I. Conductors and Insulators :

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1 Chapter 6 : Conductors - Insulators - Capacitors We have, till now, studied the electric charges and the interactions between them but not evoked how the electricity can be transfered? which meterials can do that? can we store electric charges? in this frame we can distinguish conductor meterials that can transfert electric charges and insulators that can not do that. Also, we can evoke an electronic device that can store or accumulate electricity that will be called : the capacitor. I. Conductors and Insulators : 1. Conductors : Conductors are materials that permit electrons to flow freely from particle to particle. An object made of a conducting material will permit charge to be transferred across the entire surface of the object. If charge is transferred to the object at a given location, that charge is quickly distributed across the entire surface of the object. The distribution of charge is the result of electron movement. If a charged conductor is touched to another object, the conductor can even transfer its charge to that object (figure 1) Figure 1 : uniform distribution of charge onconductors 27

2 2. Insulators : In contrast to conductors, insulators are materials that prevent the free flow of electrons from atom to atom and molecule to molecule. If charge is transferred to an insulator at a given location, the excess charge will remain at the initial location of charging. The particles of the insulator do not permit the free flow of electrons; subsequently charge is seldom distributed evenly across the surface of an insulator. While insulators are not useful for transferring charge, they do serve a critical role in electrostatic experiments and demonstrations. Conductive objects are often mounted upon insulating objects. This arrangement of a conductor on top of an insulator prevents charge from being transferred from the conductive object to its surroundings. This arrangement also allows for a student (or teacher) to manipulate a conducting object without touching it. 3. Examples of conductors and insulators : Examples of conductors include metals, aqueous solutions of salts (i.e., ionic compounds dissolved in water), graphite, and the human body. Examples of insulators include plastics, Styrofoam, paper, rubber, glass and dry air. The division of materials into the categories of conductors and insulators is a somewhat artificial division. It is more appropriate to think of materials as being placed somewhere along a continuum. Those materials that are super conductive (known as superconductors) would be placed at on end and the least conductive materials (best insulators) would be placed at the other end. Metals would be placed near the most conductive end and glass would be placed on the opposite end of the continuum. The conductivity of a metal might be as much as a million trillion times greater than that of glass. II. Capacitors : A capacitor is an electric component that consists of two parallel conductor plates generally shaped of metals separated by an insulator called dielectric. The dielectric is characterized by its dielectric constant ε. The distance between plates is denoted by d. The surface of every plate is generally denoted by A. The following figure shows the details of the structure of a plate capacitor : 28

3 Figure 2 : structure of a plate capacitor In the present course, we will be limited to this kind of capacitors. 1. Dielectric used for capacitors : The dielectric that fills the space between plates is an insulator with the permettivity ε, this latter constant measure the degree of insulation. For vacuum, this constant is denoted by ε 0 that is equal to ε 0 = 8, F.m -1. We define the relative permittivity denoted by ε r of every dielectric that shows the degree of insulation of the considered dielectric compared to vacuum and given by : ε r is without unit and equal to 1 in the case of vacuum or air. The following table shows some relative dielectric constants of usual insulators : Table 1 : relative permettivity of usual insulators 29

4 2. Capacitance of plate capacitor : The capacitance of a capacitor is a physical quantity that measures the ability of the capacitor to store an amount of electric charge. The capacitance depend of geometric caracteristic of the capacitor plates and of the dielectric filling the space between. The capacitance of the considered plate capacitor is given by : C ε A d ε ε A d Where A is the surface of every plate, d is the distance separating them and ε is the permittivity of the dielectric also called dielectric constant. We can easily notice that the dielectric is supposed to increase the capacitance of the capacitor as ε r > 1. In the international system of units (SI), the unit of capacitance is the Farad denoted by F. Example : Let's calculate the capacitance of a plate capacitor whose the surface is A = 0.1 cm 2 and the plates are separated by the distance d = 0,01mm and filled with paper. C 3 8,85.10,.,. 2,6.10 F 26 pf 3. Polarized capacitor : When there is a potential difference (voltage) across the conductors (plates) (e.g., when a capacitor is attached across a battery), an electric field develops across the dielectric, causing positive charge Q to collect on one plate and negative charge Q to collect on the other plate. If a battery has been connected to a capacitor for a sufficient amount of time, no current can flow through the capacitor. However, if a time-varying voltage is applied across the leads of the capacitor, a displacement current can flow. a- Capacitor polarization In order to polarize a capacitor, it must be connected to a battery in continuous current mode, the plate connected to the positive pole of the battery will be positively charged wheras the plate connected to the negative pole of the battery will be negatively charge ( figure 3). 30

5 Figure 3 : capacitor polarized by a battery b- Capacitance An ideal capacitor is characterized by a single constant value for its capacitance. Capacitance is expressed as the ratio of the electric charge Q on each conductor to the potential difference V between them. Typical capacitance values range from about 1 pf (10 12 F) to about 1 mf (10 3 F). The capacitance of a plarized capacitor is given by : The potential of the positively charged plate which holds the charge Q is V whereas the potential of the negatively charged plate holding the charge Q isv -. The voltage between plates is thus given by : V = V - V - The electric field between the plates is uniform and oriented towards the negative plate and, as seen in chapter 5, the magnitude!"# of the electric field between plates is given by : c. Stored energy $"# % &' % We consider a capacitor with the capacitance C, a voltage V is applied on this capacitor. The absolute charge on the plates is Q (figure 4 ). Figure 4 : polarized plate capacitor 31

6 The positive plate (down) holds the charge Q and its surface is A, we can then evoke the surface density ( ) ) for positive plate and '( * * for negative plate. The electric energy that is stored by the polarized capacitor is given by :, In the (SI) of units, the unit of the energy is the Joule denoted by J. Example 1 : A plate capacitor is polarized under the voltage V = 12 V. The capacitance is C = 1µF. Calculate the electric energy stored by this capacitor. Solution The energy stored by the capacitor is : W 0 0, J72 μj Example 2 : A plate capacitor is fabricated so that the plates with the surface of A = 6 cm 2 are separated by the distance d = 0,08 mm. The space between plates is filled by air (figure 5). Figure 5 Once polarized, the positive plate holds the surface charge density σ = 2, C.m -2. a. Calculate the capacitance C of this capacitor. b. Calculate the electric charge Q held by the positive plate. c. Calculate the energy stored by the capacitor. 32

7 Solution a. C ε 5 6 8, ,7. 6,63.10 F. b. σ 9 5 Q σ A2,4.10= 6.10 = 14, C. c. W 0 9 > > B? 1,1C. DD 15,63.10E J. d. Symbol and photos : In the electronics, the symbol of the capacitor in circuits is shown in the following figure 6 : - Non polarized capacitor polarized capacitor Here are some photos of capacitors Figure 6 : symbol of the capacitor in electric circuits 33

8 III. Capacitors associations Capacitors are one of the standard components in electronic circuits. Moreover, complicated combinations of capacitors often occur in practical circuits. It is, therefore, useful to have a set of rules for finding the equivalent capacitance of some general arrangement of capacitors. It turns out that we can always find the equivalent capacitance by repeated application of two simple rules. These rules related to capacitors connected in series and in parallel. 1. Parallel connection : We consider two capacitors connected in parallel: i.e., with the positively charged plates connected to a common ``input'' wire, and the negatively charged plates attached to a common ``output'' wire (figure. 7). C 1 C 2 Figure 7 : two capacitors connected in parallel What is the equivalent capacitance between the input and output wires? In this case, the potential difference V across the two capacitors is the same, and is equal to the potential difference between the input and output wires. The total charge Q, however, stored in the two capacitors is divided between the capacitors, since it must distribute itself such that the voltage across the two is the same. Since the capacitors may have different capacitances, C 1 and C 2, the charges Q 1 and Q 2 may also be different. The equivalent capacitance C eq of the pair of capacitors is simply the ratio 9, F where Q = Q 1 Q 2 is the total stored charge. It follows that : G HI J K J -J. K J - K J. K Giving : C eq = C 1 C 2 We can generalize this rule to N capacitors connected in parallel (see figure 8). 34

9 A A C 1 C 2 C N C equ B B Figure 8 : N capacitors connected in parallel The equivalent capacitance for N capacitors connected in parallel is then :,M -. O P Q O QR- Example : The equivalent capacitance in the following parallel connection 1 µf 2 µf is C 0S 12 3 μf 2. Series connection : we consider two capacitors connected in series: i.e., in a line such that the positive plate of one is attached to the negative plate of the other (see Figure 9). In fact, let us suppose that the positive plate of capacitor 1 is connected to the ``input'' wire, the negative plate of capacitor 1 is connected to the positive plate of capacitor 2, and the negative plate of capacitor 2 is connected to the ``output'' wire. C 2 C 1 Figure 9 : Two capacitors connected in series 35

10 What is the equivalent capacitance between the input and output wires? In this case, it is important to realize that the charge Q stored in the two capacitors is the same. This is most easily seen by considering the ``internal'' plates: i.e., the negative plate of capacitor 1, and the positive plate of capacitor 2. These plates are physically disconnected from the rest of the circuit, so the total charge on them must remain constant. Assuming, as seems reasonable, that these plates carry zero charge when zero potential difference is applied across the two capacitors, it follows that in the presence of a non-zero potential difference the charge Q on the positive plate of capacitor 2 must be balanced by an equal and opposite charge -Q on the negative plate of capacitor 1. Since the negative plate of capacitor 1 carries a charge -Q, the positive plate must carry a charge Q. Likewise, since the positive plate of capacitor 2 carries a charge Q, the negative plate must carry a charge -Q. The net result is that both capacitors possess the same stored charge Q. The potential drops, V 1 and V 2, across the two capacitors are, in general, different. However, the sum of these drops equals the total potential drop V applied across the input and output wires: i.e., V = V 1 V 2. The equivalent capacitance of the pair of capacitors is again C 0S 9 F. Thus, giving -,M ,M or G HI G -.G. G - &G. We can generalize this rule to N capacitors connected in series (figure 10). C 1 C 2 C equ A B A B C N Figure 10 : N capacitors connected in series The equivalent capacitance for N capacitors connected in series is then : - O,M O P - Q QR- 36

11 Example : The equivalent capacitance in the following seies connection 1,5 µf 2 µf is V C? TU,E C 1 0S 1 0,85 μf V IV. Exercises : Exercise 1 : Calculate the surface A of the plates of a capacitor that are separated by the distance d = 10-4 m and filled by nylon so that the capacitance is C = 2 nf. Exercise 2 : We consider a capacitor that consists of two circular metal plates, each with a radius of 5 cm. The plates are parallel to each other and separated by a distance of 1 mm. You connect a 9 volt battery across the plates. The space between plates contains the air. 1. Calculate the capacitance of the capacitor. 2. Calculate the charge on each plate. 3. Calculate the excess number of electrons on the negative plate. 4. Calculate the stored energy in the capacitor. Exercise 3 : Calculate the equivalent capacitance in the following connection 0,5 µf 1 µf 1,2 µf 1,5 µf 37

12 Chapter 6- Conductors-Insulators-Capacitors Phys ا ر ا. د Historic outline Leyden jar A Leyden jar, or Leiden jar, is a device that "stores" static electricity between two electrodes on the inside and outside of a glass jar. It was the original form of a capacitor (originally known as a "condenser"). It was invented independently by German cleric Ewald Georg von Kleist on 11 October 1745 and by Dutch scientist Pieter van Musschenbroek of Leiden (Leyden) in The invention was named for the city. The Leyden jar was used to conduct many early experiments in electricity, and its discovery was of fundamental importance in the study of electricity. Previously, researchers had to resort to insulated conductors of large dimensions to store a charge. The Leyden jar provided a much more compact alternative. 38