Electric Potential and Electrical Potential Energy

Conservative Nature of Electric Forces

Electric forces are conservative, meaning the work done by these forces is independent of the path taken. This property allows us to define a potential energy associated with electric forces, similar to gravitational potential energy.

• Electrical potential energy is a scalar quantity.

• Electrical potential (often called voltage) is defined as electrical potential energy per unit charge.

• The voltage between two points is the difference in electrical potential between them.

When a test charge q0 is placed in an electric field E, it experiences a force F = q0E. The work done in moving the charge is:

• By definition,

This represents the change in electrical potential energy.

Potential Energy of Point Charges

The potential energy required to bring a charge q0 from infinity to a distance r from a point charge q is given by:

• Where k is Coulomb's constant,

Example: Electric Potential Energy of a Group of Charges

To find the electric potential energy of a system of point charges, sum the potential energies for each pair:

Electrical Potential and Units

Definition and SI Units

Electrical potential (V) is defined as the potential energy per unit charge:

• SI unit: volt (V), where

The potential difference is the work per unit charge required to move a test charge from point A to point B:

Electron-Volt

An electron-volt (eV) is a unit of energy commonly used in atomic and nuclear physics:

• 1 eV is the energy gained by a charge of magnitude e (elementary charge) when moved through a potential difference of 1 volt.

Electric Potential in a Uniform Field

Parallel-Plate Capacitor

Inside a parallel-plate capacitor, the electric field is uniform and the potential drops linearly in the direction of the field lines.

• Potential difference:

• Where E is the electric field strength and d is the plate separation.

Equipotential Surfaces

Equipotential surfaces are surfaces where every point has the same electric potential. In a uniform field, planes perpendicular to the field are equipotential surfaces.

• Points B and C are at the same potential, lower than point A.

• Moving along an equipotential surface requires no work.

Charged Particle in a Uniform Field

Motion and Energy Conservation

A positive charge released from rest in a uniform electric field moves in the direction of the field. The change in potential is negative along the field lines, and conservation of energy can be used to find its speed.

Example: Proton in a Capacitor

A proton fired from the midpoint of a capacitor may not have enough speed to reach the positive plate, but its speed as it collides with the negative plate can be calculated using energy conservation.

Electric Potential Due to Point Charges

Path Independence

The potential difference between two points due to a point charge is independent of the path taken.

Electric Potential of a Point Charge

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

Superposition Principle: Multiple Charges

The total electric potential due to several point charges is the sum of the potentials from each charge:

Electric Potential of a Dipole

An electric dipole consists of two equal and opposite charges separated by a distance. The potential at a point depends on the positions and the dipole orientation.

Electric Potential from Continuous Charge Distributions

Uniformly Charged Ring

For a continuous charge distribution, the electric potential is calculated by integrating over the charge:

Electric Potential Due to Conductors

Charged Conductors

Inside a charged conductor, the electric field is zero and the potential is constant. The potential outside depends on the geometry and charge distribution.

Connected Charged Spheres

When two charged spheres are connected, their potentials equalize. The relationship between their charges and radii is:

Applications and Phenomena

Corona Discharge

If the electric field near a conductor is strong enough, it can ionize air molecules, leading to corona discharge. This process is important in phenomena like lightning and sparks at gas stations.

Millikan Oil-Drop Experiment

Measuring the Elementary Charge

The Millikan oil-drop experiment determined the value of the elementary charge by balancing gravitational and electric forces on tiny oil droplets.

• Setup: Oil droplets are sprayed into a chamber, and their motion is observed under the influence of an electric field.

• Forces: (electric), (gravitational), and (drag).

Summary Table: Key Concepts in Electric Potential

Additional info:

• Some examples and diagrams were expanded for clarity and completeness.

• Images were included only when directly relevant to the explanation.