Capacitor | Capacitance | Capacitance Formula

capacitor

What is Capacitor

A capacitor is a two-terminal electrical device that stores energy in the form of an electric charge. A capacitor consists metal surfaces separated by a layer of insulating medium called dielectric as shown in figure “1a”.
Capacitance is defining as that property of circuit element (capacitor) to store energy in the form of an electrical charge. A significant and distinguishing feature of capacitance is that its influence in an electric circuit is manifested only when there exists a changing potential difference across the terminals of the circuit element.
The SI Unit of Capacitance is Farad, denoted by “F” and the symbol of capacitor is shown in figure “1b”

Capacitor

The various Capacitance Formulae can be derived from geometrical view point, circuit view point and energy view point

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Geometrical View Point

let a voltage “V” is applied across the terminals of capacitor as shown in figure 2 , there will be accumulation of charge (Q) on plates of the capacitor. This charge will be
\;\;\;\;\;Q = CV\;\;\;\;\;(1)
where C is capacitance of Capacitor

According to Gauss’s flux theory, the charge accumulated on the surface is
\;\;\;\;\;Q = εAE\;\;\;\;\;(2)
where
ε = permittivity of dielectric material
A = area of plates, and
E = Electric field intensity between the plates of capacitor.

\;\;\;\;\;But E = \frac{V}{d}\;\;\;\;\;(3)\\
Where “V” is the voltage between the plates of capacitor and “d” is the distance between the plates of capacitor
Putting the value “E” from equation (3) into equation (2).
\\\;\\\;\;\;\;\;Q = \frac{\varepsilon AV}{d}\;\;\;\;\;(4)\\
Comparing Equation (1) and (4)
\\\;\\\;\;\;\;\;CV = \frac{\varepsilon AV}{d}\\
\;\;\;\;\;C = \frac{\varepsilon A}{d}\;\;\;\;\;(5)\\

Where ε = ε­0εr
ε­0 is the permittivity of vacuum or free space. This is equal to 8.854\times 10^{12}
εr is the relative permittivity.
The unity of permittivity (ε) is Farad per meter
The capacitance of parallel plate capacitor can be calculated by equation (5)

Circuit View point

Let “I” will be the current flowing in the circuit , “V” is the voltage across the capacitor of capacitance “C” as shown in figure 2 above
we know \\\;\\\;\;\;\;\;I = \frac{dq}{dt} \;\;\;\;\;\;(6)\\\;\\
Putting the value of charge (Q) from equation (1) into equation (6), then\\\;\\\;\;\;\;\;I=\frac{d(cv)}{dt}\\\;\\\;\;\;\;\;I=c\frac{dV}{dt}\;\;\;\;\;\;(7)\\\;\\
This equation shows relationship between current flowing through a capacitor having capacitance “C” and the potential difference (V) across the capacitor.
When the voltage across the capacitor is constant, then \;\;\frac{dV}{dt}=0\;\;\\I=C(0)=0\\ that means will be no current flow in the circuit. Thus capacitor for DC Voltage acts an open circuit.

According to above equation (7)  the abrupt change in capacitor voltage is not admissible because a finite change in voltage “V” in zero time gives value of infinity for dv/dt. thus the capacitor current becomes infinite, this is physical impossible  

From equation (7)\\\;\\\;\;\;\frac{dV}{dt}=\frac{1}{C}I\\\;\\\;\;\;dV = \frac{1}{C}Idt\;\;\;\;\;\;\;\;(8)\\\;\\Intergrating both sides between the limits “t” and “0”\\\;\\\;\;\;\int_{V(0))}^{v(t))}dV=\frac{1}{c}\int_{0}^{t}Idt\;\;\;\;\;\;\;\;(9)

Where V(t) is the voltage at time “t” and V(0) is initial voltage. Assuming initial voltage is zero and voltage at time “t” is V (i.e. V(t) = V), then \\\;\\\;\;\;\;V = \frac{1}{C}\int_{0}^{t}Idt\;\;\;\;\;(10)

According to equation above if there is a finite change in current in zero time, the integral combination becomes zero. Hence the capacitor voltage cannot change instantaneously. Thus step change in current through capacitor is allowable. The graphical representation of the capacitance from circuit viewpoint is shown below.\\\;\\\;\;\;\;\;\;\;\;\;\;capacitance formula

Energy View point

Consider the circuit shown in figure 2 and assuming the capacitor has zero initial voltage and current ‘I” is allowed to flow for time “t”.Then energy delivered to the capacitor is\\\;\\\;\;\;\;E = \int_{0}^{t}VIdt\;\;\;\;(11)

Putting the value of “Idt” from equation (7) into equation (11)\\\;\\\;\;\;\;E = \int_{0}^{t}CVdV\\\;\\\;\;\;\;E=\frac{1}{2}CV^{2}\;\;\;\;\;\;\;(12)\\The above equation gives relationship between capacitance and Energy stored in the capcitor

How Does Capacitor Works

When a DC voltage is placed across a capacitor as show in figure 3 below, one plate of capacitor is connected to the positive end and the other plate to the negative end. The positive (+ve) charge gets accumulates on one plate which is connected to positive terminal of DC Voltage Source and and negative (-ve) charge accumulates on the other plate connected to negative terminal of DC Voltage Source. The charge on two plates are equal and opposite. Thus total charge on both the plates is zero. Due to this charge potential difference is established between the plate of the capacitor. Due the potential difference an electric field is created between the plates.

capacitor

when the capacitor reaches to steady state condition the flow of the current stops. At this point the potential difference between the plates of capacitor and the DC Voltage source are equal. The time taken by capacitor to reach steady state condition is called charging time of the capacitor.

Now if the battery is removed the capacitor. The capacitor behaves as a voltage source. if the capacitor is connected across the load / Resistance, the current start flowing to the load till the voltage across the plates of capacitor becomes zero. This time is called discharging time of the capacitor

Uses of Capacitor

  1. They are used for providing power to electronic devices during temporary power outages or when they need additional power.
  2. The capacitor are used filter out AC signals from DC Signals.  This effect of a capacitor is majorly used in reducing the noise.They are used in rectifiers to block AC component in rectifier output.
  3. They are used as sensors in various things like measuring humidity, mechanical strain, and fuel levels.
  4. In the field of information technology capacitors are used to represent binary information as bits.

Read Also

  1. Electrical Resistor | Electrical Resistance
  2. Explain Kirchhoff’s circuit laws
  3. Ohm’s Law Definition | What is Ohm’s law explain

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