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1.11
Thus, electrons act as a medium for transferring energy from source to conductor. The electrostatic
field present everywhere in the system is a facilitator of this energy transfer process. The non-
electrostatic field in the source transfers the source energy into charge carriers flowing through it in
the form of potential energy of the charged particles in an electrostatic field. The charged particles
carry this potential energy with them into the conductor. The non-electrostatic force (i.e., the average
effect of inelastic collisions) absorbs the potential energy of charges and transfers it to the lattice.
The electrostatic field that is present within the conductor facilitates this process by converting the
potential energy of charged particles into kinetic energy before they can deliver it to atoms through
inelastic collision process.
Thus,electrostaticfieldpermeatingthroughoutthesystemisanecessaryrequirement
for conduction and energy transfer process to take place in an electrical system. The
required electrostatic field is created by surface charge distributions on conducting
surfaceseverywhereinthesystem.
1.2.5
two-terminal resistance Element
Consider the steady voltage source with resistive load across
it shown in Fig. 1.2-3.
Let us work out the electrostatic potential difference
between e and f.
Work to be done against electrostatic force to carry a unit
positive test charge around a closed path is zero. Therefore,
the work to be done to take
+
1 C charge from f to e must be
the same whether we move it through a path that lies inside
the conducting substance or outside. But the electrostatic
field is given by E
J
s
=
s
inside the conductor. Therefore,
V
J dl
ef
f
e
= −
∫
1
s
i
with dl oriented from f to e. The value of
this integral will be same for any path through the conducting
substance. However, evaluation of the integral to yield a closed-form result will be possible only in
simple cases where the geometry of conductor has some kind of symmetry or other.
We consider a simple case of a conductor with uniform cross-section. The total current may be
assumed to distribute itself uniformly throughout the cross-section in such a conductor. This results
in a current density vector that has a constant magnitude of I/A (A is the area of cross-section) and
direction parallel to the axis of conductor. This is a satisfactory assumption everywhere except at the
connection ends. With this assumption, with l as the length of conductor and A as its uniform cross-
sectional area, we get,
V
J dl
l
A
I
l
A
f
e
ef
I
= −
=
∫
1
s
s
r
i
or
V.
(1.2-2)
Fig. 1.2-3
Pertainingtovoltage
acrossatwo-
terminalresistance
B
c
d
e
f
a
b
A
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1.12
CircuitVariablesandCircuitElements
Eqn. 1.2-2 relates the electrostatic potential difference across the connection points of a piece of
conductor with uniform cross-section to the current flow through it. The proportionality constant
is dependent on material property (conductivity or resistivity) and geometry of the conductor. This
proportionality constant is called the resistance parameter R.
R
l
A
l
A
=
=
s
r
Ohm
(1.2-3)
However, actual connection point between the resistive material and external circuit may not be
accessible for observation of voltage. We measure the voltage across a resistance by connecting a
voltmeter to the connecting wire on either side of the element. Assume that the voltmeter is connected
across a-c. Then, the voltmeter will read the electrostatic potential difference V
ac
. But,
V
V
J dl
J dl
ac
ef
a
e
d
c
=
+ −
−
∫
∫
1
1
s
s
i
i
evaluated over ppaths through
connecting wire
.
Therefore, a unique voltage difference can be assigned to the conducting body only if the
conductivity of connecting wires is infinitely large. However, it is to be noted that this does not imply
thick connecting wires. In fact, Circuit Theory assumes that connecting wires have zero resistance
and negligible thickness. The reason behind the assumption of negligible cross-section for connecting
wires will be explained in a later section.
With this assumption, the electrostatic field inside
connecting wires will be zero (since conductivity is infinite).
Then, the electrostatic potential difference between the ends
of conducting body has a unique value irrespective of which
pair of points (a and b) on the connecting wire are chosen to
measure it.
Now a unique voltage and current variable pair can be
assigned to the conducting body and its electrical behaviour
can be described entirely in terms of these two variables. This
model of a conducting body is called the two-terminal resistance
element model. The symbol and element relation is shown in
Fig. 1.2-4.
Ohm’s Law, which is an experimental law, states that the voltage drop across a two-
terminalresistancemadeofalinearconductingmaterialandmaintainedataconstant
temperatureisproportionaltothecurrententeringtheelementatthehigherpotential
terminal.
Resistivity and Conductivity are functions of temperature. If the temperature range considered
is small, resistivity may be approximated as
r
(T)
=
r
(T
0
)[1
-
a
(T
-
T
0
)] where
r
(T
0
) is the known
resistivity at temperature T
0
and
a
is the temperature coefficient of resistivity.
Fig. 1.2-4
Two-terminal
resistance
R
v = Ri
(Ohm’s Law)
v
i
+
–
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AVoltageSourcewithaResistanceConnectedatitsTerminals
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