Electric Potential in a Parallel-Plate Capacitor

Potential Difference in a Capacitor

The electric potential inside a parallel-plate capacitor is determined by the uniform electric field between its plates. The potential is defined to be zero at the negative plate.

• Uniform Electric Field: The electric field E inside a parallel-plate capacitor is constant in magnitude and direction.

• Potential Calculation: The potential difference V between two points separated by distance s is found by integrating the electric field:

• Boundary Conditions: If V = 0 at the negative plate, the potential increases linearly to V = Ed at the positive plate, where d is the plate separation.

• Physical Meaning: The potential difference is directly related to the work needed to move a charge between the plates.

• Example: For a capacitor with E = 1000\,\text{V/m} and d = 0.01\,\text{m}, the potential difference is V = 10\,\text{V}.

Finding the Electric Field from the Potential

Relationship Between Field and Potential

The electric field is related to the spatial rate of change (slope) of the electric potential.

• Definition: The electric field in the s-direction is given by the negative gradient of the potential:

• Physical Interpretation: The field points in the direction of decreasing potential.

• Example: For a point charge, if V = \frac{q}{4\pi\epsilon_0 r}, then E_r = -\frac{dV}{dr} = \frac{q}{4\pi\epsilon_0 r^2}.

Electric Field of a Ring of Charge

On-Axis Potential and Field

The electric field along the axis of a ring of charge can be derived from its potential.

• Potential on Axis:

• Field Calculation: The field at position z is found by differentiating the potential with respect to z.

Graphical Analysis: Field from Potential Slope

Interpreting Graphs

The electric field is the negative of the slope of the potential graph. Regions of constant potential correspond to zero electric field.

• Piecewise Slope: For different regions, calculate E_x = -\Delta V / \Delta x.

• Example: If \Delta V = 20\,\text{V} over \Delta x = 0.02\,\text{m}, then E_x = -1000\,\text{V/m}.

• Zero Field: Where the potential is flat, the electric field is zero.

Equipotential Surfaces and Field Geometry

Properties of Equipotentials

Equipotential surfaces are always perpendicular to electric field lines. The field points in the direction of decreasing potential, and the field strength is inversely proportional to the spacing between equipotentials.

• Key Properties:

  • Field lines are perpendicular to equipotentials.

  • Equal potential differences between surfaces.

  • Closer equipotentials indicate stronger fields.

Kirchhoff's Loop Law

Conservation of Energy in Circuits

Kirchhoff's loop law states that the sum of all potential differences around any closed loop in a circuit is zero.

• Mathematical Statement:

• Application: Used to analyze complex circuits and determine unknown voltages or currents.

Conductors in Electrostatic Equilibrium

Properties of Conductors

When a conductor is in electrostatic equilibrium, several important properties hold:

• All excess charge resides on the surface.

• The surface is an equipotential.

• The electric field inside the conductor is zero.

• The external field is perpendicular to the surface.

• The field is strongest at sharp corners.

Example: A metal sphere near a flat plate shows equipotential surfaces matching the electrode shapes.

Sources of Electric Potential

Charge Separation and Potential Difference

A separation of charge creates an electric potential difference. Everyday examples include static electricity from walking on a carpet, which can cause sparks due to the potential difference created.

Batteries and Emf

Electromotive Force (Emf)

Batteries use chemical reactions to separate charges, creating a potential difference (emf) between terminals.

• Definition: Emf is the work done per charge to move positive charges from the negative to the positive terminal.

• Terminal Voltage: In an ideal battery, the terminal voltage equals the emf.

• Batteries in Series: The total potential difference is the sum of individual emfs:

Capacitance and Capacitors

Definition and Properties

Capacitance is a measure of a system's ability to store charge per unit potential difference.

• Formula:

• SI Unit: Farad (F), where .

• Geometric Nature: Capacitance depends only on the geometry (area and separation) of the electrodes.

• Applications: Capacitors are used in circuits for energy storage, filtering, and timing.

• Example: Keyboard switches use capacitors whose capacitance changes when keys are pressed.

Charging a Capacitor

When a capacitor is connected to a battery, current flows briefly until the capacitor is fully charged and reaches electrostatic equilibrium.

• Capacitance always refers to the charge per voltage on a fully charged capacitor.