# Capacitor

Capacitance of a capacitor;

The ability of a capacitor to store charge is known as its capacitance. Its unit is coulomb per volt or CV-1

The factors on which the capacitance of a capacitor depends

1. Distance between the plates

2. Area between the plates

3. Dielectric

Principle of a capacitor:

A small device used to store huge amount of electric charge in a small room is called capacitor.

Take an insulated metal plate A. Charge the plate to its maximum potential. Now take another insulated plate B. Take the plate B nearer to plate A. You will observe that negative charge will be produce on the plate near to plate A and the same amount of positive charge will be produced on the other side of plate B.

Figure: Induced positive and negative charge

Dielectric strength of a dielectric:

The maximum value of electric field intensity (or potential gradient) that can be applied to the dielectric without its electric breakdown is called dielectric strength of the dielectric.

Relative permittivity:

Relative permittivity is the factor by which the electric field between the charges is decreased relative to vacuum. Likewise, relative permittivity is the ratio of the capacitance of a capacitor using that material as a dielectric, compared to a similar capacitor that has vacuum as its dielectric.

Air or vacuum has minimum value of permittivity. The absolute (or actual) permittivity of air or vacuum is 8.854 x 10-12 C2 N-1 m-2.

Polar and non – polar molecules;

A water molecule, abbreviated as H2O, is an example of a polar covalent bond. The electrons are unequally shared, with the oxygen atom spending more time with electrons than the hydrogen atoms. Since electrons spend more time with the oxygen atom, it carries a partial negative charge.

Polarization of dielectric medium:

When a dielectric is placed between charged plates, the polarization of the medium produces an electric field opposing the field of the charges on the plate. Dielectric polarization occurs when a dipole moment is formed in an insulating material because of an externally applied electric field. When a current interacts with a dielectric (insulating) material, the dielectric material will respond with a shift in charge distribution with the positive charges aligning with the electric field and the negative charges aligning against it. By taking advantage of this response, important circuit elements such as capacitors can be made.

A formula of the capacitance of a parallel plate capacitor:

The ability of a capacitor to store charge is known as its capacitance. Its unit is coulomb per volt or CV-1

Figure; Parallel Plate Capacitor

E =${\rm{\: }}\frac{\sigma }{{{{\rm{E}}_0}}},{\rm{\: }}$assuming the plates are kept in vaccum or air.

Now, $\sigma = \frac{{{\rm{charge}}}}{{{\rm{area}}}} = \frac{{\rm{q}}}{{\rm{A}}}$

∴${\rm{E}} = \frac{{\rm{q}}}{{{{\rm{E}}_{0{\rm{\: A}}}}}}$

If V is the potential difference across the plates of the capacitor, then, we have

${\rm{E}} = \frac{{\rm{V}}}{{\rm{D}}}$

∴$\frac{{\rm{V}}}{{\rm{d}}} = \frac{{\rm{q}}}{{{{\rm{E}}_{{\rm{o\: A}}}}}}{\rm{\: }}$

Or $\frac{{\rm{q}}}{{\rm{v}}} = \frac{{{{\rm{E}}_0}{\rm{A}}}}{{\rm{d}}}$

By definition it is given by C=$\frac{{{{\rm{E}}_0}{\rm{A}}}}{{\rm{d}}}$.

Charging and discharging action of a capacitor;

When a Capacitor is connected to a circuit with Direct Current (DC) source, two processes, which are called "charging" and "discharging" the Capacitor, will happen in specific conditions.

In Figure 3, the Capacitor is connected to the DC Power Supply and Current flows through the circuit. Both Plates get the equal and opposite charges and an increasing Potential Difference, vc, is created while the Capacitor is charging. Once the Voltage at the terminals of the Capacitor, vc, is equal to the Power Supply Voltage, vc = V, the Capacitor is fully charged and the Current stops flowing through the circuit, the Charging Phase is over.

Figure 3: The Capacitor is charging

A Capacitor is equivalent to an Open-Circuit to Direct Current, R = ∞, because once the Charging Phase has finished, no more Current flows through it. The Voltage vc on a Capacitor cannot change abruptly.

When the Capacitor disconnected from the Power Supply, the Capacitor is discharging through the Resistor RD and the Voltage between the Plates drops down gradually to zero, vc = 0, Figure 4.

Figure 4: The Capacitor is discharging

In Figures 3 and 4, the Resistances of RC and RD affect the charging rate and the discharging rate of the Capacitor respectively.

The resultant capacitance of two capacitors when they are grouped (i) in series and (ii) in parallel:

A small device used to store huge amount of electric charge in a small room is called capacitor.

I. Capacitors in Series;

In series, capacitors will each have the same amount of charge stored on them because the charge from the first one travels to the second one, and so on.The total charge stored is the charge that was moved from the cell, which equals the charge that arrived at the first capacitor, which equals the charge that arrived at the second, etc.

The voltage of the circuit is spread out amongst the capacitors (so that each one only gets a portion of the total).

So from the diagram

${\rm{V}} = \frac{{\rm{Q}}}{{\rm{C}}}$

${\rm{V}} = {{\rm{V}}_1} + {\rm{\: }}{{\rm{V}}_2} + {{\rm{V}}_3} \ldots \ldots ..1$

Now,${\rm{\: }}{{\rm{V}}_1} = \frac{{\rm{q}}}{{{{\rm{C}}_1}}}$, ${{\rm{V}}_2} = \frac{1}{{{{\rm{C}}_2}}}$, ${{\rm{V}}_3} = \frac{1}{{{{\rm{C}}_3}}}$

${\rm{V}} = {\rm{q}}\left( {\frac{1}{{{{\rm{C}}_1}}} + \frac{1}{{{{\rm{C}}_2}}} + \frac{1}{{{{\rm{C}}_3}}}} \right) \ldots \ldots \ldots 2$

Then we have

${\rm{V}} = \frac{{\rm{q}}}{{\rm{C}}}$…………3

From 2 and 3

$\frac{1}{{\rm{C}}} = \frac{1}{{{{\rm{C}}_1}}} + \frac{1}{{{{\rm{C}}_2}}} + \frac{1}{{{{\rm{C}}_3}}} \ldots \ldots ..4$

For n capacitor in series C is given by

$\frac{1}{{\rm{C}}} = \frac{1}{{{{\rm{C}}_1}}} + \frac{1}{{{{\rm{C}}_2}}} + \frac{1}{{{{\rm{C}}_3}}} \ldots \ldots \ldots \ldots \ldots ..\frac{1}{{{{\rm{C}}_{\rm{n}}}}}.$

This is the equation for capacitors in series.

II. Capacitors in Parallel

Two small capacitors in parallel can be thought of as being the same as one big capacitor:

There is just as much 'plate' on the left hand side for the charge to flow into in both of these diagrams.

So adding capacitors in parallel will increase the space available to store charge and will therefore increase the capacitance of the combination.

The pd across each capacitor is the same as the total pd. Let's call it V.

QT = Total charge stored = Q1 + Q2 +Q3

Using Q=VC

VCT = VC1 + VC2 + VC3

As the capacitors or in parallel they each have the same voltage across them, so cancel the V's.

CT = C1 + C2 + C3 for capacitors in parallel.

Expression of the energy stored in a capacitor:

The maximum value of electric field intensity (or potential gradient) that can be applied to the dielectric without its electric breakdown is called dielectric strength of the dielectric.

The energy stored on a capacitor can be expressed in terms of the work done by the battery. Voltage represents energy per unit charge, so the work to move a charge element dq from the negative plate to the positive plate is equal to V dq, where V is the voltage on the capacitor. The voltage V is proportional to the amount of charge which is already on the capacitor.

Element of energy stored: ${\rm{dU}} = {\rm{Vdq}} = \frac{{\rm{q}}}{{\rm{C}}}{\rm{dq}}$

If Q is the amount of charge stored when the whole battery voltage appears across the capacitor, then the stored energy is obtained from the integral:

$\mathop \smallint \limits_0^{\rm{Q}} \frac{{\rm{q}}}{{\rm{C}}}{\rm{dq}} = \frac{1}{2}\frac{{{{\rm{Q}}^2}}}{{\rm{C}}}$

This energy expression can be put in three equivalent forms by just permutations based on the definition of capacitance C=Q/V.

${\rm{U}} = \frac{1}{2}\frac{{{{\rm{Q}}^2}}}{{\rm{C}}} = \frac{1}{2}{\rm{QV}} = \frac{1}{2}{\rm{C}}{{\rm{V}}^2}$

Principle of a capacitorand the various factors on which the capacity of parallel plate capacitor depends;

A small device used to store huge amount of electric charge in a small room is called capacitor.

Take an insulated metal plate A. Charge the plate to its maximum potential. Now take another insulated plate B. Take the plate B nearer to plate A. You will observe that negative charge will be produce on the plate near to plate A and the same amount of positive charge will be produced on the other side of plate B.

Figure: Induced positive and negative charge

The various factors on which the capacity of parallel plate capacitor depends on

1. Distance between the plates:

C=${\rm{\: }}\frac{{{\rm{E\: A}}}}{{\rm{d}}}$

2. Area between the plates:

${\rm{C}} \propto \frac{1}{{\rm{d}}}$

3. Dielectric:

${\rm{c}} \propto {\rm{E}}.$

Polar Dielectrics:

Polar dielectrics are those in which the possibility of center coinciding of the positive as well as negative charge is almost zero i.e. they don’t coincide with each other. The reason behind this is their shape. They all are of asymmetric shape. Some of the examples of the polar dielectrics is NH3, HCL, water etc.

Non Polar dielectrics:

In case of non polar dielectrics the centres of both positive as well as negative charges coincide. Dipole moment of each molecule in non polar system is zero. All those molecules which belong to this category are symmetric in nature. Examples of non polar dielectrics are: methane, benzene etc.

When a dielectric is placed between charged plates, the polarization of the medium produces an electric field opposing the field of the charges on the plate. Dielectric polarization occurs when a dipole moment is formed in an insulating material because of an externally applied electric field. When a current interacts with a dielectric (insulating) material, the dielectric material will respond with a shift in charge distribution with the positive charges aligning with the electric field and the negative charges aligning against it. By taking advantage of this response, important circuit elements such as capacitors can be made.

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