Chapter 23: Electric Potential – Study Notes

Electric Potential and Electric Potential Energy

Introduction to Electric Potential

Electric potential is a fundamental concept in electrostatics, describing the potential energy per unit charge at a point in an electric field. It is a scalar quantity that helps in understanding the energy changes experienced by charges in electric fields.

• Electric Potential Energy (U): The energy a charged object possesses due to its position in an external electric field.

• Electric Potential (V): The potential energy per unit charge, characteristic of the field itself, independent of any test charge.

• Unit: Volt (V), where 1 V = 1 J/C.

• Relationship: $V = \frac{U}{q_0}$

Work and Electric Potential

When a charge moves in an electric field, work is done by or against the field, resulting in a change in electric potential energy. The electric field is a conservative force, so the work done is path-independent.

• Work Done by the Field: $W = -\Delta U = q_0 \int_{A}^{B} \vec{E} \cdot d\vec{s}$

• Potential Difference: $\Delta V = V_B - V_A = -\int_{A}^{B} \vec{E} \cdot d\vec{s}$

• For a uniform field: $\Delta V = -E d$ (where $d$ is the displacement parallel to $\vec{E}$)

Energy and the Direction of Electric Field

The direction of the electric field determines how the potential energy of a charge changes as it moves. For a positive charge, moving in the direction of the field decreases potential energy; for a negative charge, it increases potential energy.

• Conservation of Energy: The loss in potential energy equals the gain in kinetic energy for a charge moving under the influence of the field alone.

• Work-Energy Theorem: $K_A + U_A = K_B + U_B$

Equipotential Surfaces

Equipotential surfaces are surfaces where the electric potential is constant. No work is required to move a charge along an equipotential surface, and the electric field is always perpendicular to these surfaces.

• Equipotential Surface: All points have the same potential.

• Relation to Field Lines: Equipotentials are perpendicular to electric field lines.

Electric Potential Due to Point Charges and Distributions

Potential Due to a Point Charge

The electric potential at a distance $r$ from a point charge $q$ is given by:

• $V = k_e \frac{q}{r}$, where $k_e = \frac{1}{4\pi\varepsilon_0}$

• The reference point is usually taken at infinity ($V = 0$ at $r = \infty$).

Potential Due to Multiple Point Charges

The total electric potential at a point due to several point charges is the algebraic sum of the potentials due to each charge (superposition principle):

• $V = k_e \sum_i \frac{q_i}{r_i}$

• Potential is a scalar, so vector addition is not required.

Potential Due to a Dipole

An electric dipole consists of two equal and opposite charges separated by a distance. The potential varies rapidly between the charges and falls off quickly with distance.

Potential for Continuous Charge Distributions

For continuous charge distributions, the potential at a point is found by integrating over the charge distribution:

• $V = k_e \int \frac{dq}{r}$

• For a ring, disk, or rod, the integration is performed over the geometry of the object.

Electric Potential Energy of Systems of Charges

Two Point Charges

The electric potential energy of a system of two point charges $q_1$ and $q_2$ separated by a distance $r$ is:

• $U = k_e \frac{q_1 q_2}{r}$

• If the charges have the same sign, $U > 0$ (work is required to bring them together); if opposite, $U < 0$ (work is required to keep them apart).

Multiple Charges

For more than two charges, the total potential energy is the sum over all unique pairs:

• $U = k_e \sum_{i

Electric Field from Electric Potential

Relationship Between E and V

The electric field is related to the spatial rate of change of the electric potential:

• In one dimension: $E_x = -\frac{dV}{dx}$

• In three dimensions: $E_x = -\frac{\partial V}{\partial x}$, $E_y = -\frac{\partial V}{\partial y}$, $E_z = -\frac{\partial V}{\partial z}$

Conductors and Equipotential Surfaces

Properties of Conductors in Electrostatic Equilibrium

Conductors exhibit unique properties in electrostatic equilibrium:

• The electric field inside a conductor is zero.

• All excess charge resides on the surface.

• The electric field just outside the surface is perpendicular to the surface.

• The potential is constant everywhere on the surface and inside the conductor.

Cavities in Conductors

A cavity within a conductor, with no internal charges, is a field-free region. The potential is constant throughout the cavity and the conductor.

Applications of Electric Potential

Millikan Oil-Drop Experiment

This classic experiment measured the elementary charge $e$ by balancing gravitational and electric forces on tiny charged oil droplets.

• Demonstrated the quantization of electric charge: $q = n e$

• Measured $e = 1.60 \times 10^{-19}$ C

Van de Graaff Generator

A Van de Graaff generator uses a moving belt to transfer charge to a high-potential electrode, generating large voltages for particle acceleration and other applications.

Electrostatic Precipitator

Electrostatic precipitators use electric fields to remove particulate matter from gases, commonly used in industrial air cleaning.

Xerographic Copiers and Laser Printers

These devices use the principles of electric potential and photoconductivity to transfer toner to paper, forming images and text.

Summary Table: Key Equations and Concepts

Concept	Equation	Description
Electric Potential (point charge)	$V = k_e \frac{q}{r}$	Potential at distance $r$ from charge $q$
Potential Difference (uniform field)	$\Delta V = -E d$	Between two points separated by $d$
Potential Energy (two charges)	$U = k_e \frac{q_1 q_2}{r}$	Energy of two point charges
Work and Potential	$W = -q (V_B - V_A)$	Work to move $q$ from $A$ to $B$
Electric Field from Potential	$E_x = -\frac{dV}{dx}$	Field as gradient of potential
Electron Volt	$1\ \mathrm{eV} = 1.60 \times 10^{-19}\ \mathrm{J}$	Energy gained by $e$ through $1\ \mathrm{V}$