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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 shortcircuit, 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 UniversityWest Lafayette.
 Fall '11
 NA
 Resistance

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