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LECTURE 12
KIRCHHOFF’S RULES & MULTILOOP CIRCUITS
Lecture 12
2
¨
Reading chapter 25-4 and 25-5.
¤ Combination
of resistors
n Series
n Parallel
¤ Kirchhoff’s rules
¤ Ammeters, voltmeters, and ohmmeters
Quiz: 1
3
Resistors in series
4
¨
¨
¨
The current through each resistor must be the
same.
The potential drop across a single equivalent
resistor must equal the sum of the potential
drops across the individual resistors.
In general, for n resistors in series, the
equivalent resistance is
Quiz: 2
5
Resistors in parallel
6
¨
¨
¨
The potential drop across each resistor is the
same as that across a single equivalent resistor.
The total current is equal to the sum of the
current through each resistor.
In general, for n resistors in parallel, the
equivalent resistance can be calculated with
Demo: 1
7
¨
Series and parallel light bulbs
¤ Demonstration
of the different current in the two circuits, three light bulbs in
series and in parallel, by the relative brightness of the bulbs.
Example: 1
8
¨
Find R3 so that the equivalent
resistance between terminals a
and b, Rab, is equal to R1.
Demo: 2
9
¨
Conservation of current
¤ Demonstration
of the current flowing into a junction equaling the current
flowing out the junction.
I2
I1 = I2 + I3
I1
I3
Kirchhoff’s rules
10
¨
Junction rule: the sum of the currents into a junction must
equal the sum of the currents out of the junction.
¤
¨
𝐼" = 𝐼$ + 𝐼&
Loop rule: The sum of the changes in potential around any
closed loop must equal zero.
¤
¤
¤
¤
∆𝑉)* + ∆𝑉+, = 0
The potential drops passing through a resistor in the direction of
the current.
The potential increase going from the negative terminal to
positive terminal of a battery.
+ℇ − 𝐼𝑅 = 0
Quiz: 3 & 4
11
What is the current through R?
12
V
V
V
V
V
V
V
Good loop
13
V
V
V
V
V
V
V
+V + V + V − V − IR = 0 → I = 2V R
Example: 2
14
¨
Assume that R1 = 1.0 Ω, R2 = 2.0 Ω,
E1 = 2.0 V, and E = E2 = E3 = 4.0 V.
a)
b)
Calculate the current through each ideal
battery.
Calculate Va - Vb.
Quiz: 5
15
Galvanometers
16
¨
¨
A galvanometer is a device that detects small currents passing through
it.
The scale reading is proportional to the current passing through.
Ammeters
17
¨
¨
An ammeter is a device that measures currents passing through it.
To measure the current in the resistor, ammeter has to be connected in
series.
¨
Ammeters have very small resistance to minimize the effect of itself on the
current through the circuit.
We can construct an ammeter by connecting a galvanometer in parallel
with a shunt resistor, which has a small resistance Rp.
The majority of the current passes through Rp.
¨
The current flowing through the galvanometer is given by
¨
¨
Voltmeters
18
¨
¨
¨
¨
A voltmeter is a device that measures the potential difference
across its inputs.
The voltmeter has very large resistance so that very small
amount of current would flow through it, reducing the effect of
itself on the circuit.
We can construct a voltmeter from a galvanometer connected in
series with a large resistance.
The current through the galvanometer is given by
Ohmmeters
19
¨
¨
¨
An ohmmeter is a device that measures the resistance
across its inputs.
We can construct an ohmmeter from a galvanometer, a
battery, and a resistor, connected in series.
Rs is chosen so that when a and b are shorted
(negligible resistance), the current through the
galvanometer gives a full-scale deflection.
𝐼12 = 4
ℰ
5 647 6489
Example: 3
20
¨
R = 100 kΩ
a)
b)
If the voltmeter is ideal (infinite internal
resistance) what is the measured voltage
across R?
If the voltmeter has an internal resistance
of R V = 600 Ω what is the measured
voltage across R?