Capacitance and Electric Energy Storage

Capacitance Overview

• Capacitance: Ability of a capacitor to store electric charge.
• Capacitors: Composed of two conductors close together, not touching.
  • Connected to a battery, the charge on plates is proportional to voltage.

Key Concepts

• Capacitance (C): Defined as the charge (Q) per unit voltage (V).
• Unit of Capacitance: Farad (F) - 1 F = 1 C/V.

Determination of Capacitance

• Parallel-Plate Capacitor:
  • Electric field (E) between plates: $E = \frac{Q}{\varepsilon<em>0 A}$ where ( \varepsilon0 ) is the permittivity of empty space.
  • Potential difference (V) derived from:
    $V<em>{ba} = \frac{Q d}{\varepsilon</em>0 A}$
• Formula for Capacitance: $C = \frac{Q}{V}$
  • Substitute V:
    $C = \frac{\varepsilon_0 A}{d}$

Example Calculation - Parallel-Plate Capacitor

1. Geometry: Plates are 20 cm x 3.0 cm apart with a 1.0 mm air gap.
2. Capacitance Calculation: Use the formula for capacitance with the area and distance values.
3. Charge Calculation: Using a 12-V battery, the charge on each plate can be calculated:
   $Q = C \times V$
4. Electric Field Calculation: Evaluating the electric field between plates.
5. Capacitance for 1 F: Estimate the area required to achieve a capacitance of 1 F.

Capacitors in Series and Parallel

• Parallel: Each capacitor experiences the same voltage.
  • Effective Capacitance:
    $C<em>{eq} = C</em>1 + C<em>2 + C</em>3$
• Series: Each capacitor carries the same charge:
  • Effective Capacitance:
    $\frac{1}{C<em>{eq}} = \frac{1}{C</em>1} + \frac{1}{C<em>2} + \frac{1}{C</em>3}$

Example Calculation - Series and Parallel Capacitors

• Determine capacitance for given configurations to find equivalent capacitance and charge/voltage across each.

Electric Energy Storage

• Charged capacitors store electric energy as work is done to charge them.
• Energy Stored (U): $U = \frac{1}{2} C V^2$
  • For a flash unit with a 150-μF capacitor at 200 V, calculate energy and power output when discharged over 1 ms.

Energy Changes with Plate Separation

• Discuss how energy changes if plate separation is increased while maintaining charge.
• Compare initial and final energies when plates pull apart with static charge.

Dielectrics

• Definition: An insulator characterized by dielectric constant K.
  • Increases capacitance when inserted into a capacitor.
  • Dielectric Strength: Maximum field a dielectric can withstand without breakdown.

Effect of Dielectric on Capacitors

• Two scenarios for inserting a dielectric:
  1. Connected to Battery: Constant voltage; capacitance and charge increase.
  2. Disconnected: Constant charge; capacitance increases, and voltage drops.

Example Calculation with Dielectrics

• Given a dielectric with constant K = 3.4, determine new capacitance, charge, field strength, and energy after dielectric is removed with charge preserved.

Summary of Important Equations

• Capacitance: $C = \frac{Q}{V}$
• Capacitors in Series: $\frac{1}{C<em>{eq}} = \frac{1}{C</em>1} + \frac{1}{C_2} + …$
• Capacitors in Parallel: $C<em>{eq} = C</em>1 + C_2 + …$
• Energy Stored: $U = \frac{1}{2} C V^2$
• Electric Field with Dielectric: If introduced, the field reduces due to induced polarization.