📘 Electrostatic Potential and Capacitance
Physics • Class XII
Introduction to Potential Energy
Potential energy is introduced as the work done by an external force in moving a body against a force (e.g., spring force, gravitational force). This work is stored as potential energy, which converts to kinetic energy when the external force is removed. The conservation of energy principle states that the sum of kinetic and potential energies is conserved.
Electrostatic Potential Energy
Coulomb force between stationary charges is also a conservative force, similar to gravitational force. Electrostatic potential energy can be defined for a charge in an electrostatic field, analogous to potential energy in a gravitational field.
Work Done in Moving a Charge
When moving a test charge q from point R to point P in an electric field due to a charge Q, an external force F_ext is applied to counter the electric force F_E. The work done by the external force is stored as potential energy of the charge q.
Characteristics of Electric Potential Energy Difference
The potential energy difference between two points is defined as the work required to move charge q from one point to another without acceleration. This work depends only on the initial and final positions of the charge, not on the path taken.
Conservative Forces and Potential Energy
A conservative force is characterized by the path-independence of work done. The concept of potential energy is meaningful only when work does not depend on the path taken. The work done by an electrostatic field can be demonstrated using Coulomb’s law.
Electrostatic Potential
Electrostatic potential (V) is defined as the work done per unit charge in bringing a unit positive charge from infinity to a point in an electric field. The potential difference is the key physical quantity, not the absolute potential values.
Potential Energy Difference
Potential energy difference is defined in terms of work. It is significant only as a difference, not as an absolute value. An arbitrary constant can be added to potential energy without affecting the potential energy difference. A common choice is to set the potential energy to zero at infinity.
Potential Due to a Point Charge
For a point charge Q at the origin, the potential at point P with position vector r is determined by the work done in bringing a unit positive test charge from infinity to P. The potential V at point P is given by V(r) = Q/(4πε₀r).
Example Calculation
To calculate the potential at point P due to a charge of 4 × 10⁻⁷ C located 9 cm away, use V = Q/(4πε₀r). For a charge of 2 × 10⁻⁹ C brought from infinity to point P, the work done W = qV. The work done is path independent.
Potential Due to an Electric Dipole
An electric dipole consists of two charges, q and -q, separated by a distance 2a. The dipole moment vector p is defined as p = q × 2a, pointing from -q to q. The electric potential due to a dipole is determined by the superposition principle, where the potential is the sum of potentials due to the charges q and -q.
Electric Potential Due to a Dipole
The formula for potential V at a point P is given by: V = - (1/(4πε₀)) * (q/r₁ + (-q)/r₂), where r₁ and r₂ are the distances from the charges to point P. For large distances (r >> a), the potential simplifies to V = (1/(4πε₀)) * (p·r̂)/r².
Potential Due to a System of Charges
For a system of charges q₁, q₂, ..., qₙ with position vectors r₁, r₂, ..., rₙ, the potential V at a point P is the algebraic sum of the potentials due to each charge. The potential due to each charge is given by: Vᵢ = (1/(4πε₀)) * (qᵢ/rᵢₚ). Thus, the total potential V at point P is: V = V₁ + V₂ + ... + Vₙ.
Continuous Charge Distribution and Electric Potential
To determine the electric potential due to a continuous charge distribution characterized by charge density ρ(r), divide it into small volume elements of size Dv, each carrying a charge ρDv. The total potential is found by integrating the contributions from all volume elements.
Electric Potential of a Uniformly Charged Spherical Shell
For a uniformly charged spherical shell, the electric field outside the shell behaves as if the entire charge is concentrated at the center. The potential outside the shell is given by V = (1 / (4πε₀)) * (q / r) for r ≥ R. Inside the shell, the electric field is zero, leading to a constant potential equal to its surface value: V = (1 / (4πε₀)) * (q / R).
Example: Finding Zero Electric Potential
Given two charges, 3 × 10⁻⁸ C and -2 × 10⁻⁸ C, located 15 cm apart, the point where the electric potential is zero can be found. Taking the positive charge at the origin, the potential at point P on the x-axis can be calculated.
Example: Analyzing Potential Differences and Work Done
In analyzing potential differences and work done in electric fields, consider the following: (a) The potential difference V_P - V_Q is positive since V_P > V_Q. (b) A small negative charge moves from higher to lower potential energy, indicating a positive potential energy difference. (c) Work done by the field in moving a positive charge from Q to P is negative, as work is done against the field.
Equipotential Surfaces
An equipotential surface is a surface with a constant value of potential at all points. For a single charge q, the potential V is given by V = (1/(4πε₀)) * (q/r). Equipotential surfaces for a single point charge are concentric spherical surfaces centered at the charge.
Relation Between Electric Field and Potential
Consider two closely spaced equipotential surfaces A and B with potential values V and V + dV. The work done in moving a unit positive charge from surface B to surface A against the electric field is |E| * dl, which equals the potential difference VA - VB. This leads to the relation |E| = - (dV/dl).
Potential Energy of a System of Charges
To calculate the work done in building a configuration of two charges q₁ and q₂, initially at infinity, we consider the work done by an external agency to bring the charges to their locations.
Potential Energy of Two Charges
The potential energy U of a system of two charges q₁ and q₂ is given by: U = (1 / (4πε₀)) * (q₁ * q₂ / r₁₂). This expression remains unchanged regardless of the order in which the charges are brought together due to the path-independence of work for electrostatic forces.
Significance of Charge Signs
If q₁ * q₂ > 0, the potential energy is positive, indicating repulsive forces. If q₁ * q₂ < 0, the potential energy is negative, indicating attractive forces. Positive work is required to separate like charges, while negative work is needed to bring unlike charges together.
Potential Energy of Multiple Charges
For a system of three charges q₁, q₂, and q₃ located at r₁, r₂, and r₃, the total potential energy U is calculated by summing the work done in bringing each charge from infinity to its respective position: U = (1 / (4πε₀)) * (q₁ * q₂ / r₁₂ + q₁ * q₃ / r₁₃ + q₂ * q₃ / r₂₃).
Electrostatic Potential in a Conductor
The electrostatic potential is constant throughout the volume of the conductor and equal on its surface. Since the electric field is zero inside and has no tangential component on the surface, no work is done moving a test charge within the conductor.
Electric Field Inside a Conductor
Inside a conductor, the electrostatic field is zero. This is a defining property of conductors, as free electrons redistribute themselves to ensure no electric field exists within.
Electric Field at the Surface of a Charged Conductor
The electrostatic field at the surface of a charged conductor must be normal (perpendicular) to the surface at every point. If it were not, free charges would experience a force and move, violating the static condition.
Charge Distribution in a Conductor
In a static situation, excess charge resides only on the surface of a conductor. A neutral conductor has equal positive and negative charges throughout.
Electric Field and Surface Charge Density
The electric field at the surface of a charged conductor is given by the equation: E = σ/ε₀, where σ is the surface charge density and ε₀ is the permittivity of free space.
Electrostatic Shielding
Electrostatic shielding refers to the phenomenon where the electric field inside a cavity of a conductor is zero, regardless of the cavity's size, shape, or external electric fields.
Dielectrics and Polarisation
Dielectrics are non-conducting substances with negligible charge carriers. When a dielectric is placed in an external electric field, the absence of free charge carriers means that the charge distribution does not adjust as it does in conductors.
Molecular Behavior in Dielectrics
Molecules can be classified as polar or non-polar: Non-polar molecules have coinciding positive and negative charge centers, while polar molecules have separated charge centers, creating a permanent dipole moment.
Induced Dipole Moments in Non-Polar Molecules
In an external electric field, non-polar molecules experience displacement of positive and negative charges in opposite directions until the external force is balanced by the restoring force, resulting in an induced dipole moment.
Polarization in Polar Molecules
Polar molecules develop a net dipole moment in an external field due to the alignment of individual dipole moments, which are randomly oriented in the absence of an external field.
Capacitors and Capacitance
A capacitor consists of two conductors separated by an insulator. The capacitance C is defined as C = Q/V, where Q and -Q are the charges on the conductors and V is the potential difference.
Capacitance Calculation Example
For typical values (A = 1 m², d = 1 mm), the capacitance can be calculated as follows: C = ε₀A/d = (8.85 × 10⁻¹² F/m)(1 m²)/(0.001 m) = 8.85 × 10⁻⁹ F = 8.85 nF.
Energy Stored in a Capacitor
The energy stored in a capacitor can be determined by considering two conductors with charges Q and -Q. The energy stored is related to the configuration of the conductors and their charges.
Work Done in Charge Transfer
To calculate the total work done in transferring charge, we consider the work done in transferring an infinitesimal amount of charge. The work done in transferring a small charge dQ' is expressed as dW = V' dQ'.
Transient Period in Capacitor Systems
A transient period occurs before the system stabilizes, during which a transient current flows from the first capacitor to the second. Energy is lost in the form of heat and electromagnetic radiation.
Electrostatic Force and Potential Energy
Electrostatic force is a conservative force. The work done by an external force in moving a charge q from point R to point P is given by q(V_P - V_R), representing the difference in potential energy between the two points.
Superposition Principle for Potential
For a configuration of multiple charges q₁, q₂, ..., qₙ, the potential at point P is given by the superposition principle: V_P = (1 / (4πε₀)) * Σ(q_i / r_iP).
Key Physical Quantities
Important physical quantities related to electrostatics include: Potential (V), Capacitance (C), Polarisation (P), and Dielectric constant (K).
Examples of Electrostatic Effects
Several examples illustrate electrostatic principles, including the attraction of small bits of paper by a charged comb and the use of metallic ropes to ground static electricity in vehicles carrying flammable materials.