Chapter 25: The Electric Potential

Introduction

This chapter introduces the concept of electric potential and electric potential energy, fundamental ideas in understanding how charges interact and how energy is stored and transferred in electric fields. The chapter builds on previous knowledge of work, energy, and electric forces.

Energy in Physical Systems

Kinetic and Potential Energy

• Kinetic Energy (K): The energy of motion, given by the sum of the kinetic energies of all particles in a system: .

• Potential Energy (U): The energy stored due to the interaction between particles, such as gravitational or electric potential energy.

• Change in Potential Energy: The change in potential energy, , is equal to the negative of the work done by the interaction forces:

• If all forces are conservative (e.g., gravity, electric force), the total energy is conserved (remains constant over time).

Work Done by a Constant Force

Definition and Calculation

• The work done by a constant force as a particle moves through a displacement is:

• is the angle between the force and the displacement vectors.

• Work is maximized when force and displacement are parallel ().

Gravitational Analogy

Potential Energy in a Gravitational Field

• For gravity, the work done by the gravitational force as an object moves from height to is:

• The change in gravitational potential energy is:

• As an object falls, it loses potential energy and gains kinetic energy.

Electric Potential Energy in a Uniform Electric Field

Behavior of Charges in a Uniform Field

• A positive charge in a uniform electric field experiences a force .

• As the charge moves a distance in the direction of the field, the work done by the field is:

• The change in electric potential energy is:

• For a positive charge moving with the field, decreases and increases (the charge speeds up).

• For a negative charge, the potential energy increases as it moves with the field (it slows down).

Potential Energy of Two Point Charges

Interaction Energy

• The electric potential energy of two point charges and separated by distance is:

• This energy is positive for like charges (repulsion) and negative for opposite charges (attraction).

• As , (reference point).

Conservative Nature of the Electric Force

• The work done by the electric force depends only on the initial and final positions, not the path taken.

• This property allows the definition of electric potential energy.

Electric Potential (V)

Definition and Units

• Electric potential at a point is the potential energy per unit charge:

• Unit: volt (V), where .

• Electric potential is a property of the source charges and is independent of the test charge used to measure it.

Electric Potential Due to Point Charges

• The electric potential at a distance from a point charge is:

• The potential is a scalar and obeys the superposition principle: the total potential is the sum of the potentials from all charges.

Electric Potential in a Parallel-Plate Capacitor

• For a uniform electric field between plates separated by distance :

• The electric field can be found from the potential difference:

• Units: .

Potential Energy of Multiple Point Charges

• For a system of point charges, the total potential energy is the sum over all unique pairs:

$U = \sum_{i

• is the distance between charges and .

Electric Potential of a Continuous Charge Distribution

• For a continuous distribution, the potential at point is:

• Integrate over the entire charge distribution, where is the distance from to .

Electric Potential of a Ring of Charge

• For a ring of total charge and radius , the potential at a point on the axis a distance from the center is:

• At large distances (), this reduces to the potential of a point charge .

Potential Energy of a Dipole in an Electric Field

• An electric dipole with moment in a uniform field has potential energy:

• is the angle between and .

• The energy is minimum when the dipole is aligned with the field.

Problem-Solving Strategies

• Define the system and identify if energy is conserved.

• Draw before-and-after diagrams to clarify initial and final states.

• Apply conservation of energy: or .

• For continuous charge distributions, divide the charge into small elements and integrate.

Examples and Applications

• Proton Approaching a Charged Sphere: Use conservation of energy to find the speed needed for a proton to reach a charged sphere, treating the sphere as a point charge if it is much more massive than the proton.

• Escape Speed of Electron and Positron: Calculate the minimum speed required for two oppositely charged particles to escape each other's attraction, using at infinity as the reference.

• Potential of Two Charges: The net potential at a point is the sum of the potentials from each charge, as potential is a scalar quantity.

• Potential of a Ring of Charge: The potential on the axis of a ring is derived by summing contributions from each segment, resulting in a simple formula due to symmetry.

Key Terms

• Electric Potential (V): Energy per unit charge at a point in space.

• Potential Difference (ΔV): The change in electric potential between two points.

• Equipotential Surface: A surface on which the electric potential is constant.

• Electric Dipole: A pair of equal and opposite charges separated by a distance.

• Superposition Principle: The total potential is the sum of potentials from all sources.

Summary Table: Key Equations

Additional info: Some context and explanations have been expanded for clarity and completeness, as is standard in academic study guides.