Capacitors store electrical energy in their electric fields. This energy is proportional to the square of the charge and inversely proportional to capacitance. Understanding how capacitors store energy is crucial for many electrical applications.

The energy stored in a capacitor relates to the work done to charge it against the electric field. This concept connects to broader ideas of electric potential energy and fields, which are fundamental in electrostatics and circuit theory.

Energy Stored in a Capacitor

Energy storage in capacitors

• Energy stored in a capacitor directly proportional to the square of the charge on the capacitor (doubling charge quadruples energy)
• Energy stored in a capacitor inversely proportional to the capacitance (halving capacitance doubles energy)
• Formula for energy stored in a capacitor: $U = \frac{1}{2} \frac{Q^2}{C}$
  • $U$ = energy stored in joules (J)
  • $Q$ = charge in coulombs (C)
  • $C$ = capacitance in farads (F)
• Alternative formula using voltage across the capacitor: $U = \frac{1}{2} CV^2$
  • $V$ = voltage in volts (V)
• The energy stored is a form of electrostatic potential energy

Capacitor energy and electric fields

• Energy stored in a capacitor equals work done to charge it
  • External source (battery) moves charges from one plate to the other against the electric field
• Electric field between capacitor plates is uniform and perpendicular
  • Electric field magnitude: $E = \frac{V}{d}$ ($d$ = distance between plates)
• Electric field energy density (energy per unit volume): $u = \frac{1}{2} \varepsilon_0 E^2$
  • $\varepsilon_0$ = permittivity of free space ($8.85 \times 10^{-12}$ F/m)
• Total capacitor energy is energy density times volume between plates: $U = u \cdot Ad = \frac{1}{2} \varepsilon_0 E^2 Ad$
  • $A$ = area of each plate
• The potential difference between the plates determines the amount of energy stored

Applications of capacitor energy

• Parallel capacitor combination:
  1. Total capacitance is sum of individual capacitances: $C_{total} = C_1 + C_2 + ... + C_n$
  2. Voltage across each capacitor same and equal to source voltage
  3. Total energy stored is sum of energies in each capacitor: $U_{total} = U_1 + U_2 + ... + U_n$
• Series capacitor combination:
  1. Reciprocal of total capacitance is sum of reciprocals of individual capacitances: $\frac{1}{C_{total}} = \frac{1}{C_1} + \frac{1}{C_2} + ... + \frac{1}{C_n}$
  2. Charge on each capacitor same and equal to total charge
  3. Total energy stored is sum of energies in each capacitor: $U_{total} = U_1 + U_2 + ... + U_n$
• Defibrillators deliver controlled electric shock to heart to restore normal rhythm
  • Use capacitors to store large energy (100-400 J) and release quickly
  • Energy delivered: $U = \frac{1}{2} CV^2$ ($C$ = defibrillator capacitance, $V$ = voltage charged to)

Dielectrics and Capacitor Energy

• Dielectrics are insulating materials placed between capacitor plates
• Dielectrics increase the capacitance of a capacitor
• The work required to charge a capacitor with a dielectric is less than without
• Dielectrics affect the potential difference between the plates
• The energy stored in a capacitor with a dielectric is influenced by its capacitance