Lecture Notes on Capacitors

Introduction to Capacitors

• A capacitor stores electrical charge.
• It consists of two metal plates separated by an insulator (e.g., air, paper, water).
• Unlike a battery, a capacitor stores charge by moving electrons from one plate to another.

Capacitance

• Capacitance (C) is measured in farads (F).
• Equation: ( q = Cv ), where:
  • ( q ) = charge (coulombs)
  • ( C ) = capacitance (farads)
  • ( v ) = voltage (volts)
• Capacitance is a measure of charge efficiency (charge per volt).
  • Higher capacitance means more charge can be stored per volt.
• Example: Capacitor A (10 farads) vs. Capacitor B (2 farads)
  • A stores more charge than B at the same voltage.

Factors Affecting Capacitance

• Construction of the Capacitor
  • Capacitance depends on the capacitor's construction, not the voltage.
  • Formula: ( C = \varepsilon_0 \frac{A}{d} )
    • ( A ) = area of plates
    • ( d ) = distance between plates
    • ( \varepsilon_0 ) = permittivity of free space
• Increasing the plate area increases capacitance.
• Increasing the distance decreases capacitance.

Dielectric and Its Impact

• A dielectric is an insulating material that increases capacitance.
• Modified formula: ( C = k \varepsilon \frac{A}{d} )
  • ( k ) = dielectric constant (e.g., ( k = 1 ) for vacuum, ( 4.3 ) for quartz)
• Adding a dielectric increases capacitance and decreases voltage.

Charge and Voltage

• Voltage (V): ( V = \frac{W}{q} ) (work per unit charge)
• Charge on a capacitor is quantized and related to the electron's charge.
• Electric Field: ( E = \frac{V}{d} )
  • Related to surface charge density and permittivity.

Energy Stored in a Capacitor

• Energy formulas:
  • ( U = \frac{1}{2} qV )
  • ( U = \frac{1}{2} CV^2 )
  • ( U = \frac{q^2}{2C} )

Charging and Discharging a Capacitor

• Charging: Electrons move from one plate to another when connected to a battery.
  • The battery pumps electrons, causing one plate to be positive and the other negative.
• Discharging: Occurs when electrons flow back, equalizing charge across plates.
  • Capacitors release stored energy through a circuit (e.g., lighting a bulb).

Practical Considerations

• Common capacitors range from microfarads to picofarads.
• Supercapacitors can have capacitance in farads but are rare.

Summary

• Understanding capacitors involves knowledge of electric charge, voltage, capacitance, and energy storage.
• Key relationships and formulas govern how capacitors are used and applied in circuits.