Potential Due to Point Charges and Distributions

Introduction

Key Concepts

Electric Potential Defined

Electric potential, often referred to simply as potential, is a scalar quantity that represents the work done by an external force in bringing a unit positive charge from infinity to a specific point in an electric field without any acceleration. It provides a measure of the potential energy per unit charge at a point in space.

The electric potential \( V \) at a point due to a point charge \( Q \) is given by the equation: $$ V = \frac{k \cdot Q}{r} $$ where:

• \( V \) is the electric potential in volts (V).
• \( k \) is Coulomb's constant, approximately \( 8.988 \times 10^9 \, \text{N.m}^2/\text{C}^2 \).
• \( Q \) is the charge in coulombs (C).
• \( r \) is the distance from the charge to the point in question in meters (m).

Electric Potential Due to Multiple Point Charges

The principle of superposition allows us to calculate the electric potential due to multiple point charges by algebraically summing the potentials due to each charge individually. If there are multiple charges \( Q_1, Q_2, \ldots, Q_n \), the total electric potential \( V \) at a point is: $$ V = \sum_{i=1}^{n} \frac{k \cdot Q_i}{r_i} $$ where \( r_i \) is the distance from the \( i^{th} \) charge to the point of interest.

Electric Potential Energy

Electric potential energy is the energy a charge possesses due to its position in an electric field. For a charge \( q \) at a point with electric potential \( V \), the potential energy \( U \) is: $$ U = q \cdot V $$ This relationship shows that potential energy depends directly on both the charge and the surrounding electric potential.

Electric Potential and Electric Field Relationship

The electric field \( \vec{E} \) is related to the electric potential \( V \) by the negative gradient of the potential: $$ \vec{E} = -\nabla V $$ In one dimension, this simplifies to: $$ E = -\frac{dV}{dx} $$ This equation indicates that the electric field points in the direction of decreasing potential and its magnitude is the rate at which the potential changes with position.

Electric Potential Due to Continuous Charge Distributions

For continuous charge distributions, the electric potential is calculated by integrating the contributions of infinitesimal charge elements over the entire distribution. For a linear charge distribution with charge density \( \lambda \), the potential \( V \) at a distance \( r \) is: $$ V = \int \frac{k \cdot \lambda \, dl}{r} $$ Similarly, for surface \( \sigma \) and volume \( \rho \) charge distributions, the integrals are: $$ V = \int \frac{k \cdot \sigma \, dA}{r} \quad \text{and} \quad V = \int \frac{k \cdot \rho \, dV}{r} $$ respectively.

Potential of a Dipole

An electric dipole consists of two equal and opposite charges separated by a distance \( d \). The potential \( V \) at a point on the axial line (the line extending through both charges) a distance \( r \) from the center of the dipole is: $$ V = \frac{k \cdot p}{r^2} $$ where \( p = Q \cdot d \) is the dipole moment. This potential decreases with the square of the distance from the dipole.

Equipotential Surfaces

Equipotential surfaces are surfaces where the electric potential is constant. No work is done in moving a charge along an equipotential surface because the potential difference is zero. For point charges, equipotential surfaces are spheres centered around the charge. For multiple charges, the shape of equipotential surfaces becomes more complex, reflecting the combined effects of all charges.

Calculating Potential in Different Geometries

Electric potential calculations vary with geometry:

• Point Charge: As previously described, the potential decreases with distance as \( 1/r \).
• Infinite Line Charge: The potential at a distance \( r \) from an infinite line charge with linear charge density \( \lambda \) is: $$ V = \frac{2k \cdot \lambda}{r} $$
• Infinite Plane Charge: The potential near an infinite plane with surface charge density \( \sigma \) is: $$ V = 2k \cdot \sigma \cdot x $$ where \( x \) is the perpendicular distance from the plane.

Potential Difference and Voltage

The potential difference between two points is the work done per unit charge in moving a charge between those points. It is a measure of the energy change and is commonly referred to as voltage. The potential difference \( \Delta V \) between points \( a \) and \( b \) is: $$ \Delta V = V_b - V_a $$ This concept is fundamental in circuits and energy transfer in electric systems.

Energy Stored in an Electric Field

The energy \( U \) stored in an electric field created by a continuous charge distribution is calculated by integrating the potential energy: $$ U = \frac{1}{2} \int \rho V \, dV $$ For point charges, this simplifies to: $$ U = \frac{k \cdot Q_1 \cdot Q_2}{r} $$ where \( Q_1 \) and \( Q_2 \) are the interacting charges separated by distance \( r \).

Applications of Electric Potential

Electric potential is utilized in various applications, including:

• Capacitors: Understanding potential helps in determining the energy stored and the capacitance of capacitors.
• Electric Circuits: Voltage drops and potential differences are critical for analyzing and designing circuits.
• Electrostatics: Calculating forces and interactions between charges relies on potential concepts.

Calculus in Electric Potential

Advanced calculations of electric potential, especially in continuous charge distributions, often require calculus. Integrals are used to sum contributions over charge distributions, and differential equations may be employed when dealing with boundary conditions and potential functions in space.

Boundary Conditions and Potential Problems

Solving for electric potential often involves applying boundary conditions, such as specifying the potential on conductors or at infinity. Techniques like separation of variables and the method of images are used to solve potential problems in different configurations.

Gauss's Law and Electric Potential

While Gauss's Law is primarily used to determine electric fields, it can also aid in calculating electric potential for highly symmetric charge distributions. By first finding the electric field and then integrating to find the potential, one can leverage symmetry to simplify problems.

Potential in Conductors

Inside a conductor in electrostatic equilibrium, the electric potential is constant. This is because the free charges within the conductor move to cancel any internal electric field, resulting in no potential difference within the material.

Capacitance and Potential

Capacitance is defined as the ability of a system to store charge per unit potential difference: $$ C = \frac{Q}{V} $$ Understanding potential is essential for determining the capacitance of different configurations, such as parallel plates, spherical capacitors, and cylindrical capacitors.

Potential and Electric Potential Gradient

The electric potential gradient describes how rapidly the potential changes in space. It is directly related to the electric field, as the field is the negative gradient of the potential. This relationship is crucial in determining forces on charges and predicting electric field behavior from known potentials.

Comparison Table

Summary and Key Takeaways