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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 UniversityWest Lafayette.
 Fall '11
 NA
 Energy

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