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CHAPTER 11
CURRENT AND CIRCUITS
In the last session, we mainly dealt with charges that are slow moving or at rest. In
this session we focus our attention on charges in motion or electric current. Although
an electric current is a stream of moving charges, not all moving charges constitute
an electric current. There must be a net flow of charge through a given surface, if we
are to say that current passes through that surface. For example, the flow of water
through a garden hose does not result in an electric current, although billions of
protons are moving in the same direction. It is because there is an exact amount of
electrons moving in the same direction, thus there is not net transport of charge.
Electric potential
Let us consider a particle with charge q 0 . Similarly to the mechanics case, we must
be able to assign to this test charge an electric potential energy U. If the test charge
moves from an initial point i to a final point f, we can define the difference in its
electric potential energy as
U  U f  Ui  Wif .
Here Wif is the work done by the electrical forces acting on the particle as it moves
from i to f. We define the electric potential V (or simply potential) at a point as the
potential energy per unit charge at that point. In other words V can be written as
V
U
.
q0
The SI unit is joule per coulomb (J/C) or volt (V). The potential difference between
any two points i and f can then be written as
V  Vf  Vi  
Wif
q0
.
Electric current
i
i
a)
i
Battery
b)
Fig. 11.1 A loop of copper, with and without a battery.
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Fig. 11.1 a shows that an isolated conducting loop of copper is all at the same
potential. Although conducting electrons are present, no net electric force acts on
them and thus there is no current. On the other hand, Fig. 11.1 b shows that if we
insert a battery in the loop, the loop is no longer at the same potential. The
conducting electrons are then forced to move in one direction, reaching a steady-state
flow and thus establishing a current.
If charge dq passes through a hypothetical cross section of the loop in time dt, then
the current through that section is defined as
i
dq
.
dt
The SI unit for current is the coulomb per second or the ampere (A):
1 ampere = 1 A = 1 coulomb per second = 1 C/s.
For the direction of the current we follow the historical convention:
The current arrow is drawn in the direction in which positive carriers
would move, even if the actual carriers are not positive.
Resistance
If we apply the same potential difference between the ends of geometrically similar
rods of copper and of glass, very different currents result. This is due to the
characteristic of the conductor we call the resistance. We determine the resistance of
a conductor between any two points by applying a potential difference V between
those points and measuring the current i that results. The resistance R is then
R
V
.
i
The SI unit for resistance is volt per ampere. This combination occurs so often that
we give it the name ohm (symbol  ), given as
1 ohm = 1  = 1 volt per ampere = 1 V / A.
A conductor whose function in the circuit is to provide a specified resistance is called
resistor.
Another way of calculating the resistance is given by the formula
R
L
.
A
Here  is called the resistivity of the material of the conductor and its SI unit is the
.m , and its values range from 1.62  10 8 for silver (good conductor) to ~ 10 16 for
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fused quartz (insulator). L is the length (in meters m) of the conducting wire and A is
its cross-sectional area (in m 2 ).
Ohm’s law
As we now know a resistor is a conductor with a specified resistance. This means
that it has the same resistance even if the magnitude and direction of an applied
potential difference change. We can say that
Ohm’s law is an assertion that the current flowing through a device is
directly proportional to the potential difference applied to the device.
But some of the devices do not obey that rule (like a pn junction diode), whereas
other do (like a resistor). We can state then that
A conducting device obeys Ohm’s law when its resistance is
independent of the magnitude and polarity of the applied potential
difference.
Power in electric circuits
If we consider a simple circuit consisting of a battery (symbol
) that is
connected by wires of negligible resistance to an unspecified conducting device. The
battery maintains a potential difference of magnitude V across its terminals, and thus
across the device. Since there is a conducting path between the two terminals of the
battery, a steady current i flows around the circuit. The amount of charge dq that
moves between those terminals in time interval dt is equal to idt. Thus the variation
in the electric potential energy is
dU  dqV  idtV .
The power associated with this energy variation is given by dU dt , which can be
written as
P  iV .
The unit of the power is the volt-ampere, which can be written as
J
C
J
1 V.A = 1  1  = 1 = 1W.
 C  s 
s
For a resistor R, we can write the power as
P  i2 R
or
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P
V2
.
R
The last two equations apply only to the transfer of electric potential energy to
thermal energy in the resistor, whereas the original equation could be applied to all
kinds of electric energy transfers.
Kirchhoff’s loop rule
Let us consider a closed circuit. We start at any point in the circuit and mentally
proceed around the circuit in either direction, adding algebraically the potential
differences that we encounter. When we arrive at our starting point we must have
returned to our starting potential. So we can state Kirchhoff’s loop rule as
The algebraic sum of the changes in potential encountered in a
complete traversal of any circuit must be zero.
A rule that helps finding the potential differences in more complex circuits is the
resistance rule and it states that
If you mentally pass through a resistance in the direction of the
current, the change in potential is -iR; in the opposite direction it is
+iR.
Resistances in series
Let us consider n of resistances connected in series to an ideal battery (Fig.11.2).
R1
R3
R2
V
Fig. 11.2 Resistances in series.
We state then that
Connected resistances are said to be in series when a potential
difference that is applied across the combination is the sum of the
resulting potential differences across the individual resistances.
This is equivalent to saying that the currents through the resistances are equal. The
equivalent resistance Req is then calculated as the sum of all the resistances in the
circuit. That is
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n
Req  R1  R2    Rn   R j .
j 1
Kirchhoff’s junction rule
A point in the circuit is considered as a junction if it is the connection point of more
than two branches in the circuit.
a
f
b
c
e
d
Fig. 11.3 Junctions in a circuit.
Fig. 11.3 shows that the points b and e are junctions, whereas the other points are not.
If charge travels from a towards b, at b the charge will be divided outgoing towards c
and e, and because of the conservation of charge, no charge is lost. Now if we
consider the currents instead of the charge we can state Kirchhoff’s junction rule as
The sum of the currents approaching any junction must be equal to
the sum of the currents leaving that junction.
Resistances in parallel
Let us consider n resistances connected in parallel to an ideal battery (Fig. 11.4).
V
R1
R2
R3
Fig. 11.4 Resistances in parallel.
We can say that
Connected resistances are said to be in parallel when a potential
difference that is applied across the combination is the same as the
resulting potential difference across the individual resistances.
The equivalent resistance Req is then given by
n
1
1
1
1
1


  
 .
Req R1 R2
Rn j 1 R j
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Measuring instruments
The ammeter
The ammeter is an instrument used to measure the current in a circuit. To measure
the current in a device, the ammeter should be inserted in series with the device in the
circuit. It is essential that the resistance of the ammeter is much smaller than any
other resistance in the circuit. Otherwise, the very presence of the meter will change
the current to be measured.
The voltmeter
The voltmeter is an instrument used to measure the potential differences (or voltage
drops). To measure the potential difference across a device in a circuit, the voltmeter
should be inserted in parallel to that device. In the case of a voltmeter, it is essential
that its resistance is much larger than any other resistance in the circuit, in order not
to change the potential difference being measured.
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