PH 106
Dr. T. Mewes
Chapter 28:
Direct current circuits
Direct current
A current in a circuit that is constant in magnitude and direction is called a direct
current (DC). A battery hooked up to a circuit will produce a direct current.
Electromotive force
Historically a battery is called a source of electromotive force or a source of emf.
The emf ε of a battery is the maximum voltage the battery can provide between its
terminals. For an ideal battery, with no internal resistance r, the terminal voltage would
be equal to its emf.
However real batteries have an internal resistance r:
internal resistance
+-
r
ε
ΔV
Therefore:
ΔV = ε − Ir (1)
where Ir is the voltage drop over the internal resistance r of the battery. Or in other
words the emf ε is equivalent to the open circuit voltage, that is, the terminal voltage
when the current is zero.
Now a resistor with a load resistance RLoad is connected to the battery:
internal resistance
+-
r
ε
ΔV
RLoad
1/6
PH 106
Dr. T. Mewes
The voltage drop over RLoad is ΔV , by using Ohm’s law we get for the current through
the load:
ΔV
I=
(2)
RLoad
Now use (1):
ε − Ir
I=
(3)
RLoad
and solve for I:
I=
ε
RLoad + r
(4)
This equation shows that the current in a DC circuit not only depends on the load
resistance RLoad but also on the internal resistance r of the battery. However often the
load resistance is much bigger than the internal resistance r:
RLoad >> r : I ≈
ε
(5)
RLoad
Power output of a battery:
We can solve equation (4) for ε:
ε = IRLoad + Ir
(6)
By multiplying (6) with I we get:
Iε = I 2 RLoad + I 2 r (7)
This equation shows us that the total power output of the battery Iε is split up in the
power delivered to the load I 2 RLoad and the power delivered to the internal resistance
of the battery I 2 r .
Series combination of resistors:
ΔV
+ -
ΔV
+ I
I
R1
R2
Req
The current I1 that passes through resistor R1 is the same as the current I2 through
resistor R2:
I1 = I 2 = I
(8)
2/6
PH 106
Dr. T. Mewes
The total voltage drop over both resistors is ΔV and is the sum of the voltage drop over
the individual resistors:
ΔV = ΔV1 + ΔV 2
(9)
By using Ohm’s law for each of the resistors we get:
ΔV = I1R1 + I 2 R2 = I ( R1 + R2 ) = IReq
(10)
With the equivalent resistance Req given by:
Req = R1 + R2 (11)
More generally:
The equivalent resistance of a series combination of N resistors is the sum of the
individual resistances:
N
Req = R1 + R2 + ... + RN = ∑ Ri
(12)
i =1
Parallel combination of resistors:
ΔV
+ I
ΔV
+ -
I2
I1
R2
Req
R1
Here the voltage drop ΔV is the same over both resistors. Furthermore the total current
I entering the junction of both resistors has to be the sum of the currents through each
resistor:
I = I1 + I 2
(13)
For the total current I we can use Ohm’s law with the equivalent Req:
ΔV
I=
(14)
Req
with (13) we have:
1
1
ΔV ΔV1 ΔV2 ΔV ΔV
I=
=
+
=
+
= ΔV ( + ) (15)
Req
R1
R2
R1 R2
R1 R2
Therefore
1
1
1
=
+
Req R1 R2
(16)
More generally:
For a parallel combination of N resistors the inverse equivalent resistance is equal to the
sum of the inverses of the individual resistances:
N
1
1
1
1
1
=
+
+ ... +
=∑
Req R1 R2
RN i =1 Ri
(17)
3/6
PH 106
Dr. T. Mewes
One notices that the equivalent resistance of a parallel combination of resistors is
always less than the smallest resistance in the group.
Kirchhoff’s rules:
Junction rule:
At any junction the sum of the currents must be equal to zero:
N
∑ Ii = 0
(18)
i =1
Currents entering the junction are entered in the junction rule with a positive sign, while
those leaving the junction are entered with a negative sign.
Electrical current
Water current
I2
I1
I3
So for the figure above we have:
I1 + (− I 2 ) + (− I 3 ) = I1 − I 2 − I 3 = 0 (19)
Loop rule:
The sum of the potential difference across all elements around any closed circuit loob
must be zero:
N
∑ ΔVi = 0
(20)
i =1
Example:
ΔV
+ -
a
I
R1
b
d
R2
I
c
Using the loop rule gives us:
ΔVda + ΔVab + ΔVbc + ΔVcd = 0
(21)
The voltage across the battery ΔVda is equal to the terminal voltage ΔV , i.e. ΔVda = ΔV .
Furthermore there is no voltage drop between points b and c (no resistance): ΔVbc = 0 .
4/6
PH 106
Dr. T. Mewes
The voltage drops across the resistors are:
ΔVab = − IR1 and ΔVcd = − IR2
Using all this in equation (21) we get:
ΔV − IR1 + 0 − IR2 = 0
(22)
or by solving for I:
ΔV
(23)
I=
R1 + R2
Which is something we already know – namely that the equivalent resistance of two
resistors in series is the sum of the individual resistances.
So lets complicate the problem a little more:
ΔV1= 6 V
d
a
+ -
R1
R2
+
b
c
ΔV2= 12 V
Using the loop rule gives us:
ΔVda + ΔVab + ΔVbc + ΔVcd = 0
(24)
The voltage across the battery ΔVda is equal to the terminal voltage ΔV1 , i.e. ΔVda = ΔV1 .
But now there is a voltage drop between points b and c because we added another
battery so: ΔVbc = −ΔV2 .
The voltage drops across the resistors are:
ΔVab = − IR1 and ΔVcd = − IR2
Using all this in equation (24) we get:
ΔV1 − IR1 − ΔV2 − IR2 = 0
(25)
or by solving for I:
ΔV − ΔV2
(26)
I= 1
R1 + R2
So with this (not so clever) arrangement of batteries and resistors it is only the voltage
difference between the two batteries that generates the current – in particular there is no
current for ΔV1 = ΔV2 .
By reversing the polarity of the lower battery one would instead get:
ΔV + ΔV2
(27)
I= 1
R1 + R2
5/6
PH 106
Dr. T. Mewes
Electrical meters
Ammeter:
ΔV
+ I
I
A
R1
A device that measures current is called an ammeter. The ammeter must be connected
in series with the other elements in the circuit. An ideal ammeter should have zero
resistance so that the measured current is not altered. A typical resistance for an
ammeter is 60 Ω, which leads to problems when trying to measure circuits with
comparable or lower resistances.
Voltmeter:
ΔV
+ -
R1
V
A device that measures the potential difference between two points is called a voltmeter.
The voltmeter must be connected in parallel with the circuit element over which the
potential difference is to be measured. An ideal voltmeter should have an infinite
resistance, so that no current flows through the voltmeter. However, real voltmeters
always have a finite resistance.
6/6