lecture11 - 11. Lecture 11 11.1 Inductors A solenoid a part...

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Unformatted text preview: 11. Lecture 11 11.1 Inductors A solenoid a part of a circuit is also called an inductor. Its purpose is to store energy similarly to a capacitor. However it stores energy in a magnetic field as opposed to a capacitor which stores energy in an electric field. This makes its electrical characteristic different from the capacitor. Indeed, we know that the magnetic field produced inside the solenoid is given by µ0 IN ￿ B= z ˆ (11.1) ￿ where N is the number of turns, ￿ is the length of the solenoid and I is the current. If the current changes, the magnetic field and therefore its flux changes (see fig65) producing a voltage between the terminals of the solenoid. Using Faraday’s law, Lenz law and looking at the figure we conclude that ￿ ∆Φ ∆|B | N ∆I ∆I = NA = N Aµ0 =L ∆t ∆t ￿ ∆t ∆t where A is the cross section area of the solenoid. We defined the quantity Va − Vb = (11.2) N 2A (11.3) ￿ which is given in terms of the area A, the length ￿ and the number of turns. It is a property of the solenoid and is called the inductance. It is measured in Henrys (H): L = µ0 Vs (11.4) A as can be seen from eq.(11.3). Going back to that formula we see that if we have a DC current, namely ∆I = 0 then ∆V = 0, that is, it essentially acts as a cable. However ∆t any change in current will generate a potential difference that opposes that change. To compare with a capacitor we recall that 1H = 1 Q = C ∆V (11.5) If a current goes into a capacitor it will increase its charge so we find ∆Q ∆V =I=C (11.6) ∆t ∆t Comparing with eq.(11.3) we see that the current is proportional to the change in voltage whereas for a solenoid it is the other way around. Although both capacitors and solenoids both store energy, this important difference in their voltage to current relation makes solenoids play a different role in electric circuits. An important application of solenoids is in transformers which we now study. – 76 – Figure 65: A coil through which a variable current circulates produces a voltage across its terminals given by Faraday’s law. 11.2 Transformers A transformer consists of two solenoids as indicates in fig.66. Through one solenoid, called the primary, we pass a time dependent current which creates a time dependent magnetic flux through the other solenoid called the secondary. This time dependent flux induces a voltage in the secondary which then acts as a battery. The interesting point however is that the voltage in the secondary is not the same as in the primary. Indeed using Faraday’s law we find for the voltage in the primary and secondary: ∆B ∆t ∆B V c − V d = N2 A ∆t V a − V b = N1 A – 77 – (11.7) (11.8) Their ratio is given by Va − Vb N1 = (11.9) Vc − Vd N2 So we see that by having different number of turns in the secondary we can increase or decrease the voltage. In the demo we see a transformer that increase the voltage from 120V to 15, 000V producing an interesting display. Stepping up the voltage is not just for show, it is very important in power transmission. When transmitting power through a line, we can consider that the resistance of the line is in series with the load. Therefore the current going through both is the same. To minimize the power lost in the line we need to minimize the current since the power lost is P = I 2 /R. On the other hand we need to maintain the same power at the load. The only way is to increase the voltage of the line and for that one can use a transformer to step up the voltage to typical values of a few hundred kilovolts. Figure 66: Two coils wrapped around each other constitute a transformer. If a variable current goes through one of them, the other acts as a battery with a variable voltage. The ratio of the voltages in the primary and secondary is given by the ration in the number of turns of each of them. – 78 – Figure 67: Demo. A transformer can be used to increase the voltage of an oscillating (alternate) current. The increased voltage can be used to create an interesting display. – 79 – ...
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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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