lecture5 - 5. Lecture 5 5.1 Resistivity For a conductor of...

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Unformatted text preview: 5. Lecture 5 5.1 Resistivity For a conductor of uniform cross sectional area A it turns out experimentally that the resistance is given by L R=ρ (5.1) A where L is the length and ρ is the resistivity (measured in Ω · m), a property of the material. So, the longer the cable and the smaller the cross section the larger the resistance. It seems that the lower the resistance the better, however in electric circuits sometimes a resistance plays an important role so there are special components called resistors which, although being conductors have a relatively high resistance. We already discussed them in the previous lecture. A typical application is to limit the current in a circuit. Namely, for a given V, a very small resistance will create a large current I which could damage some component. In fact it is easy to see that energy is being dissipated in a conductor since charge is moving from a value of the potential to another (see fig.28). The energy difference ∆U gained when transferring a charge Q across a potential difference ∆V is ∆ U = Q∆ V (5.2) Such energy is converted into heat. In a given time interval ∆t an amount of charge Q = I ∆t is transferred so the energy is ∆U = I ∆V ∆t (5.3) or ∆U = I ∆V (5.4) ∆t Here P is the power, namely energy per unit time that is dissipated in the resistor. It is converted into heat. Such heat has to be removed by cooling the equipment either passively (letting air dissipate the heat) or actively (for example with a cooling fan). Using Ohm’s law we have alternative equivalent expressions for the power P= P = I · ∆V = I 2 R = (∆V )2 R (5.5) For a given ∆V if the resistance R is very small the power dissipated is very large. This is called a short-circuit, the large amount of heat produced usually melts the insulation and even the conductor itself with the consequent fire risk. All materials conduct a small amount of electricity but an idea of the difference between an insulator and a – 36 – conductor can be seen in the following table where the resistivity of various material is given. Material Resistivity (Ω · m) Silver 1.59 × 10−8 Copper 1.68 × 10−8 Aluminum 2.82 × 10−8 Tungsten 5.60 × 10−8 Zinc 5.90 × 10−8 Nickel 6.99 × 10−8 Iron 1.0 × 10−7 Germanium 4.6 × 10−1 seawater 2 × 10−1 Silicon 6.40 × 102 Glass 1010 to 1014 Between copper and glass there is a factor of at least 1018 . We should also mention that resistivity depends on the temperature. The table are typical values at 20 oC . Figure 28: A battery connected to a conductor produces a steady flow of current. The electrons move opposite to the direction of the current. When going across the potential they gain energy which is converted into heat by collision with the atoms. 5.2 Capacitors A capacitor is a device that can be used to store electric energy that can be used later – 37 – on. It has numerous applications some of which we are going to discuss in the rest of this lecture or later in the course. In fact if you open any electronic device one easily finds several of them. In its most simple form it has two parallel surfaces of area A separated by a distance d. One surface has charge Q and the other −Q. Since the electric field of a flat surface we already discussed, we can easily find that the electric field cancels Q ￿ outside the capacitor and inside it is given by |E | = Aε0 . Notice that σ = Q and the A factor of two goes away because we have two planes. Given the electric field we use the formula for the potential difference: Qd ￿ ∆ V = −|E |∆ x = − Aε0 (5.6) The surface with positive charge has the largest potential. We also see that the potential difference is proportional to Q. If we define the capacity C (do not confuse with the symbol for Coulomb!) through the equation Q = C ∆V (5.7) then we find the capacity equal to C= Aε0 d (5.8) From the definition (5.7) we see that the unit of capacity is C/V (here C is Coulombs!) a C unit known as Faraday (F). That is 1F = 1 V . Although this is just an example most of the capacitors in practical applications are similar. Commonly the planes are very thin and an insulator is in the middle. The foils are then rolled so that capacitor occupies less space but it essentially works in the same way. If we want to have large capacity we need d very small and that’s why thin insulators sheets between the plates are used and also large area which is why we need to roll it to occupy less volume. In practical circuits a Faraday is a large unit so capacity is usually measured in µF = 10−6 F , that is micro Faraday or even nF = 10−9 F or pF = 10−12 F . In the picture (fig. 30) we see some examples. – 38 – Q>0 V > V2 1 d Q −Q V2 V1 Area = A E Figure 29: A simple capacitor consists of two oppositely charged surfaces. This configuration stores energy. Figure 30: Examples of capacitors. The electrolytic ones have to be connected in a particular polarity. The − signs indicate that the corresponding terminal should always have lower potential, namely if connected to a battery it should be connected to the negative terminal. – 39 – ...
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This note was uploaded on 12/07/2011 for the course PHY 219 taught by Professor Na during the Fall '11 term at Purdue University-West Lafayette.

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