#### Dielectrics and Capacitors, Physics tutorial

Capacitance and Permittivity:

The charge provided to an isolated conductor can be regarded as being 'stored' on it. The amount of charge it will take based on the electric field thus created at the surface of the conductor. If this is too great there is a breakdown in the insulation of the surrounding medium, resultant in sparking and discharge of the conductor. The change in potential due to a specific charge based on the size of the conductor, the material surrounding it and the proximity of other conductors.

The idea which an insulated conductor in specific condition has a certain capacitance, or charge storing capability is much helpful and is deduced as follows: When the potential of an insulated conductor change by 'V' if given a charge Q, the capacitance 'C' of the conductor is:

C = Q/V

In words, the capacitance is the charge needed to cause unit change in the potential of a conductor.

The unit of C is coulomb per volt (CV-1) as well termed as a Farad (F) in honor of Michael Faraday. The farad is an extremely large unit. Therefore the microfarad (1 uF = 10-6 F), the nanofarad (1nF = 10-9F) or the picofarad (1PF = 10-12 F) are usually employed.

The Parallel Plate capacitor:

The most general kind of capacitor comprises principle of two conducting plates parallel to one other and separated through a distance that is small as compared with the linear dimensions of the plates.

Let us suppose that the plates are in vacuum, the surface area of each and every plate being A. When a charge +Q lines on the upper plate, the net flux from it to the lower plates is:

EA = Q/εo

However, E is as well given by, E = Vab/l

Here Vab is the potential difference between the plates and l is their separation.

Vab = El = Ql/εoA

Therefore the capacitance of a parallel-plate capacitor in vacuum is:

C = Q/Vab = εoA/l

Note:

The above equation is not strictly true due to the reason of non-uniformity of the field at the edge of the plates. As εo A and l are constants for a particular capacitor, the capacitance is a constant independent of the charge on the capacitor and is directly proportional to the area of the plates and inversely proportional to their separation.

Energy of a Charged Capacitor:

The procedure of charging a capacitor comprise of transferring charge from the plate at lower potential to the plate at higher potential. The charging method thus needs the outlay of energy. Let us suppose that the charging method is taken out by starting with both plates fully uncharged, and then repeatedly eliminating small positive charges from one plate and transferring them to the other plate. The final charge 'Q' and the final potential difference 'V' are associated by means of:

Q = CV

As the potential difference rises in proportion with the charge, the average potential difference Vav throughout the charging process is just one-half the maximum value, or

Vav = Q/2C

This is the average work per unit charge in such a way that the total work needed is just Vav multiplied through the net charge, or

W = Vav Q = Q2/2C

By using the relation Q = CV, we have

W = Q2/2C = 1/2 CV = 1/2 QV joules (J)

Energy Density:

We might consider the stored energy to be positioned in the electric field between the plates of the capacitor. The capacitance of a parallel-plate capacitor in vacuum is:

C = εo A/l

The electric field fills the space among the plates, of volume Aι, and is represented by:

E = V/l

The energy per unit volume, or the energy density, is represented by:

Energy Density = (1/2) (CV2)/Al

Energy Density = [(1/2) εo AE2l2]/Al2

Energy Density = (1/2) εo E2

Combination of Capacitors:

1) Capacitors in Parallel:

The figure represents three capacitors of capacitances C1, C2 and C3 that are connected in parallel. The applied potential difference, 'V' is similar across each however the charges are different and are given by:

Q1 = VC1; Q2 = VC2; Q3 = VC3

The total charge, Q on the three capacitors is:

Q = Q1 + Q2 + Q3

Q = V (C1 + C2 + C3)

When C is the capacitance of the single equivalent capacitor, it would have charge Q when the potential difference across it is V.

Hence Q = VC

And C = C1 + C2 + C3

We can as well notice that Q1: Q2: Q3 = C1: C2: C3

In another words, the charges on capacitors in parallel are in the ratio of their capacitances.

The capacitors in figure above are in series and encompass capacitances C1, C2 and C3.

Assume a potential difference of V volt applied across the combination causes the motion of charge from plate Y to plate A in such a way that a charge +Q appear on A and an equivalent however opposite charge -Q appears on Y.

This charge -Q will induce a charge +Q on the plate X when the plates are large and close altogether. The plates X ad M and the connection between them form an insulator conductor whose total charge should be zero and so +Q and X induces a charge -Q on M. In turn this charge induces +Q on L and so forth.

Capacitors in series therefore all have the similar charge and the potential difference across each is represented by:

V1 = Q/C1   V2 = Q/C2   V3 = Q/C3

The total potential difference 'V' across the network is:

V = V1 + V2 + V3

V = Q/C1 + Q/C2 + Q/C3

V = Q (1/C1 + 1/C2 + 1/C3)

If 'C' is the capacitance of the single equivalent capacitor, it would encompass a charge 'Q' when the potential difference across it is V.

Thus, Q/C = Q (1/C1 + 1/C2 + 1/C3)

1/C = (1/C1 + 1/C2 + 1/C3)

Permittivity:

We derived the coulomb's law of force among the two point charges Q1 and Q2 at a distance 'r' apart. Therefore,

F = (Q1 Q2)/(4πεor2)

Here εo is the permittivity of free space. This provides us an idea that the force depended on the intervening media.

Relative Permittivity, Er:

Experiment represents that inserting an insulator or dielectric among the plates of a capacitor raises its capacitances. When Co is the capacitance of a capacitor when a vacuum is between its plates and 'C' is the capacitance of the similar capacitor having a dielectric filling the space among the plates, the relative permittivity Er of the dielectric is stated by:

Er = c/co

Taking a parallel-plate capacitor as an illustration, we have:

Er = c/co = E/εo

Here 'E' is the permittivity of the dielectric and 'εo' is that of a vacuum (that is, of free space). The expression for the capacitance of a parallel-plate capacitor having a dielectric of relative permittivity Er can thus be written as:

C = (Er εo A)/l

Relative permittivity consists of no units, dissimilar E and Eo which have. It is a pure number with no dimensions.

For air at atmospheric pressure Er = 1.0005 that is close adequate to unity and so for most purposes Eair = Eo.

Dielectric Strength and Breakdown:

Whenever a dielectric material is subjected to an adequately strong electric field, it becomes a conductor. This phenomenon is termed as dielectric breakdown. The onset of conductor, related by cumulative ionization of molecules of the material, is often quite rapid, and might be characterized by means of spark or arc discharges.

If a capacitor is subjected to extreme voltage, an arc might be formed via a layer of dielectric, burning or melting a hole in it, permitting the two metal foils to come in contact, making a short circuit, and rendering the device permanently useless as a capacitor.

The maximum electric field a material can endure devoid of the occurrence of breakdown is termed as dielectric strength.

Types of Capacitor:

Capacitors might be categorized into two broad groups, that is fixed and variable capacitors. They might be further categorized according to their construction and use. The capacitors are employed in electric circuits for different purposes. Different kinds have different dielectric. The choice of kind depends on the value of capacitance and stability (that is, the capability to retain the similar value having age, temperature change and so on) required and on the frequency of any alternating current which will flow in the capacitor.

1) Dielectric Capacitor:

Dielectric Capacitors are generally of the variable kind were a continuous variation of capacitance is needed for tuning transmitters, receivers and transistor radios. The variable dielectric capacitors are multi-plate air-spaced types which have a set of fixed plates (that is, the stator vanes) and a set of movable plates (that is, the rotor vanes) that move in between the fixed plates.

The place of the moving plates with respect to the fixed plates finds out the overall capacitance value. The capacitance is usually at maximum if the two sets of plates are fully meshed altogether. High voltage kind tuning capacitors have relatively big spacing or air-gaps between the plates having breakdown voltages arriving many thousands of volts.

2) Film Capacitor:

Film Capacitors are the most generally available of all kinds of capacitors, comprising of a relatively big family of capacitors having the difference being in their dielectric properties. Such comprise polyester (Mylar), polystyrene, polycarbonate, polypropylene, metalized paper, Teflon and so on. Film kind capacitors are available in capacitance ranges from as small as 5pF to as large as 100uF based on the real kind of capacitor and its voltage rating.

Film capacitors as well come in an assortment of shapes and case styles that comprise:

• Wrap & Fill (Oval & Round): In this kind, capacitor is wrapped in a tight plastic tape and encompasses the ends filled by epoxy to seal them.
• Epoxy Case (Rectangular & Round): In this kind, the capacitor is encased in a molded plastic shell that is then filled by epoxy.
• Metal Hermetically Sealed (Rectangular & Round): In this kind, the capacitor is encased in a metal tube or can and again sealed by epoxy.

3) Ceramic Capacitors:

Disc Capacitors or Ceramic Capacitors are generally made by coating the two sides of small porcelain or ceramic disc having silver and are then stacked altogether to build a capacitor. For extremely low capacitance values a single ceramic disc of around 3 to 6 mm is utilized. Ceramic capacitors encompass a high dielectric constant (High-K) and are available so that the relatively high capacitance's can be acquired in a small physical size.

They represent big non-linear changes in capacitance against temperature and as an outcome are employed as de-coupling or by-pass capacitors as they are as well non-polarized devices. Ceramic capacitors contain values ranging from some picofarads to one or two microfarads however their voltage ratings are usually quite low.

4) Electrolytic Capacitors:

Electrolytic Capacitors are usually utilized if extremely large capacitance values are needed. Here rather than employing a very thin metallic film layer for one of the electrodes, a semi-liquid electrolyte solution in the form of a jelly or paste is employed that serves up as the second electrode (generally the cathode).

The dielectric is an extremely thin layer of oxide that is grown electro-chemically in production having the thickness of the film being less than ten microns. This insulating layer is so thin that it is possible to make capacitors having a large value of capacitance for a small physical size as the distance between the plates, 'd' is extremely small.

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