PES 1120 Spring 2014, Spendier
Lecture 23/Page 1
Today:
- light bulb example
- multiloop circuits
- Devices that measure current and voltage
- RC circuits (time-varying currents)
Las time we stopped with:
n
1
1
Resistor in parallel:

Req
i 1 Ri
n
Resistor in series: Req   Ri
i 1
and the demo of light bulbs in series and parallel
DEMO: Light bulbs, series vs parallel combinations
Example 1:
Two identical light bulbs are to be connected to a source with ε = 8 V and negligible internal
resistance. Each light bulb has a resistance R = 2  . Find
i) the current through each bulb,
ii) the potential difference across each bulb, and
iii) the power delivered to each bulb and
iv) the entire network
if the bulbs are connected
a) in series and
b) in parallel.
c) If your goal is to the maximum amount of light, which circuit combination would you
choose?
d) Suppose one of the bulbs burns out; that is its filament breaks and current can no longer
flow through it. What happens to the other bulb on the series case? In the parallel case?
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Lecture 23/Page 2
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c) Both the potential difference across each bulb and the current through each bulb are twice
as great as in the series case. Hence the power delivered to each bulb is four times greater,
and each bulb glows more brightly than in the series case. If the goal is to produce the
maximum amount of light from each bulb, a parallel arrangement is superior to a series
arrangement.
d) In the series case the same current flows through both bulbs. If one of the bulbs burn out,
there will be no current at all in the circuit, and neither bulb will glow. In the parallel case the
potential difference across either bulb remains equal to 8V even if one of the bulbs burn out.
Hence the current through the functional bulb remains equal to 4 A. This principle is used in
household wiring systems.
PES 1120 Spring 2014, Spendier
Lecture 23/Page 4
More on Multi-loop circuits: Kirchhoff’s Circuit Rules

1) Loop rule
V  0
closed loop
2) Junction rule
I
in
  I out
Problem-Solving Strategy: Applying Kirchhoff’s Rules
Kirchhoff’s rules can be used to analyze multiloop circuits. The steps are summarized
below:
(1) Draw a circuit diagram, and label all the quantities, both known and unknown. The
number of unknown quantities is equal to the number of linearly independent
equations we must look for.
(2) Assign a direction to the current in each branch of the circuit. (If the actual direction
is opposite to what you have assumed, your result at the end will be a negative number.)
(3) Apply the junction rule to all but one of the junctions. (Applying the junction rule to
the last junction will not yield any independent relationship among the currents.)
(4) Apply the loop rule to the loops until the number of independent equations obtained
is the same as the number of unknowns. For example, if there are three unknowns, then
we must write down three linearly independent equations in order to have a unique
solution.
Traverse the loops using the convention below for ΔV:
The same equation is obtained whether the closed loop is traversed clockwise or
counterclockwise. (The expressions actually differ by an overall negative sign. However,
using the loop rule, we are led to 0 = -0, and hence the same equation.)
(5) Solve the simultaneous equations to obtain the solutions for the unknowns.
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Lecture 23/Page 5
Example 2: The Figure shows a multiloop circuit containing three ideal batteries and five
resistances with the following values:
R1 = 2.0  , R2 = 4.0  , ε1 = 3.0 V, ε2 = 6.0 V
Find the magnitude and direction of the current in each of the three branches.
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One more circuit symbol: Grounding a circuit
For Figure (a), point a is directly connected to ground, as indicated by the common
symbol
. Grounding a circuit usually means connecting the circuit to a conducting
path to Earth’s surface (actually to the electrically conducting moist dirt and rock below
ground). Here, such a connection means only that the potential is defined to be zero at the
grounding point in the circuit (the wire connected to ground). Thus in Figure above, the
potential at a is defined to be Va = 0. Figure (b) is the same circuit except that point b is
now directly connected to ground. Thus, the potential there is defined to be Vb = 0.
Measurement Devices
Voltmeter
- Measures the voltage across a circuit element. Voltmeters must be
placed in parallel. To keep any current from flowing into them, voltmeters have nearly
infinite resistance.
Ammeter
- Measures the current through a circuit element. Ammeters must be
placed in series. To avoid having a large effect on the circuit, ammeters have almost no
resistance.
Ohmmeter - measures the resistance of any element connected between its terminal.
Any instrument that measures voltage or current will disturb
the circuit under observation. Some devices, known as
ammeters, will indicate the flow of current by a meter
movement or a digital display. There will be some voltage
drop due to the resistance of the flow of current through the
ammeter. An ideal ammeter has zero resistance, but in the
case of your multimeter (single device can act as a ammeter
or a voltmeter) the resistance is 1Ω .