Dr.Alain Brizard
College Physics II (PY 211)
DC Circuits
Textbook Reference:Chapter 26 – sections 13.
• EMF and Terminal Voltage
A real battery is described in terms of its electromotive force (or emf) E and its internal
resistance r.When current I is drawn from a battery,its terminal voltage V is less than
its emf E
V = E I r
as a result of its internal resistance r.
For example,when 10 A is drawn from a 12V battery with an internal resistance of 0.1
Ω,the terminal voltage is 11 V = 12 V−(10 A) ∙ (0.1 Ω).On the other hand,when 1 A is
drawn from the same battery,the terminal voltage is now 11.9 V = 12 V−(1 A) ∙ (0.1 Ω).
When a resistor R is placed in a circuit with a battery with emf E and internal resistance
r,the total current I ﬂowing through the circuit is determined by Ohm’s Law as
V = E I r = I R → I =
E
r +R
and V =
E R
r +R
where V is the voltage across the resistor R.Hence,if R r then V E while,if
R = r then V = E/2.Unless otherwise stated,we shall normally assume that the internal
resistance of batteries appearing in circuits is negligible.
1
• Resistors in Series and in Parallel
When resistors (R
1
,R
2
,∙ ∙ ∙) are connected in series to a battery with terminal voltage
V,the same current I ﬂows through each resistor R
n
,which experiences a potential drop
V
n
= I R
n
.Since the sum of all potential diﬀerences (V
1
,V
2
,∙ ∙ ∙) add up to V,we may then
replace all resistors connected in series by a single equivalent resistor R
ser
deﬁned as
V =
n
V
n
=
n
I R
n
= I
n
R
n
= I R
ser
→ R
ser
=
n
R
n
.
For example,when N identical resistors R are placed in series,the equivalent resistance is
R
ser
= N R.
When resistors (R
1
,R
2
,∙ ∙ ∙) are connected in parallel to a battery with terminal voltage
V,each resistor R
n
experiences the same potential V and,thus,the current ﬂowing through
the resistor is I
n
= V/R
n
.Since the sum of all currents (I
1
,I
2
,∙ ∙ ∙) add up to the current I
drawn from the battery,we may then replace all resistors connected in parallel by a singleresistor R
par
deﬁned as
I =
n
I
n
=
n
V
R
n
= V
n
1
R
n
=
V
R
par
→ R
par
=
n
1
R
n
−1
For example,when N identical resistors R are placed in parallel,the equivalent resistance
is R
par
= R/N.
In a hybrid connection of resistors (R
1
,R
2
,∙ ∙ ∙),the equivalent resistor R
eq
satisﬁes the
condition R
par
< R
eq
< R
ser
.For example,when one resistor R
1
is connected in series with
two resistors R
2
and R
3
connected in parallel,the equivalent resistance is
R
eq
= R
1
+
R
2
R
3
R
2
+R
3
so that,if R
1
= R
2
= R
3
= R,then R
par
= R/3 < R
eq
= 3R/2 < R
ser
= 3R.
• Kirchhoﬀ’s Rules
When a hybrid circuit has several resistors and more than one battery,Kirchhoﬀ’s First
and Second Rules provide a convenient procedure to calculate currents ﬂowing through all
paths in the circuit.
Kirchhoﬀ’s First Rule,or Junction Rule,is based on the conservation law of charge
and states that,at each junction in the circuit,the sum of all currents that ﬂow into the
junction is equal to the sumof all currents ﬂow out of the junction.For example,if current
I
1
ﬂows into a junction and two currents I
2
and I
3
ﬂow out of the junction,the Junction
Rule states that I
1
= I
2
+I
3
and,hence,out of three currents (I
1
,I
2
,I
3
),only two currents
are independent.
2
Kirchhoﬀ’s Second Rule,or Loop Rule,is based on the conservation law of energy and
states that,along each closed loop in the circuit,the sumof the potential diﬀerences must
be equal to zero.By convention,the potential drops across a resistor as we move along
the current ﬂowing through it or the potential increases as we move againts the current.
Hence,for a closed path abcda,the Loop Rule states that
0 = V
ab
+ V
bc
+ V
cd
+ V
da
,
where some potential diﬀerences must be positive while others must be negative.
Consider,for example,the circuit shown in the Figure below.
Here,there are two junctions at points c and f and three loops abcfa,abcdefa,and cdefc.
The junction rule at point c is I
1
= I
2
+I
3
while the junction rule at point f is I
2
+I
3
= I
1
.
Hence,only two currents (I
2
and I
3
,say) are independent.
By following each closed path in the counterclockwise direction,the loop rules,on the
other hand,yield
0 = V
1
− I
3
R
3
− I
1
R
1
0 = V
1
− I
2
R
2
+ V
2
− I
1
R
1
0 = V
2
− I
2
R
2
+ I
3
R
3
By choosing the ﬁrst and third loop equations and substitute the junction equation I
1
=
I
2
+I
3
,we obtain the coupled linear equations
I
2
R
1
+ I
3
(R
1
+R
3
) = V
1
I
2
R
2
− I
3
R
3
= V
2
,
which can be solved for I
2
and I
3
as
I
2
=
V
1
R
3
+ V
2
(R
1
+R
3
)
R
2
(R
1
+R
3
) +R
1
R
3
and I
3
=
V
1
R
2
− V
2
R
1
R
2
(R
1
+R
3
) +R
1
R
3
3
This means that the current I
3
is positive if V
1
R
2
> V
2
R
1
or negative if V
1
R
2
< V
2
R
1
.
Lastly,the current I
1
is determined from the junction rule
I
1
= I
2
+ I
3
=
V
1
(R
2
+R
3
) + V
2
R
3
R
2
(R
1
+R
3
) +R
1
R
3
.
Reading Assigment:Sec.265 DC Ammeters and Voltmeters
4
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