📘 Electric Charges and fields
Physics • Class XII
Introduction to Electric Charges
Electric charges can be observed through phenomena such as sparks from synthetic clothing and lightning during thunderstorms. These occurrences are due to the discharge of electric charges accumulated from rubbing insulating surfaces, commonly referred to as static electricity. Electrostatics is the study of forces, fields, and potentials arising from static charges.
Historical Background of Electric Charge
The discovery of electric charge dates back to Thales of Miletus in 600 BC, who noted that amber rubbed with wool or silk attracts light objects. The term 'electricity' is derived from the Greek word 'elektron', meaning amber. Various materials can attract light objects when rubbed together.
Types of Electric Charge
There are two kinds of electric charges: positive and negative. Key observations include: - Like charges repel each other. - Unlike charges attract each other. The property that differentiates these charges is known as the polarity of charge. For example, a glass rod rubbed with silk acquires a positive charge, while the silk acquires a negative charge.
Charge Neutralization
When an electrified object comes into contact with the material it was rubbed against, the charges are neutralized. This means that the previously charged objects no longer exhibit attraction or repulsion towards other objects. Benjamin Franklin named the charges positive and negative, with glass and cat's fur being positive and plastic and silk being negative, respectively.
Detection of Electric Charge
A gold-leaf electroscope is a simple apparatus used to detect electric charge. It consists of a vertical metal rod housed in a container, which can indicate the presence of charge on a body.
Electrification of Bodies
To electrify a neutral body, charge must be added or removed. A body is considered charged when there is an excess or deficit of charge. In solids, electrons are the charges transferred between bodies. A body can be positively charged by losing electrons or negatively charged by gaining electrons. For example, rubbing a glass rod with silk transfers electrons from the rod to the silk, resulting in a positively charged rod and a negatively charged silk. No new charge is created during this process.
Conductors and Insulators
Substances that allow electricity to pass through easily are called conductors (e.g., metals, human and animal bodies, earth). Insulators (e.g., glass, porcelain, plastic, nylon, wood) resist the passage of electricity. When charge is transferred to a conductor, it distributes evenly over its surface, while charge on an insulator remains localized.
Basic Properties of Electric Charge
Electric charge has three basic properties: quantisation, additivity, and conservation. Charge is a scalar quantity, meaning it has no direction. It is invariant under rotation and for frames of reference in relative motion.
Additivity of Charges
The total charge of a system containing two point charges, q1 and q2, is obtained by adding them algebraically. This principle allows for the calculation of total charge in systems with multiple charges.
Total Charge of a System
The total charge of a system containing n charges (q1, q2, q3, ..., qn) is calculated by summing the individual charges: q_total = q1 + q2 + q3 + ... + qn. Charge has magnitude but no direction, similar to mass. Unlike mass, which is always positive, charge can be positive or negative.
Conservation of Charge
Charge is conserved in an isolated system. When two bodies are charged by rubbing, electrons transfer from one body to another, resulting in one body gaining charge and the other losing charge. The total charge in the isolated system remains constant despite redistribution among the bodies.
Quantisation of Charge
Charge is quantised, meaning all free charges are integral multiples of a basic unit of charge denoted by e. The relationship is given by q = ne, where n is any integer (positive or negative). The charge of an electron is -e and that of a proton is +e. The quantisation of charge was suggested by Faraday's laws of electrolysis and demonstrated by Millikan in 1912.
Charge Calculation
If a body contains n1 electrons and n2 protons, the total charge is calculated as: Total Charge = n2 × e + n1 × (-e) = (n2 - n1) e. Since n1 and n2 are integers, the total charge is always an integral multiple of e.
Charge Quantization at Macroscopic and Microscopic Levels
At the macroscopic level, charges are typically in the range of a few milliCoulombs (mC), making the discrete nature of charge (quantization) negligible. However, at the microscopic level, where charges are in the order of tens or hundreds of e, quantization becomes significant and cannot be ignored.
Coulomb's Law
Coulomb's law describes the force between two point charges. It states that the force (F) between two point charges q₁ and q₂, separated by a distance r in vacuum, is given by the formula: F = k * (q₁ * q₂) / r², where k is Coulomb's constant.
Coulomb's Experiment
Coulomb used a torsion balance to measure the force between two charged metallic spheres. When the separation between the spheres is much larger than their radius, they can be treated as point charges.
Unit of Charge
Coulomb's law allows for the definition of a unit of charge. The constant k in the equation can be chosen arbitrarily, affecting the size of the unit of charge. In SI units, k is approximately 9 × 10^9 Nm²/C², leading to the definition of one coulomb (1 C) as the charge that experiences a force of 9 × 10^9 N when placed 1 m from another charge of the same magnitude in a vacuum.
Permittivity of Free Space
In later formulations, the constant k is expressed as k = 1/(4πε₀), where ε₀ is the permittivity of free space. Coulomb's law can then be rewritten as E = (1/(4πε₀)) * (Q/r²) * r̂.
Vector Notation of Coulomb's Law
Since force is a vector, Coulomb's law is often expressed in vector notation. The force on charge q1 due to charge q2 is denoted as F₁₂, and the position vectors of the charges are r₁ and r₂.
Coulomb’s Force Law
Coulomb’s force law describes the force between two point charges q1 and q2 located at positions r1 and r2. The force F21 is given by the equation: F21 = (1/(4πε0)) * (q1 * q2 / r21^2) * ɵ21.
Newton’s Third Law and Coulomb’s Law
The force F12 on charge q1 due to charge q2 can be derived from Coulomb’s law by interchanging the charges: F12 = -F21. This demonstrates that Coulomb’s law is consistent with Newton’s third law of motion.
Comparison of Electric and Gravitational Forces
The electric force between an electron and a proton is significantly stronger than the gravitational force between them. The ratio of the magnitudes of electric force (Fe) to gravitational force (Fg) for two protons is approximately 1.3 × 10^36.
Example Calculations
Example calculations show the electric force between an electron and a proton at a distance of 1 Å (10^-10 m) and the accelerations of both due to their mutual attraction.
Electric Field Concept
An electric field is created by a point charge Q placed at the origin. If another charge q is placed at a distance r from Q, it experiences a force due to the electric field created by Q.
Introduction to Electric Fields
The concept of an electric field was introduced to explain the force acting on a charge when another charge is present. A charge Q produces an electric field in its surrounding area, which acts on another charge q placed at a point P.
Electric Field Definition and Equation
The electric field E produced by charge Q at a point r is defined by the equation: E(r) = (1/(4πε₀)) * (Q/r²) * r̂.
Force Between Charges
The force F exerted by charge Q on charge q is given by: F = (1/(4πε₀)) * (Qq/r²) * r̂.
Electric Field and Test Charge
The electric field E due to charge Q is defined as the force that a unit positive charge would experience if placed at that point. The charge Q is referred to as the source charge, while charge q is the test charge.
Independence of Electric Field
The electric field E due to charge Q is independent of the test charge q because the ratio F/q remains constant regardless of the value of q.
Electric Field Due to a Point Charge
The magnitude of the electric field E due to a point charge depends only on the distance r from the charge.
Electric Field Due to a System of Charges
For a system of charges q1, q2, ..., qn with position vectors r1, r2, ..., rn relative to an origin O, the electric field at a point P (position vector r) is defined as the force experienced by a unit test charge placed at that point.
Physical Significance of Electric Field
The electric field is introduced as a convenient way to characterize the electrical environment of a system of charges. It indicates the force a unit positive test charge would experience if placed at a point in space without disturbing the system.
Electric Field Lines
Electric field lines visually represent the electric field as a vector quantity. They originate from a point charge and point radially outward. The strength of the electric field is indicated by the density of the field lines.
Electric Flux
Electric flux is defined as the volume crossing an area per unit time. If the normal to the surface makes an angle θ with the direction of flow, the projected area is affected by this angle.
Electric Flux Definition
Electric flux (Df) through an area element (DS) is defined by the equation Df = E · DS = E DS cos(q), where E is the electric field, DS is the area element, and q is the angle between E and DS.
Understanding Area Elements
An area element should be treated as a vector, having both magnitude and direction. The direction is specified by the normal to the plane of the area element.
Orientation and Electric Field Lines
The number of electric field lines crossing a unit area normal to the field is a measure of the electric field strength.
Units of Electric Flux
The unit of electric flux is N·C⁻¹·m².
Total Electric Flux Calculation
To calculate the total flux (Φ) through a surface (S), divide the surface into small area elements, calculate the flux at each element, and sum them up.
Electric Dipole Definition
An electric dipole consists of a pair of equal and opposite point charges (q and -q) separated by a distance (2a). The direction from -q to q defines the dipole direction.
Electric Field of an Electric Dipole
The electric field (E) of an electric dipole can be calculated using Coulomb’s law and the superposition principle. The field is analyzed for two cases: points on the dipole axis and points in the equatorial plane.
Dipole Moment Definition
The dipole moment vector (p) of an electric dipole is defined as: p = q × 2a ˆp.
Dipole in a Uniform External Field
A permanent dipole with dipole moment p in a uniform external electric field E experiences forces on its charges but no net force due to the uniformity of the field. However, a torque is generated due to the separation of charges.
Effects of Non-Uniform Electric Fields
In a non-uniform electric field, the dipole experiences a net force in addition to the torque.
Continuous Charge Distribution
Continuous charge distributions are used when dealing with large numbers of charges, such as on the surface of a charged conductor.
Surface Charge Density
Surface charge density (σ) is defined as the charge (DQ) per unit area (ΔS).
Linear and Volume Charge Densities
Linear charge density (λ) is defined as the charge (DQ) per unit length (Δl) of a wire, while volume charge density (ρ) is defined as the charge (DQ) per unit volume (ΔV).
Electric Field from Continuous Charge Distribution
The electric field due to a continuous charge distribution can be calculated using Coulomb’s law.
Gauss's Law
Gauss's Law relates electric flux (Φ) through a closed surface to the charge (q) enclosed.
Application of Gauss's Law
When applying Gauss's law, the surface chosen is referred to as the Gaussian surface.
Understanding Electric Flux
The total electric flux through a closed surface can be zero if no charge is enclosed.
Electric Field Due to an Infinite Wire
The electric field produced by an infinitely long thin straight wire is radial.
Electric Field Due to a Uniformly Charged Infinite Plane Sheet
The electric field due to an infinite plane sheet with uniform surface charge density (σ) is constant and directed parallel to the x-axis.
Field Due to a Uniformly Charged Thin Spherical Shell
The electric field due to a uniformly charged thin spherical shell of radius R exhibits spherical symmetry.
Electric Field Outside the Shell
For a point P outside the shell, the Gaussian surface is a sphere of radius r centered at O.
Electric Field Inside the Shell
For a point P inside the shell, the Gaussian surface is again a sphere centered at O.
Example: Electric Field in a Charged Atom Model
An early atomic model features a positively charged nucleus of charge Ze, surrounded by a uniform density of negative charge up to radius R.
Properties of Electric Charge
Conductors allow movement of electric charge, while insulators do not. In metals, mobile charges are electrons; in electrolytes, both positive and negative ions are mobile.
Electrostatic Force Calculations
To calculate the electrostatic force between two charged spheres, use Coulomb's Law.
Quantization of Electric Charge
The statement 'electric charge of a body is quantized' means that electric charge exists in discrete amounts, typically multiples of the elementary charge (e).
Electric Field and Forces
The electric field (E) at a point due to point charges can be calculated, and the force (F) on a test charge placed in the field can be determined using F = qE.
Electric Flux and Charge Relationship
A point charge causing an electric flux of -1.0 × 10^3 Nm²/C through a spherical Gaussian surface can be analyzed.
Charge and Electric Flux of a Uniformly Charged Sphere
A uniformly charged conducting sphere with a surface charge density can be analyzed to find the total charge and the electric flux leaving the surface.
Linear Charge Density from Electric Field
An infinite line charge producing an electric field can be used to calculate the linear charge density.
Electric Field between Parallel Plates
Two large, thin metal plates with surface charge densities of opposite signs can be analyzed to determine the electric field in different regions.