Class 12 Physics: Electric Charges and Fields Complete Study Guide

Electric Charge, Basic Properties, and Coulomb's Law

Electric charge is defined as an intrinsic property of the elementary particles of matter which gives rise to electric forces between various objects. This property is measured in the SI unit of the Coulomb, abbreviated as $C$, and is categorized as a scalar quantity.

There are several basic properties governing electric charges. The quantization of charge states that the total charge $Q$ on a body is always an integral multiple of a basic unit of charge denoted by $e$, represented by the formula Q = \binomp ne. The conservation of charge principle dictates that the total charge of an isolated system remains unchanged with time. Through the additivity of charge, the total charge of a system is determined by the algebraic sum of all individual charges within that system. Furthermore, it is a fundamental observation that like charges repel one another while unlike charges attract each other.

Coulomb’s Law states that the force of attraction or repulsion between two stationary point charges is directly proportional to the product of the magnitude of those charges and inversely proportional to the square of the distance between them. This force acts along the line joining the two charges. The constant of proportionality is given as $\frac{1}{4\times\text{π}\times\text{ε}_0}$, which has a numerical value of $9 \times 10^9 \text{ N m}^2/\text{C}^2$. An important characteristic of this interaction is that the force between two specific charges remains unaffected by the presence of other charges in the vicinity. The ratio of the force in a vacuum to the force in a medium is defined by the relative permittivity or dielectric constant, expressed as $\frac{F_{\text{vacuum}}}{F_{\text{medium}}} = \text{ε}_r$.

The Superposition Principle and Electric Fields

The Superposition Principle states that the total force acting on a specific charge due to a collection of other charges is equal to the vector sum of the individual forces acting on it. For example, the total force on a charge $q_1$ due to other charges $q_2, q_3, \text{... } q_n$ is calculating by summing the forces: $F_{\text{total}} = \text{Σ}F$. In this process, the order in which the vector sum of forces is calculated does not matter, as the result will always provide the total force.

Electric field intensity at a point located at a distance $r$ from a point charge $q$ in air is defined by the equation $E = \frac{1}{4\times\text{π}\times\text{ε}_0} \times \frac{q}{r^2}$. Electric field lines serve as a visual representation of these fields and possess several basic characteristics. Field lines always start from positive charges and end at negative charges. These electrostatic field lines do not form any closed loops, and two field lines can never cross each other.

Electric Dipoles and Torque

An electric dipole consists of two equal and opposite charges separated by a distance. Every dipole is associated with a dipole moment, denoted as $\text{ρ}$, whose magnitude is equal to the product of the magnitude of either charge $q$ and the distance $2a$ between the charges, expressed as $P = q \times (2a)$. The direction of the dipole moment vector points from the negative charge $-q$ toward the positive charge $+q$.

The electric field generated by a dipole varies depending on the point of observation. On the axial line, also known as the "end on position," the field is given by $E = \frac{1}{4\times\text{π}\times\text{ε}_0} \times \frac{2P}{r^3}$. On the equatorial line, referred to as the "broad on position," the electric field is expressed as $E = \frac{1}{4\times\text{π}\times\text{ε}_0} \times \frac{P}{r^3}$.

When a dipole is placed in an external electric field, it experiences torque. This occurs because the forces acting on the charges are equal in magnitude, unlike in direction, and parallel, but act at different points. This alignment forms a couple which rotates the dipole in an anticlockwise direction. Torque is defined as the moment of the couple and is calculated using the formula $\text{τ} = P \times E$ or $\text{τ} = P \times E \times \text{sin}(\text{θ})$.

Electric Flux, Gauss's Law, and Charge Densities

Electric flux over an area within an electric field represents the total number of electric field lines crossing that area, formulated as Φ = E ⋅ ΔS. Gauss’s Law provides a method for calculating this flux, stating that the total normal electric flux over a closed surface $S$ (known as a Gaussian surface) in a vacuum is equal to $\frac{1}{\text{ε}_0}$ times the total charge $Q$ contained inside the surface. This is represented by the integral $\text{Φ} = \frac{Q}{\text{ε}_0}$.

Calculations involving electric fields often utilize different charge densities depending on the geometry of the charge distribution. Volume charge density is given as $\text{ρ} = \frac{q}{V}$. Surface charge density is defined as $\text{σ} = \frac{q}{A}$. Linear charge density is defined as $\text{λ} = \frac{q}{L}$.

Applications of Gauss's Law

Gauss's Law is applied to determine the electric fields produced by various charged configurations. For an infinitely long, thin, uniformly charged straight wire, the electric field is calculated as $E = \frac{\text{λ}}{2\times\text{π}\times\text{ε}_0 \times r}$, showing that the field $E$ is proportional to $\frac{1}{r}$.

For a uniformly charged thin spherical shell, the electric field varies based on the position relative to the shell. At a point outside the shell where the distance $r$ identifies as $r > R$, the field is $E = \frac{1}{4\times\text{π}\times\text{ε}_0} \times \frac{Q}{r^2}$. Exactly on the surface where $r = R$, the field is $E = \frac{1}{4\times\text{π}\times\text{ε}_0} \times \frac{Q}{R^2}$. Inside the shell, where $r < R$, the electric field is always zero ($E = 0$). For a uniformly charged solid sphere, the electric field is given by the expression $E = \frac{\text{ρ} \times r}{3\times\text{ε}_0}$.

Finally, for a uniformly charged infinite thin plane sheet of charge, the electric field is expressed as $E = \frac{\text{σ}}{2\times\text{ε}_0}$.