Capacitors

Textbook

• University Physics Volume 2: Chapter 8 - Capacitance
• University Physics Volume 2: Chapter 10.5 - RC Circuits

What is a Capacitor and what is Capacitance

A capacitor is an electronic component that stores electrical energy in an electric field, created between two conductive plates separated by an insulating material (called a dielectric). Capacitance is the measure of a capacitor's ability to store charge per unit voltage, typically measured in farads (F).

Capacitors are widely used in various applications, including:

• **Energy storage** (e.g., in power supplies)
• **Filtering** (e.g., in electronic circuits to smooth out fluctuations in voltage)
• **Timing circuits** (e.g., controlling signal timing in oscillators)
• **Signal coupling and decoupling** (e.g., in audio equipment).

Their ability to store and release energy quickly makes them essential in electronics.

Formulas

Capacitance Definition

The capacitance C of a capacitor is defined as the ratio of the charge Q stored on one plate to the voltage V across the plates. $C={\frac {Q}{V}}$

• Where:*
• *C* is the capacitance (in farads, F)
• *Q* is the charge (in coulombs, C)
• *V* is the voltage (in volts, V)

Capacitance of a Parallel Plate Capacitor

For a parallel plate capacitor, the capacitance depends on the area A of the plates, the separation d between them, and the permittivity ε of the dielectric material between the plates. $C={\frac {\varepsilon A}{d}}$

• Where:*
• *C* is the capacitance (in farads, F)
• *ε* is the permittivity of the dielectric material (in farads per meter, F/m)
• *A* is the area of one plate (in square meters, m²)
• *d* is the separation between the plates (in meters, m)

Energy Stored in a Capacitor

The energy E stored in a charged capacitor is proportional to its capacitance and the square of the voltage across it. $E={\frac {1}{2}}CV^{2}$

• Where:*
• *E* is the energy stored (in joules, J)
• *C* is the capacitance (in farads, F)
• *V* is the voltage (in volts, V)

Energy Density in a Capacitor

The energy density u represents the energy stored per unit volume in the electric field between the plates. $u={\frac {1}{2}}\varepsilon E^{2}$

• Where:*
• *u* is the energy density (in joules per cubic meter, J/m³)
• *ε* is the permittivity of the material (in farads per meter, F/m)
• *E* is the electric field strength (in volts per meter, V/m)

Equivalent Capacitance in Series

For capacitors connected in series, the reciprocal of the total (or equivalent) capacitance is the sum of the reciprocals of the individual capacitances. ${\frac {1}{C_{\text{eq}}}}={\frac {1}{C_{1}}}+{\frac {1}{C_{2}}}+\cdots +{\frac {1}{C_{n}}}$

• Where:*
• *C_eq* is the equivalent capacitance (in farads, F)
• *C_1, C_2, \dots, C_n* are the individual capacitances (in farads, F)

Equivalent Capacitance in Parallel

For capacitors connected in parallel, the total capacitance is the sum of the individual capacitances. $C_{\text{eq}}=C_{1}+C_{2}+\cdots +C_{n}$

• Where:*
• *C_eq* is the equivalent capacitance (in farads, F)
• *C_1, C_2, \dots, C_n* are the individual capacitances (in farads, F)

Capacitor Discharge (Voltage over Time)

During the discharge of a capacitor through a resistor, the voltage decreases exponentially over time. $V(t)=V_{0}e^{-{\frac {t}{RC}}}$

• Where:*
• *V(t)* is the voltage at time t (in volts, V)
• *V_0* is the initial voltage (in volts, V)
• *R* is the resistance (in ohms, Ω)
• *C* is the capacitance (in farads, F)
• *t* is time (in seconds, s)

Capacitor Charging (Voltage over Time)

During charging, the voltage across a capacitor increases exponentially, approaching its final value. $V(t)=V_{0}\left(1-e^{-{\frac {t}{RC}}}\right)$

• Where:*
• *V(t)* is the voltage at time t (in volts, V)
• *V_0* is the final voltage (in volts, V)
• *R* is the resistance (in ohms, Ω)
• *C* is the capacitance (in farads, F)
• *t* is time (in seconds, s)

Simulations

• Charging And Discharging A Capacitor
• An Example Of A Natural Capacitor
• Factors Affecting Capacitance
• PhET Capacitor Lab Simulations

Other Links

• Charged Parallel Plates
• RC circuit simulator