Difference between revisions of "Capacitors"
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*[https://openstax.org/books/university-physics-volume-2/pages/8-introduction University Physics Volume 2: Chapter 8 - Capacitance] | *[https://openstax.org/books/university-physics-volume-2/pages/8-introduction University Physics Volume 2: Chapter 8 - Capacitance] | ||
*[https://openstax.org/books/university-physics-volume-2/pages/10-5-rc-circuits University Physics Volume 2: Chapter 10.5 - RC Circuits] | *[https://openstax.org/books/university-physics-volume-2/pages/10-5-rc-circuits University Physics Volume 2: Chapter 10.5 - RC Circuits] | ||
= 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. | |||
<math> C = \frac{Q}{V} </math> | |||
*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. | |||
<math> C = \frac{\varepsilon A}{d} </math> | |||
*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. | |||
<math> E = \frac{1}{2} C V^2 </math> | |||
*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. | |||
<math> u = \frac{1}{2} \varepsilon E^2 </math> | |||
*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. | |||
<math> \frac{1}{C_{\text{eq}}} = \frac{1}{C_1} + \frac{1}{C_2} + \cdots + \frac{1}{C_n} </math> | |||
*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. | |||
<math> C_{\text{eq}} = C_1 + C_2 + \cdots + C_n </math> | |||
*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. | |||
<math> V(t) = V_0 e^{-\frac{t}{RC}} </math> | |||
*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. | |||
<math> V(t) = V_0 \left( 1 - e^{-\frac{t}{RC}} \right) </math> | |||
*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) | |||
= Videos = | = Videos = |
Revision as of 10:11, 18 September 2024
Textbook
- University Physics Volume 2: Chapter 8 - Capacitance
- University Physics Volume 2: Chapter 10.5 - RC Circuits
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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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)
Videos
Capacitors
Capacitors in Series and in Parallel
Build your own capacitor
Simulations
- Charging And Discharging A Capacitor
- An Example Of A Natural Capacitor
- Factors Affecting Capacitance
- PhET Capacitor Lab Simulations