A capacitor is a component that stores charge and electrical energy. We will study capacitance, energy, and the effects of dielectrics.
Basic Knowledge
Capacitor: A component consisting of two conductor plates facing each other.
Capacitance C [F = C/V]:Q = CV (Q: stored charge, V: voltage between the plates)
Capacitance of a parallel-plate capacitor:C = ε₀S/d (S: plate area, d: distance between plates)
When a dielectric with relative permittivity ε_r is inserted: C = ε_r ε₀S/d
Electrostatic energy:U = Q²/(2C) = CV²/2 = QV/2
Capacitor connections:
・Parallel: C = C₁ + C₂ (common voltage)
・Series: 1/C = 1/C₁ + 1/C₂ (common charge)
Capacitance C (capacitance) (The ability to store charge. Unit: F = C/V)
Farad F (Unit of capacitance. 1 F = 1 C/V)
Dielectric and relative permittivity ε_r (Inserting an insulator between the plates increases C by a factor of ε_r)
Electrostatic energy U (Energy stored in a capacitor. U = CV²/2)
Series and parallel connections (Calculation of equivalent capacitance. The reverse pattern of resistors)
Further Study (Background and Significance)
Capacitor connections are the reverse pattern of resistor connections: resistor series → capacitor parallel, and resistor parallel → capacitor series, so the formulas are reversed.
When a dielectric is inserted, the result differs depending on whether the battery remains connected (constant voltage) or the capacitor is isolated (constant charge). With constant voltage, Q increases according to Q = CV, and the energy also increases. With constant charge, the voltage decreases and the energy decreases (the difference becomes the work done by the force pulling in the dielectric).
Supercapacitors are used in regenerative braking in electric vehicles and other applications. They charge and discharge faster than batteries.
- Q = CV (basic equation)
- C = ε₀S/d (parallel plates)
- U = CV²/2 = Q²/2C (energy)
- Parallel: C_total = ΣC (common voltage)
- Series: 1/C_total = Σ(1/C) (common charge)
- Dielectric: C → ε_r × C
- Energy changes differently depending on the connection
Points to Note (Easily Confused)
① Series and parallel connections of capacitors are the reverse of those of resistors. ② Energy is CV²/2, so it is proportional to V². ③ In a series connection, the charge stored on each capacitor is the same (common Q). ④ The conditions after inserting a dielectric differ depending on whether the battery is connected, so the conditions must be checked.
Practice
- When a voltage of 100 V is applied to a capacitor with capacitance 4.0 μF, find the stored charge and electrostatic energy.
- Find the equivalent capacitance when 2.0 μF and 3.0 μF capacitors are connected in parallel and in series.
- What happens to the capacitance when the distance between the plates of a parallel-plate capacitor is doubled?