Capacitors and Dielectrics Notes

Capacitance and Dielectrics

The effectiveness of a capacitor depends on the material used between its plates, known as the dielectric. Different materials can be used as dielectrics, including air, oil-coated paper, polystyrene (Styrofoam), glass, and water. The medium, whether solid, air, or liquid, maintains separation of charges, storing energy by maintaining an electric field. This energy can be released quickly.

Key Factors and Formulas

The capacitance is affected by:

  • Area (A): Larger area allows for greater charge storage.
  • Distance (d): Closer plates result in a stronger electric field.

Epsilon Naught (\epsilon_0)

\epsilon_0 (epsilon naught) is the absolute permittivity of free space, a constant representing the amount of charge that can be stored in a vacuum. It is usually provided on formula sheets.

Relative Permittivity (\epsilon_r)

Insulating material is called a dielectric, allowing the capacitor to store more charge. The higher the relative permittivity (\epsilonr), the higher the charge. \epsilonr is material-dependent. The formula is:

C = \epsilon0 \epsilonr \frac{A}{d}

where

  • C is capacitance
  • \epsilon_0 is the absolute permittivity of free space
  • \epsilon_r is the relative permittivity of the dielectric material
  • A is the area of the plates
  • d is the distance between the plates

If the dielectric is just air, the formula simplifies to:

C = \epsilon_0 \frac{A}{d}

Charging and Charge Storage

When charging a capacitor:

  1. The plate connected to the negative terminal accepts electrons (stores negative charge).
  2. The plate connected to the positive terminal loses electrons (stores positive charge).

When fully charged, electron flow stops, and both plates have equal and opposite charges. The voltage across the plates matches the supply voltage, and an electric field exists between the plates. This charge can be released rapidly, like in a camera flash.

Factors Affecting Charge Storage

  1. Plate Area: A larger area increases charge storage.
  2. Distance Between Plates: Closer plates strengthen the electric field.
  3. Dielectric Constant: A higher dielectric constant enhances charge separation (polarization).

Air has a low dielectric constant, making it less effective at separating charges compared to better insulators.

All these factors increase capacitance (C), making the capacitor more powerful and able to store more charge.

Capacitors in Series and Parallel

Series

Capacitors in series have the same charge, and total capacitance is calculated using the reciprocal formula:

\frac{1}{C{total}} = \frac{1}{C1} + \frac{1}{C_2} + …

Parallel

Capacitors in parallel simply add:

C{total} = C1 + C_2 + …

Charge (Q) Calculation

Q (charge) can be calculated using:

Q = CV

where

  • C is capacitance
  • V is voltage

Electrical Energy (E)

Electrical energy stored in a capacitor can be calculated using:

E = \frac{1}{2}QV

Other forms:

E = \frac{1}{2}CV^2

This is derived by substituting Q=CV into the first equation:

E = \frac{1}{2}(CV)V = \frac{1}{2}CV^2

Energy Calculation from a Graph

If given a voltage-charge graph, the energy stored is the area under the curve.

Time Constant (τ)

The time constant ($\tau$) measures the time taken to charge to 63% or discharge to 37% (which is 100% - 63%) of the original value. Use 63% as the key number.

  • Charging: Time to reach 63% of full charge.
  • Discharging: Time to drop to 63% of the initial charge.

It takes approximately five time constants (5$\tau$) to fully charge or discharge a capacitor.

The formula for the time constant is:

\tau = RC

where

  • R is resistance
  • C is capacitance

Each time constant represents 63% of the remaining charge or discharge. This is an exponential decay so the amount decreases over time.

Charging Behavior

During charging, the voltage across the capacitor (VC) and the voltage across the resistor (VR) add up to the supply voltage (V_S):

VS = VC + V_R

As VC increases, VR decreases (since VS is constant), causing the current to decrease. The current stops when the capacitor is fully charged, and VC reaches its maximum.

Discharging Behavior

During discharging, there is no supply voltage, and the capacitor acts as the voltage source. As V_C decreases, the current also decreases until the capacitor is fully discharged.

Voltage over Time Constants Formula

V = V_{max} (1 - 0.37^n)

Where:

  • V = voltage of capacitance
  • V_{max} = supply voltage
  • n = number of time constants.