Electric Potential from the Electric Field

Definition and Calculation

The electric potential at a point in space is related to the work done by the electric field in moving a charge between two points. The potential difference between points a and b is given by:

• Formula:

• Application to Parallel-Plate Capacitor: For a uniform field between plates, set reference at the negatively charged plate. The potential at a point is:

• Uniform Field: (for negative direction) (potential varies linearly with distance in uniform field)

• Potential Difference Between Plates:

Graphical Interpretation

• The area under the vs. graph gives the potential difference:

• Example: If decreases linearly from $2000 over $1V(1) = -1000$ V.

Potential of a Point Charge

Formula and Derivation

The electric potential due to a point charge at a distance is:

• Formula:

• Derivation:

Solid Sphere with Uniform Charge

• For a sphere of radius and total charge :

• Inside ():

• Outside ():

• Potential at center ():

Electric Field from the Potential

Relationship and Calculation

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

• Formula:

• In three dimensions:

• If potential is constant in a region, the electric field is zero there.

• Adding a constant to the potential does not change the electric field.

Equipotential Surfaces

Properties and Examples

Equipotential surfaces are surfaces where the electric potential is the same at every point.

• Key Properties:

  • Electric field lines are always perpendicular to equipotential surfaces.

  • No work is required to move a charge along an equipotential surface.

  • Equipotential surfaces are not physical objects but mathematical constructs.

• Examples:

  • For a point charge, equipotential surfaces are concentric spheres centered on the charge.

  • For parallel plates, equipotential surfaces are planes parallel to the plates.

Conductors in Electrostatic Equilibrium

Properties of Conductors

• Electric field inside a conductor is zero in electrostatic equilibrium.

• Potential is constant everywhere inside and on the surface of a conductor.

• Surface charge density is highest at points of smallest radius of curvature (sharp points).

• All points on a conductor connected by a wire have the same potential.

Capacitance and Capacitors

Definition and Parallel Plate Capacitor

• Capacitance (): Units: Farad (F), where

• Parallel Plate Capacitor: where is plate area, is separation, is vacuum permittivity.

• Capacitance depends only on geometry and material, not on or .

Energy Stored in a Capacitor

• Formula:

• Energy is stored in the electric field between the plates.

• Energy density:

Dielectrics and Capacitors

Dielectric Materials

• Dielectrics are insulating materials that increase capacitance when placed between capacitor plates.

• Dielectric constant ():

• Dielectric reduces the effective electric field inside the capacitor.

• Dielectric strength: maximum field a material can withstand without breakdown.

Capacitors in Circuits

Series and Parallel Combinations

• Parallel: All capacitors share the same voltage; total charge is sum of individual charges.

• Series: All capacitors share the same charge; total voltage is sum of individual voltages.

Example Calculations

• Given capacitors in series or parallel, use above formulas to find equivalent capacitance, charge, and voltage across each.

• For a series connection to a battery:

• For a parallel connection:

Batteries and Electromotive Force (emf)

Battery Operation

• Batteries maintain a potential difference between terminals by converting chemical energy to electrical energy.

• Electromotive force (emf): Units: Volt (V)

• When connected to a load, the battery supplies current until reactants are depleted.

Summary Table: Capacitance Formulas

Key Terms

• Electric Potential (V): Energy per unit charge due to electric field.

• Capacitance (C): Ability to store electric charge per unit potential difference.

• Dielectric: Insulating material that increases capacitance.

• Equipotential Surface: Surface of constant electric potential.

• Electrostatic Equilibrium: State where charges in a conductor are at rest and field inside is zero.

• Electromotive Force (emf): Energy supplied per unit charge by a battery or source.

Additional info: Some table entries and explanations were expanded for completeness and clarity based on standard physics curriculum.