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
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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: [object Object],[object Object]
U U U = energy stored in joules (J)
Q Q Q = charge in coulombs (C)
C C C = capacitance in farads (F)
Alternative formula using voltage across the capacitor: [object Object],[object Object]
V V 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: [object Object],[object Object] (d d d = distance between plates)
Electric field energy density (energy per unit volume): [object Object],[object Object]
ε 0 \varepsilon_0 ε 0 = permittivity of free space (8.85 × 1 0 − 12 8.85 \times 10^{-12} 8.85 × 1 0 − 12 F/m)
Total capacitor energy is energy density times volume between plates: U = u ⋅ A d = 1 2 ε 0 E 2 A d U = u \cdot Ad = \frac{1}{2} \varepsilon_0 E^2 Ad U = u ⋅ A d = 2 1 ε 0 E 2 A d
The potential difference between the plates determines the amount of energy stored
Applications of capacitor energy
Parallel capacitor combination:
Total capacitance is sum of individual capacitances: [object Object],[object Object]
Voltage across each capacitor same and equal to source voltage
Total energy stored is sum of energies in each capacitor: U t o t a l = U 1 + U 2 + . . . + U n U_{total} = U_1 + U_2 + ... + U_n U t o t a l = U 1 + U 2 + ... + U n
Series capacitor combination:
Reciprocal of total capacitance is sum of reciprocals of individual capacitances: [object Object],[object Object]
Charge on each capacitor same and equal to total charge
Total energy stored is sum of energies in each capacitor: U t o t a l = U 1 + U 2 + . . . + U n U_{total} = U_1 + U_2 + ... + U_n U t o t a l = 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 = 1 2 C V 2 U = \frac{1}{2} CV^2 U = 2 1 C V 2 (C C C = defibrillator capacitance, V V 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