Formula Sheet: Capacitors

Table of Contents
1. Capacitor Fundamentals
2. Energy Storage in Capacitors
3. Capacitors in Circuits
4. RC Circuits (Time-Dependent Behavior)
5. Dielectric Properties
6. Important Relationships and Concepts
7. Special Cases and Applications
View more

Capacitor Fundamentals

Capacitance Definition

• Capacitance: \[ C = \frac{Q}{V} \]
  • \( C \) = capacitance (farads, F)
  • \( Q \) = magnitude of charge on each plate (coulombs, C)
  • \( V \) = potential difference between plates (volts, V)
  • 1 farad = 1 coulomb/volt
  • Capacitance is a measure of a capacitor's ability to store charge per unit voltage

Parallel Plate Capacitor

• Capacitance of parallel plates: \[ C = \frac{\epsilon_0 A}{d} \]
  • \( \epsilon_0 \) = permittivity of free space = 8.85 × 10^-12 F/m
  • \( A \) = area of one plate (m²)
  • \( d \) = separation distance between plates (m)
  • Capacitance increases with larger plate area
  • Capacitance decreases with greater separation distance
• With dielectric material: \[ C = \frac{\kappa \epsilon_0 A}{d} = \kappa C_0 \]
  • \( \kappa \) = dielectric constant (dimensionless, κ ≥ 1)
  • \( C_0 \) = capacitance without dielectric (vacuum/air)
  • Dielectrics increase capacitance by factor κ
  • For air/vacuum, κ = 1

Electric Field in Capacitor

• Electric field between parallel plates: \[ E = \frac{V}{d} \]
  • \( E \) = electric field strength (V/m or N/C)
  • \( V \) = voltage across plates (V)
  • \( d \) = plate separation (m)
  • Field is uniform between plates (ignoring edge effects)
• Electric field in terms of charge: \[ E = \frac{\sigma}{\epsilon_0} \]
  • \( \sigma \) = surface charge density = Q/A (C/m²)
  • Valid for parallel plate capacitor
• With dielectric: \[ E = \frac{E_0}{\kappa} \]
  • \( E_0 \) = field without dielectric
  • Dielectric reduces the electric field

Energy Storage in Capacitors

Energy Stored

• Energy stored in capacitor (three equivalent forms): \[ U = \frac{1}{2}CV^2 = \frac{1}{2}QV = \frac{Q^2}{2C} \]
  • \( U \) = stored electrical potential energy (joules, J)
  • \( C \) = capacitance (F)
  • \( V \) = voltage (V)
  • \( Q \) = charge (C)
  • Use the form most convenient for given variables
• Energy density: \[ u = \frac{U}{\text{volume}} = \frac{1}{2}\epsilon_0 E^2 \]
  • \( u \) = energy per unit volume (J/m³)
  • \( E \) = electric field strength (V/m)
  • Represents energy stored in the electric field itself
• Energy density with dielectric: \[ u = \frac{1}{2}\kappa \epsilon_0 E^2 \]
  • Energy density increases with dielectric material

Capacitors in Circuits

Capacitors in Series

• Equivalent capacitance (series): \[ \frac{1}{C_{eq}} = \frac{1}{C_1} + \frac{1}{C_2} + \frac{1}{C_3} + ... \]
  • Series capacitors add reciprocally
  • Equivalent capacitance is always less than smallest individual capacitance
• For two capacitors in series: \[ C_{eq} = \frac{C_1 C_2}{C_1 + C_2} \]
  • Simplified formula for two capacitors only
• Properties of series capacitors:
  • All capacitors have same charge: \( Q_1 = Q_2 = Q_3 = ... \)
  • Voltages add: \( V_{total} = V_1 + V_2 + V_3 + ... \)

Capacitors in Parallel

• Equivalent capacitance (parallel): \[ C_{eq} = C_1 + C_2 + C_3 + ... \]
  • Parallel capacitors add directly
  • Equivalent capacitance is greater than any individual capacitance
• Properties of parallel capacitors:
  • All capacitors have same voltage: \( V_1 = V_2 = V_3 = ... \)
  • Charges add: \( Q_{total} = Q_1 + Q_2 + Q_3 + ... \)

RC Circuits (Time-Dependent Behavior)

Charging Capacitor

• Charge as function of time: \[ Q(t) = Q_{max}(1 - e^{-t/RC}) \]
  • \( Q_{max} \) = maximum charge = \( CV_0 \)
  • \( V_0 \) = battery/source voltage
  • \( R \) = resistance (Ω)
  • \( C \) = capacitance (F)
  • \( t \) = time (s)
  • Charge increases exponentially to maximum value
• Voltage across capacitor during charging: \[ V_C(t) = V_0(1 - e^{-t/RC}) \]
  • \( V_C(t) \) = voltage across capacitor at time t
  • Approaches \( V_0 \) as \( t \to \infty \)
• Current during charging: \[ I(t) = I_0 e^{-t/RC} = \frac{V_0}{R}e^{-t/RC} \]
  • \( I_0 \) = initial current = \( V_0/R \)
  • Current decreases exponentially
  • Maximum at t = 0, approaches zero as t → ∞

Discharging Capacitor

• Charge during discharge: \[ Q(t) = Q_0 e^{-t/RC} \]
  • \( Q_0 \) = initial charge on capacitor
  • Charge decreases exponentially to zero
• Voltage during discharge: \[ V_C(t) = V_0 e^{-t/RC} \]
  • \( V_0 \) = initial voltage across capacitor
  • Voltage decreases exponentially
• Current during discharge: \[ I(t) = -I_0 e^{-t/RC} = -\frac{V_0}{R}e^{-t/RC} \]
  • Negative sign indicates opposite direction to charging current
  • Magnitude decreases exponentially

Time Constant

• RC time constant: \[ \tau = RC \]
  • \( \tau \) = time constant (seconds)
  • \( R \) = resistance (Ω)
  • \( C \) = capacitance (F)
  • Time for capacitor to charge to 63% of maximum (or discharge to 37% of initial)
• Key time constant values:
  • After 1τ: 63.2% charged (or 36.8% remaining)
  • After 2τ: 86.5% charged (or 13.5% remaining)
  • After 3τ: 95.0% charged (or 5.0% remaining)
  • After 4τ: 98.2% charged (or 1.8% remaining)
  • After 5τ: 99.3% charged (or 0.7% remaining) - considered "fully" charged/discharged

Dielectric Properties

Effect of Dielectric on Capacitor Properties

• Capacitance with dielectric: \[ C = \kappa C_0 \]
  • Capacitance increases by factor κ
• Electric field reduction: \[ E = \frac{E_0}{\kappa} \]
  • Electric field decreases by factor κ
• For battery connected (constant voltage):
  • Voltage remains constant: \( V = V_0 \)
  • Charge increases: \( Q = \kappa Q_0 \)
  • Capacitance increases: \( C = \kappa C_0 \)
  • Energy stored increases: \( U = \kappa U_0 \)
• For isolated capacitor (constant charge):
  • Charge remains constant: \( Q = Q_0 \)
  • Voltage decreases: \( V = V_0/\kappa \)
  • Capacitance increases: \( C = \kappa C_0 \)
  • Energy stored decreases: \( U = U_0/\kappa \)

Dielectric Strength

• Dielectric breakdown:
  • Maximum electric field a dielectric can withstand before conducting
  • If \( E > E_{breakdown} \), dielectric fails and conducts current
  • Different materials have different breakdown strengths
• Maximum voltage before breakdown: \[ V_{max} = E_{breakdown} \times d \]
  • \( E_{breakdown} \) = dielectric strength of material (V/m)
  • \( d \) = separation distance (m)

Important Relationships and Concepts

Power in Capacitive Circuits

• Instantaneous power: \[ P(t) = V(t) \times I(t) \]
  • Power varies with time in RC circuits
  • During charging: power delivered from source
  • During discharging: power dissipated in resistor

Charge and Current Relationship

• Current as rate of charge flow: \[ I = \frac{dQ}{dt} \]
  • Fundamental relationship between current and charge
  • Useful for deriving RC circuit equations

Voltage-Charge-Capacitance Triangle

• Three related equations:
  • \( Q = CV \)
  • \( V = Q/C \)
  • \( C = Q/V \)
  • Any one can be derived from the others

Work Done in Charging

• Work to charge capacitor: \[ W = \int_0^Q V \, dq = \int_0^Q \frac{q}{C} \, dq = \frac{Q^2}{2C} = U \]
  • Work done equals energy stored
  • Voltage increases linearly with charge during charging
  • Average voltage during charging is \( V/2 \), not \( V \)

Special Cases and Applications

Cylindrical Capacitor

• Capacitance of cylindrical capacitor: \[ C = \frac{2\pi \epsilon_0 L}{\ln(b/a)} \]
  • \( L \) = length of cylinder (m)
  • \( a \) = radius of inner conductor (m)
  • \( b \) = radius of outer conductor (m)
  • Less commonly tested on MCAT

Spherical Capacitor

• Capacitance of spherical capacitor: \[ C = 4\pi \epsilon_0 \frac{ab}{b-a} \]
  • \( a \) = radius of inner sphere (m)
  • \( b \) = radius of outer sphere (m)
  • Less commonly tested on MCAT

Energy Considerations

• Energy dissipated in resistor during discharge: \[ E_{dissipated} = \frac{1}{2}CV_0^2 \]
  • All energy initially stored in capacitor is dissipated as heat in resistor
  • Independent of resistance value
• Energy supplied by battery during charging: \[ E_{battery} = CV_0^2 \]
  • Battery supplies twice the energy stored in capacitor
  • Half is stored in capacitor, half dissipated in resistor as heat