Bridge Measurements
5. Bridge Measurements*
Objective: The objective is to learn about the null measurement concept. The team will need to
figure out how to apply this concept to measure resistivity of a copper alloy.
The learning objectives are the following:
1. To be able to explain how a bridge circuit works.
2. To know how to use a Wheatstone bridge with slide wire to determine unknown
resistances, and resistivity of a wire sample of copper alloy.
3. To develop a realistic theory of errors applied to the Wheatstone bridge.
4. To learn about galvanometers and how to use them without damaging them.
5. To understand the practical advantages of null-type measurements.
(These ideas are based on materials presented in your textbook, and underlie several
modern measurement techniques.)
Reading assignment: Review series and parallel circuits and Kirchhoff’s laws.
Read the following sections. (Section numbers may be slightly different depending on the edition
of your textbook: Check the section titles.)Study the following sections of your textbook before
coming to lab. There will be a check to see that you have done this.
Knight, Jones and Field 162): 23.3 Series and parallel circuits, also read about Kirchhoff’s laws.
Serway and Vuille (212): 18.3 Resistors in Parallel , 18.4: Kirchhoff's Rules and Complex DC
Circuits
Serway and Jewett (252): 28.2 Resistors in Series and Parallel , 28.3: Kirchhoff's Rules (11)
Pre-Lab exercises: Bring solutions with you to the laboratory. Your instructor will check to see
that you have done them. Turn the solutions in with your report.
1. At the right is a circuit
comprising a battery and two
resistors. The ground symbol at
the bottom means the bottom
wire is selected as the reference
zero volts.
a
R1
OFF
V
V
b
(a) Draw lines from the
appropriate terminals on the
multi-meter to where the wires
would go to measure the voltage
at point b.
R2
A
c

(b) The knob on top of the
battery is its positive terminal. If
the battery voltage is Vo what
voltage would be measured at
point a?
V
A
COM
A
10A
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*© William A Schwalm 2012
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Bridge Measurements
(c) What voltage would be measured at c ?
(d) How much current flows through each resistor ?
(e) Thus, what is the voltage at point b ?
Now of course in the previous exercise you were assuming that no current flows through the
meter (infinite meter resistance) and that there is no resistance inside the battery. Neither of
these is quite true. If R1 were 22 Ohms and R2 where 150 Ohms, then your formula would be
almost perfectly correct.
2. Explain in a couple of sentences what would go wrong with the calculation in the previous
problem if R1 were 22 mega Ohms and R2 were 150 mega Ohms. Think about the meter itself.
3. At the right is what is sometimes called a bridge circuit. We say the bridge is “balanced” if no
current flows through the bridge resistor Rv (i.e. through the vertical resistor between a and b).
Here you will figure out how the other resistors have to be related in order to balance the bridge.
Assume the negative terminal of the battery (short line) is at zero volts, even though there is no
ground symbol in the circuit.
Let the battery voltage be Vo.
a
(a) Assuming that no current
flows through Rv, how is the
current that flows through R1
related to the current that flows
through R2 ? Notice that
current flows through the two
in series, in this case.
R2
R1
Rv
Rs
Rx
b
(+)
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Bridge Measurements
(b) Use the previous conclusion to figure out the voltage at a, assuming no current in Rv.
the algebra, not just the result.
Show
Va =
(c) Still assuming the bridge is balanced (meaning what ? ) find the voltage at point b. Thus
Vb =
(d) If the bridge is balanced, what is the relation between Vb and Va ?
(e) Thus using the results you have gotten, prove that when the bridge is balanced,
R2 Rx
,

R1 Rs
or equivalently, show that
Rs Rx
.

R1 R2
In-class group activities: At the top of the next page is a sketch of the apparatus you have
available.
1. Wire the apparatus the way it is shown. When you are finished, have your instructor inspect
your circuit before you actually begin making measurements.
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Bridge Measurements
G
Rs
Rx
a
0…......10……….20……..30…..….40….….50….…..60….…..70……..80……..90….....100
to wall receptacle or DC power
supply
b
Over the meter stick is fastened a meter of resistive wire. When the key is depressed, the two
lengths of wire act as two resistors.
2. Draw in the space provided a schematic diagram (resistors as
circuit, showing how it corresponds to the actual equipment.
etc. ) of the bridge
3. (Phys 212 and 252) If L1 = length of the wire forming R1 and L2 = length of the wire for R2,
then it is pretty clear that the resistance is proportional to the length—more length, more
resistance—so L2 / L1 = R2 / R1. Also, L1+L2 = L = one meter. Now suppose Rs (the standard
resistance) is known rather accurately. Show that if the error in measuring L1 is one millimeter,
then the relative error in measuring the unknown resistance is
Rx
.001 m 2

.
Rx
L1 1m  L1 
(What does this tell you about measuring?)
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Bridge Measurements
Note on galvanometers:
Basically, a galvanometer is a fairly sensitive ammeter, or mille ammeter.
However, the exact calibration of the scale is not important, because the
galvanometer is just used to determine
(a) whether or not a current flows, and if so (b) in what direction.
Because a galvanometer is rather sensitive it is rather easy to damage it by feeding it too much
current. There are several ways to protect it. One way is with a fuse. Most of the ones we use
have slow-blow fuses. This keeps the wiring
from burning out, usually, but it doesn’t protect
the needle from getting bent. If you damage a
galvanometer your group will loose points. Some
galvanometers have button switches (left). No
current flows until a button is depressed. Use the
one with the largest current (or voltage) rating
first, then the middle one and then the one that says GAL (most sensitive) last, so that you can
avoid burning it out. Sometimes a galvanometer without buttons has a resistor on one terminal
(center). Here you can make contact using an alligator clip either by clipping to the resistor wire
on the outside (low sensitivity) or connecting close to the galvanometer terminal (high sensitivity).
Finally, some galvanometers have no protection (right). Just be very careful with these.
current
out
current
in
current
in
current
out
Sign convention: The usual way a
galvanometer works is that:
The needle deflects
toward the terminal
where the current
comes in.
This is illustrated at the left.
Exploring the equipment: The way the setup works is that the galvanometer will deflect if the
bridge is not balanced. Thus you are only using the galvanometer to detect a balance condition
(no current) and not to measure anything. This is called a null condition, and the method
illustrates the idea of making null measurements.
Contact is made to the wire on the meter stick (slide wire) by means of a key, or a button pressing
a knife edge to the wire. When you use it you should just tap it, not hold it down. In particular,
don’t hold it down and drag it. There are two reasons:
(a) Until the bridge is balanced, you can get too much current flowing through the galvanometer.
You don’t want this to happen for more than a split second.
(b) If you hold the edge against the wire and drag it, it shaves a little bit off the wire. You are
counting on the wire having a nice uniform cross sectional area. Shaving it on one side ruins it.
Now you should fiddle around with the apparatus and try making some measurements. See for
example if you can measure some of the resistors that have (approximate) resistances written on
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Bridge Measurements
them. Of course, you need a standard resistance that is known very accurately. This takes the
place of Rs. The unknown is Rx. Be sure everyone in the team makes at least one practice
measurement. Make notes of things you discover as you go along.
Problem solving assignment: Your team needs to find the resistivity of the copper alloy wire used
to wire a large hotel. This is so that problems in the wiring can be located by the maintenance
staff using precise resistance measurements. Here’s what you have:
1. A spool of wire made of the copper alloy, length = 1.164 Kilometers (nearly a mile). The coil
can’t be unwound, because if forms part of an experimental setup in another lab.
2. Whoever wound the coil attached a tag to it. 22 gauge copper, 3400 turns
3. Your team has the equipment to measure the resistance using a standard resistance and a
Wheatstone bridge.
4. The Chemical Rubber Company handbook gives the resistivity for pure copper as
  1.72 108  / m  1  0.00393 (T  20C) / C .
Problem
Measure as accurately as you can the resistance of
the wire using the bridge circuit and then determine the resistivity.
The value should include a  error estimate.
Measurement plan: Work out a plan for making measurements and answering the customer’s
question accurately. The plan should include details such as how you will figure out the wire
cross-section without wrecking the coil. (You can’t just work backward, because you don’t know
enough about the alloy for that.) You need to say what measurements you will make, how you
will make them (including how to use the bridge) and how you will analyze the data. You also
need to say how you will figure out the accuracy. (What error bounds will you report to the hotel’s
maintenance crew, and how will you get them?)
Record your group’s plan here and include it in your individual reports.
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Bridge Measurements
Implementation: Carry out you plan, record the necessary data here and write notes to remind
yourself what all these data mean. As usual you will need to have your plan and data checked
before you leave.
Analysis: Write up your analysis here, explaining as you go.
Conclusion: Write here a brief report of what you did and what you found. This should be in the
form of a short, professional-looking paragraph or two explaining what you did and what you
found. It should be in a form such that your supervisor could transmit it directly to the customers,
i.e. to the hotel staff. They are keenly interested in the over-all error estimate as well as the
resistivity.
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Bridge Measurements
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