbing the charges. This characterization of the electrical environment is useful for understanding the behavior of charges in their surroundings.
  2. Vector Field: The electric field is a vector field, meaning it has both magnitude and direction at every point in space. Since force is a vector quantity, the electric field provides a convenient way to describe how forces would act on test charges at various points.
  3. Independence of Test Charge: The concept of electric field is independent of the test charge used to measure it. It is a property of the system of charges itself and does not depend on the specific characteristics of the test charge. This allows for a consistent and universal description of the electric field.
  4. Beyond Electrostatics: While electric field is crucial for electrostatics (the study of stationary charges), its true physical significance becomes evident when dealing with time-dependent electromagnetic phenomena. In situations involving moving charges or changing electric and magnetic fields, the concept of the electric field plays a vital role. For example, it helps explain the time delay in the effects of motion of charges and the propagation of electromagnetic waves.
  5. Central Concept in Physics: The notion of a field, including the electric field, was first introduced by Michael Faraday and is now considered one of the central concepts in physics. Fields have independent dynamics and evolve according to their own laws. They can transport energy and information over space, even in cases where direct interactions between charges are not possible.

Electric Field Lines

Electric field lines are a visual representation that helps us understand the electric field around charged objects. Here are some key points about electric field lines:

  1. Direction of the Electric Field: Electric field lines show the direction of the electric field at various points in space. The direction of the field at a particular location is tangent to the field line at that point.
  2. Starting and Ending Points: Field lines start from positive charges and terminate at negative charges. In the absence of charges, field lines may extend to infinity. This illustrates that the electric field is directed away from positive charges and toward negative charges.
  3. Continuous Curves: Within a charge-free region, electric field lines are continuous curves without any breaks. They provide a smooth representation of the electric field.
  4. Non-Crossing: Electric field lines never cross each other. If they were to cross, it would imply that at the point of intersection, there are multiple directions for the electric field, which is not possible.
  5. No Closed Loops: Electrostatic field lines do not form closed loops. This property arises from the conservative nature of the electric field. In other words, if you start and end at the same point along a field line, no net work is done by the electric field.
  6. Density Indicates Field Strength: The density of field lines in a particular region indicates the strength of the electric field. Where the lines are closer together, the field is stronger, and where they are farther apart, the field is weaker.
  7. 3-Dimensional Representation: Electric field lines exist in three-dimensional space, even though they are often represented in two dimensions for simplicity. In three dimensions, they form curved lines that extend radially from charges.
  8. Useful Visualization Tool: Electric field lines are a useful visualization tool for understanding how charges influence their surroundings. They provide an intuitive way to grasp the direction and relative strength of the electric field in different regions.

Electric Flux

Electric flux is a concept used to quantify the electric field passing through a given area or surface. Here are the key points regarding electric flux:

  1. Analogous to Liquid Flow: To understand the concept of electric flux, consider the flow of a liquid through a surface. The rate of flow is proportional to the area of the surface that is perpendicular to the flow direction. Similarly, electric flux quantifies the flow of the electric field through a surface.
  2. Definition of Electric Flux: Electric flux (Df) through a small area element (DS) is defined as the dot product of the electric field (E) and the area element vector (DS): Df=E⋅DS=E⋅DS⋅cosθ Here, θ is the angle between the electric field vector and the normal to the area element (DS).
  3. Direction of Area Element: An area element is not just a scalar but a vector because it has both magnitude and direction. The direction of the area element vector is along its normal (perpendicular to the surface).
  4. Direction of Electric Flux: The direction of the electric flux depends on the angle (θ) between the electric field vector and the area element vector. When θ=0∘, the electric flux is at its maximum, and when θ=90∘, the electric flux becomes zero because the field lines are parallel to the area element.
  5. Closed Surface Convention: For closed surfaces, like a closed box, the convention is to define the direction of the area element vector as outward from the enclosed volume. Therefore, the electric flux through a closed surface is calculated using the outward normal.
  6. Unit of Electric Flux: The unit of electric flux is N C^(-1) m^2, which is equivalent to volt-meters (V·m).
  7. Total Electric Flux: To calculate the total electric flux (f) through a given surface (S), you can sum up the flux through all small area elements on the surface. Mathematically, it can be expressed as f=∑i​E⋅dSi. In the limit as the area elements become infinitesimally small (DS→0), this summation becomes an integral: f=∮S​E⋅dS where ∮S​ represents the surface integral over the entire surface S.

Electric flux is a useful concept in electromagnetism, particularly in Gauss’s Law, where it helps in relating the flux through a closed surface to the charge enclosed by that surface. It provides a quantitative measure of how electric field lines pass through or interact with a given area or surface.

Electric Dipole

An electric dipole consists of two point charges of equal magnitude but opposite sign, q and −q, separated by a distance of 2a. Here are some key points about electric dipoles:

  1. Definition: An electric dipole is a system of two charges, one positive and one negative, separated by a fixed distance. The separation between these charges defines the length of the dipole.
  2. Direction of the Dipole: The direction from the negative charge (-q) to the positive charge (q) is considered as the direction of the dipole. This direction is a vector quantity.
  3. Centre of the Dipole: The midpoint between the positive and negative charges is known as the centre of the dipole.
  4. Net Charge: The total charge of an electric dipole is always zero since it consists of equal and opposite charges. This means that q+(−q)=0.
  5. Electric Field of a Dipole: The electric field produced by a dipole at a point in space depends on the distance from the dipole and the angle with respect to the dipole axis.
  6. Far Field Approximation: At distances much larger than the separation between the two charges (i.e., r≫2a), the electric fields due to q and −q approximately cancel each other out. As a result, the electric field produced by the dipole falls off faster than the inverse square of the distance (1/r^2).
  7. Dipole Moment: The strength of an electric dipole is quantified by its dipole moment (p), which is defined as the product of the charge magnitude (q) and the separation distance (2a): p=q⋅2a.
  8. Direction of Dipole Moment: The direction of the dipole moment is the same as the direction from the negative charge to the positive charge.
  9. Expression for Electric Field: The electric field (E) due to an electric dipole at a point on the axis of the dipole at a distance r from the centre of the dipole is given by: E=4πϵ0​1​⋅r32p​Here, 0ϵ0​ is the permittivity of free space, p is the dipole moment, and r is the distance from the dipole along its axis.
  10. Expression for Electric Field on the Equatorial Plane: The electric field due to the dipole at a point in the equatorial plane (perpendicular to the dipole axis) is given by: E = 1/4πϵ0​1 ​⋅ p/r^3​

In this case, the electric field depends on the inverse cube of the distance from the dipole.

Electric dipoles are essential in understanding the behavior of electric fields in various physical systems, including molecules, antennas, and electromagnetic radiation. They serve as a fundamental concept in electromagnetism and are frequently used to analyze and describe charge distributions.

The Electric Field of an Electric Dipole

The electric field of an electric dipole, consisting of two point charges q and −q separated by a distance 2a, can be described in two cases:

(i) For Points on the Axis of the Dipole:

Consider a point P located at a distance r from the center of the dipole along the dipole axis. The electric field due to the charge –qE and the charge (Eq​) at point P can be calculated using Coulomb’s law. The magnitudes of E−q​ and Eq​ are given by:

E(−q)​=1/4πϵ0​ ⋅ q/(r−a)^2

E(q)​=1/4πϵ0 ​⋅ q/(r+a)^2​

Here, ϵ0​ represents the permittivity of free space.

The total electric field at point P is the vector sum of E(−q)​ and E(q)​, and it is directed along the dipole axis (p). The magnitude of the total electric field E at point P is given by:

E=E(q)​−E(−q)​=1/4πϵ0 ​⋅ 2qa/(r2−a2)^3/2​

For points much farther away from the dipole than its length (i.e., r≫a), this expression can be approximated as:

E≈1/4πϵ0​ ​⋅ 2qa/r^3​

(ii) For Points on the Equatorial Plane of the Dipole:

Consider a point P located in the equatorial plane of the dipole, which is perpendicular to the dipole axis. In this case, the magnitudes of the electric fields E(−q)​ and E(q)​ due to the charges −q and q are equal. However, their components along the dipole axis cancel each other out, leaving only the components perpendicular to the axis.

The total electric field at point P in the equatorial plane is directed opposite to the dipole moment p. The magnitude of this electric field E is given by:

E = 2 ⋅ 1/4πϵ0 ​⋅ aq/(r2−a2)^3/2​

For points much farther away from the dipole than its length (i.e., r≫a), this expression can be approximated as:

E ≈ 1/4πϵ0​ ​⋅ qa/r^3​

In both cases, at large distances (r≫a), the electric field E due to the dipole falls off as 1/r^3. This behavior is different from that of a single point charge, which has an 1/r^2 dependence on distance. The dipole moment p, defined as p=q⋅2a, quantifies the strength and direction of the electric dipole.

Physical significance of dipoles

Dipoles, whether they are permanent or induced, play a significant role in various areas of physics, chemistry, and technology due to their unique properties. Here’s the physical significance of dipoles:

  1. Polarity in Molecules: Molecules with a permanent electric dipole moment, known as polar molecules, have a separation of positive and negative charges within the molecule. This separation creates an electric field, and polar molecules can interact with other molecules through electric forces. This property is crucial in understanding chemical reactions, molecular interactions, and the behavior of substances in various environments.
  2. Chemical Bonding: Dipoles are essential in explaining the nature of chemical bonds. For example, in a water molecule (H2O), the electronegativity difference between hydrogen and oxygen atoms leads to an uneven distribution of charge, resulting in a permanent dipole moment. Understanding these dipoles helps explain the formation and strength of different types of chemical bonds, such as polar covalent bonds.
  3. Solubility and Solution Properties: Dipoles play a vital role in the solubility of substances. Polar substances tend to dissolve in polar solvents, while nonpolar substances prefer nonpolar solvents. This phenomenon is explained by the interaction between the dipoles in the solute and solvent molecules.
  4. Dielectric Materials: Materials with permanent or induced dipoles are referred to as dielectrics. Dielectrics are commonly used in capacitors, where their ability to store electric charge is a direct result of their response to an applied electric field. The presence of dipoles within dielectric materials increases the capacitance of capacitors.
  5. Dipole-Dipole Interactions: In polar molecules, the permanent dipoles interact with each other through dipole-dipole interactions. These interactions affect the physical properties of substances, such as melting and boiling points, viscosity, and phase transitions.
  6. Electric Field Sensing: Dipoles are used in various sensing applications. For instance, molecules with a permanent dipole moment can be used as sensors to detect changes in electric fields, making them valuable in devices like infrared (IR) spectroscopy instruments.
  7. Electrostatic Manipulation: In micro- and nanotechnology, researchers use external electric fields to manipulate and control the orientation and movement of dipoles. This is employed in applications such as the manipulation of nanoparticles and the development of microelectromechanical systems (MEMS).
  8. Biological Systems: In biological systems, dipoles play a role in the interaction between molecules, including proteins and nucleic acids. Understanding the electrostatic forces between charged and polar molecules is essential for elucidating biological processes and drug interactions.
  9. Material Properties: Dipoles influence the properties of various materials, including ferroelectric and ferromagnetic materials. Ferroelectric materials have spontaneous electric polarization, while ferromagnetic materials have magnetic dipoles. These properties are exploited in technologies like magnetic data storage and memory devices.
  10. Quantum Mechanics: In quantum mechanics, the concept of electric dipole moments is used to analyze the energy levels and transitions in atoms and molecules. Techniques such as nuclear magnetic resonance (NMR) and electron spin resonance (ESR) rely on the interaction of magnetic dipoles with external magnetic fields.

Dipole in a Uniform External Field

In the scenario, where a permanent dipole moment p is placed in a uniform external electric field E, the dipole experiences a torque that tends to align it with the field direction. The torque is given by:

Magnitude of torque=∣p×E∣=pEsinθ

where p is the dipole moment, E is the external electric field, and θ is the angle between p and E.

The direction of this torque is normal to the plane formed by p and E, and it tends to align the dipole with the field. When the dipole is perfectly aligned with the field (θ=0), the torque becomes zero.

If the external field is not uniform (i.e., it varies in strength or direction), then in addition to the torque, there may also be a net force on the dipole. The direction and magnitude of this force depend on the orientation of the dipole moment p with respect to the non-uniform field.

  1. Dipole Parallel to the Field: If the dipole moment p is parallel to the external field E, there is no net torque (θ=0), but there is a net force acting on the dipole in the direction of the field. This force tends to move the dipole in the direction of increasing field strength.
  2. Dipole Antiparallel to the Field: If the dipole moment p is antiparallel to the external field E, again, there is no net torque (θ=180∘), but there is a net force acting on the dipole in the direction of decreasing field strength. This force tends to move the dipole in the direction of decreasing field strength.
  3. General Orientation of Dipole: In general, when the dipole moment p makes an angle θ with respect to the field E, both a torque and a force may act on the dipole. The torque will attempt to align the dipole with the field, while the force will depend on the specific orientation and strength of the non-uniform field.

This behavior can be observed in everyday situations, such as when a charged comb induces a temporary dipole moment in a piece of paper, causing it to move in the direction of the non-uniform electric field created by the comb. The non-uniformity of the field leads to an attractive force between the induced dipole and the charged comb, resulting in motion.

Continuous Charge Distribution

When dealing with continuous charge distributions, it’s often impractical to work with discrete charges, and we use continuous functions to describe the charge distribution. This approach is particularly useful for situations where charges are distributed over extended regions, such as the surface of a charged conductor or a charged wire.

Here are the three common types of continuous charge distributions and their corresponding charge density definitions:

  1. Surface Charge Density (σ):
    • Definition: Surface charge density, denoted by σ (sigma), represents the amount of charge per unit area on a surface. It is defined as the charge (∆Q) divided by the area (∆S) of a small element on the surface.
    • Formula: σ=∆Q/∆S​
    • Unit: Coulombs per square meter (C/m²)
    • Example: When considering the charge distribution on the surface of a conductor, we use surface charge density to describe how charge is spread over that surface.
  2. Linear Charge Density (λ):
    • Definition: Linear charge density, denoted by λ (lambda), represents the amount of charge per unit length along a line. It is defined as the charge (∆Q) divided by the length (∆l) of a small element of the wire.
    • Formula: λ=∆Q/∆l​
    • Unit: Coulombs per meter (C/m)
    • Example: When dealing with a charged wire or a wire with a linear charge distribution, we use linear charge density.
  3. Volume Charge Density (ρ):
    • Definition: Volume charge density, denoted by ρ (rho), represents the amount of charge per unit volume within a region of space. It is defined as the charge (∆Q) divided by the volume (∆V) of a small element of the region.
    • Formula: ρ=∆Q/∆V​
    • Unit: Coulombs per cubic meter (C/m³)
    • Example: When considering charge distribution within a three-dimensional region or volume, we use volume charge density.

Gauss’s Law

Gauss’s Law is a fundamental principle in electrostatics that relates the electric flux through a closed surface to the total charge enclosed by that surface. Here’s a summary of the key points and implications of Gauss’s Law:

  1. Statement of Gauss’s Law (without proof):
    • Gauss’s Law relates the electric flux (φ) through a closed surface (S) to the total charge (q) enclosed by that surface:
      • φ=​q/ε0​
    • Where:
      • φ is the electric flux through the closed surface.
      • q is the total charge enclosed by the closed surface.
      • ε0​ (epsilon naught) is the electric constant or vacuum permittivity (ε0​≈8.854×10−12C2/N⋅m2).
  2. Properties and Implications:
    • Gauss’s Law is true for any closed surface, regardless of its shape or size.
    • The charge (q) on the right side of Gauss’s Law includes the sum of all charges enclosed by the surface, whether they are positive or negative.
    • When applying Gauss’s Law, you choose a closed surface called the “Gaussian surface” that encloses the region of interest.
    • Gauss’s Law is particularly useful when calculating electric fields for systems with certain types of symmetry, as it simplifies the calculation significantly.
    • If the net electric flux through a closed surface is zero, it implies that the total charge contained within the closed surface is also zero.
    • Gauss’s Law is based on the inverse square dependence on distance contained in Coulomb’s Law. Any violation of Gauss’s Law would indicate a departure from the inverse square law.
  3. Gaussian Surfaces:
    • To apply Gauss’s Law, you choose an appropriate Gaussian surface. This surface should be carefully selected based on the symmetry of the charge distribution, making the calculation of the electric field easier.
    • The Gaussian surface can pass through continuous charge distributions, but it should not pass through discrete charges, as the electric field near a discrete charge is not well-defined.
  4. Solving for Electric Fields:
    • Gauss’s Law provides a powerful tool for calculating electric fields in situations with high symmetry, such as spherical, cylindrical, or planar symmetry.
    • By selecting a suitable Gaussian surface and exploiting the symmetry of the charge distribution, you can simplify the calculation of the electric field and find it more easily than through direct integration.

Applications of Gauss’s Law

Indeed, Gauss’s Law is a powerful tool for calculating electric fields in cases of high symmetry. Let’s explore some common applications of Gauss’s Law in finding electric fields:

  1. Uniformly Charged Sphere:
    • One of the most straightforward applications of Gauss’s Law is to find the electric field both inside and outside a uniformly charged sphere. By choosing a Gaussian surface that is a concentric sphere with the same center but varying radius, you can easily show that the electric field inside the sphere is zero, and outside it, it behaves as if all the charge were concentrated at the center of the sphere.
  2. Infinite Uniformly Charged Line:
    • Gauss’s Law simplifies the calculation of the electric field around an infinitely long uniformly charged line. By choosing a cylindrical Gaussian surface that surrounds the line, the electric field can be shown to be directly proportional to the distance from the line and inversely proportional to that distance squared.
  3. Infinite Uniformly Charged Plane Sheet:
    • When dealing with an infinite uniformly charged plane sheet, Gauss’s Law allows you to quickly determine that the electric field is constant and uniform in magnitude and direction on both sides of the sheet. The field lines are perpendicular to the sheet’s surface.
  4. Charged Conducting Sphere:
    • For a charged conducting sphere, Gauss’s Law helps you find the electric field both inside and outside the sphere. Inside the sphere, the electric field is zero, while outside it behaves as if all the charge were concentrated at the sphere’s center.
  5. Charged Non-Conducting Cylinder:
    • Gauss’s Law can be applied to find the electric field both inside and outside a uniformly charged non-conducting cylinder with cylindrical symmetry. Depending on your choice of Gaussian surface, you can calculate the electric field as a function of distance from the cylinder’s axis.
  6. Electric Flux through Non-Closed Surfaces:
    • Gauss’s Law is also useful for finding the electric flux through non-closed surfaces, such as open cylinders or planes. By carefully choosing your Gaussian surface and taking into account the symmetry of the charge distribution, you can calculate the electric flux and, subsequently, the electric field.
  7. Hollow Conducting Sphere:
    • When dealing with a charged hollow conducting sphere, Gauss’s Law tells you that the electric field is zero inside the sphere and behaves as if all the charge were concentrated at the sphere’s center outside it. This is similar to the uniformly charged solid sphere case but with no charge inside.

The Electric Field Due to an Infinitely Long Straight Uniformly Charged Wire

The electric field due to an infinitely long straight uniformly charged wire is a classic example of using Gauss’s Law to simplify the calculation. Here are the key points about this situation:

  1. Radial Symmetry: The wire is an axis of symmetry. This means that the electric field, as a result of the wire’s charge distribution, must have the same magnitude at points equidistant from the wire.
  2. Direction of Electric Field: The electric field at every point around the wire is radial, meaning it points directly away from or toward the wire. The direction depends on the sign of the linear charge density, with outward if positive and inward if negative.
  3. Magnitude of Electric Field: The magnitude of the electric field depends only on the radial distance (distance from the wire). It is constant along any circle centered on the wire.
  4. Gaussian Surface: To apply Gauss’s Law, you choose a suitable Gaussian surface. In this case, you consider a cylindrical Gaussian surface surrounding the wire. The curved part of the cylinder has an electric field that is normal to the surface and constant in magnitude.
  5. Flux Calculation: The flux through the curved cylindrical part of the Gaussian surface is E multiplied by the surface area. Since the electric field is constant over this surface, the flux is straightforward to calculate.
  6. Application of Gauss’s Law: Gauss’s Law states that the electric flux through a closed surface is equal to the total charge enclosed by that surface divided by the permittivity of free space (ε₀).
  7. Resultant Electric Field: By equating the flux through the Gaussian surface to the charge enclosed divided by ε₀, you can find the electric field’s magnitude as a function of the radial distance from the wire using Gauss’s Law.

The final expression for the electric field, as derived using Gauss’s Law, is:

E=2kλ/r​

Where:

  • E is the magnitude of the electric field.
  • k is Coulomb’s constant (equal to 1/4πϵ0​​).
  • λ is the linear charge density (charge per unit length) of the wire.
  • r is the radial distance from the wire.

This formula provides the magnitude of the electric field at any point at a distance r from the infinitely long straight uniformly charged wire. The direction of the electric field is always radial, either away from or toward the wire, depending on the sign of the charge density λ.

The Electric Field Due to a Uniformly Charged Infinite Plane Sheet

The electric field due to a uniformly charged infinite plane sheet is another classic example of using Gauss’s Law to simplify calculations. Here are the key points for this scenario:

  1. Uniform Surface Charge Density: The plane sheet has a uniform surface charge density, denoted as s. This means that the charge per unit area is the same everywhere on the sheet.
  2. Direction of Electric Field: Due to the symmetry of the problem, the electric field will not depend on the coordinates in the plane of the sheet (y and z). Therefore, the electric field at every point must be parallel to the x-direction.
  3. Choice of Gaussian Surface: To apply Gauss’s Law, a suitable Gaussian surface is chosen. In this case, a rectangular parallelepiped (or a cylindrical surface) is chosen. The key is to have surfaces that are perpendicular to the direction of the electric field, as these surfaces will contribute to the flux.
  4. Flux Calculation: The flux through the two faces (1 and 2) of the Gaussian surface is calculated. Electric field lines are parallel to the other faces, so they do not contribute to the total flux. The unit vectors normal to these surfaces are in opposite directions, but their fluxes have the same magnitude.
  5. Application of Gauss’s Law: Gauss’s Law states that the electric flux through a closed surface is equal to the total charge enclosed by that surface divided by the permittivity of free space (ε₀).
  6. Resultant Electric Field: By equating the flux through the Gaussian surface to the charge enclosed divided by ε₀, you can find the electric field’s magnitude as a function of the surface charge density s using Gauss’s Law.

The final expression for the electric field, as derived using Gauss’s Law, is:

E=s/2ϵ0​​

Where:

  • E is the magnitude of the electric field.
  • s is the uniform surface charge density of the infinite plane sheet.
  • ϵ0​ is the permittivity of free space.

This formula provides the magnitude of the electric field at any point near the infinite plane sheet. The direction of the electric field is always parallel to the x-axis, away from the sheet if s is positive and toward the sheet if s is negative.

For a finite large planar sheet, this formula is approximately true in the middle regions of the sheet, away from the ends.

The Electric Field due to a Uniformly Charged Thin Spherical Shell

The electric field due to a uniformly charged thin spherical shell is a classic example illustrating the use of Gauss’s Law. Here are the key points for this scenario:

  1. Uniform Surface Charge Density: The spherical shell has a uniform surface charge density, denoted as s. This means that the charge per unit area is the same everywhere on the shell.
  2. Symmetry and Direction of Electric Field: Due to spherical symmetry, the electric field at any point (inside or outside the shell) can only depend on the radial distance (r) from the center of the shell to that point. The electric field must also be radial, meaning it points along the radius vector from the center of the shell to the point in question.
  3. Field Outside the Shell: To calculate the electric field (E) outside the shell, a Gaussian surface is chosen. In this case, it’s a sphere of radius r with its center at the center of the shell and passing through point P. The electric field is assumed to have the same magnitude (E) and direction (radial) at every point on the Gaussian surface.
  4. Flux Calculation (Outside the Shell): The flux through each element (DS) of the Gaussian surface is E⋅DS. By summing up the flux through all these elements, the total flux through the Gaussian surface is E⋅4πr^2.
  5. Application of Gauss’s Law (Outside the Shell): Gauss’s Law states that the total electric flux through a closed surface is equal to the total charge enclosed by that surface divided by the permittivity of free space (ϵ0​).
  6. Resultant Electric Field (Outside the Shell): By applying Gauss’s Law, you equate the flux through the Gaussian surface to the total charge enclosed, which is q=4πR^2.s (the total charge on the spherical shell). This results in the expression for the electric field E=q/4πϵ0​r^2​.Vectorially, this expression can be represented as E=q/4πϵ0 . 1/​r^2 . ​r^, where r^ is the radial unit vector.The electric field is directed outward if the total charge (q) is positive and inward if q is negative. This expression essentially indicates that the electric field behaves as if all the charge of the shell is concentrated at its center. This result is valid for points outside the shell.
  7. Field Inside the Shell: For points inside the shell, the Gaussian surface still encloses no charge because all of the charge is on the outer surface of the shell. Therefore, Gauss’s Law results in E=0 for points inside the shell (r<R).

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