**Lab E3: The Wheatstone Bridge**

**Introduction**

The Wheatstone bridge is a circuit used to compare an unknown resistance with a known resistance. A schematic is shown below:

The unknown resistor is R_{x}, the resistor R_{k} is
known, and the two resistors R_{1} and R_{2} have a known
ratio , although
their individual values may not be known. A galvanometer G measures the
voltage difference V_{AB} between points A and B. Either the known
resistor R_{k} or the ratio is
adjusted until the voltage difference V_{AB} is zero and no current
flows through G. When V_{AB }= 0, the bridge is said to be "balanced".

Since V_{AB} = 0, the voltage drop from C to A must equal the
voltage drop from C to B, V_{CA} = V_{CB}. Likewise, we
must have V_{AD} = V_{BD}. So we can write,

(1)

(2) .

Dividing (2) by (1), we have

(3) .

Thus, the unknown resistance R_{x} can be computed from the
known resistance R_{k} and the known ratio .
Notice that the computed R_{x} does not depend on the voltage V_{o};
hence, V_{o} does not have to be very stable or well-known. Another
advantage of the Wheatstone bridge is that, because it uses a *null measurement*,*
*(V_{AB} = 0), the galvanometer does not have to be calibrated.

In practice, the Wheatstone bridge is seldom used merely to determine
the value of a resistor in the manner just described. Instead, it is usually
used to measure small changes in R_{x} due, for instance, to temperature
changes or the motion of microscopic defects in the resistor. As an example,
suppose R_{x} = 10^{6} W and
we wanted to measured a change in R_{x} of 1W
, resulting from a small temperature change. There is no ohmmeter which
can reliably measure a change in resistance of 1 part in a million. However,
the bridge can be set up so that V_{AB} = 0 when R_{x}
is exactly 10^{6} W . Then any change
in R_{x} , ,would
result in a non-zero V_{AB}, which, as we show below, is proportional
to .

You would not weigh a cat by weighing a boat with and without a cat
on board. Likewise, you would not want to measure very small changes in
R_{x} by measuring R_{x} with and without the change. Instead,
you want to arrange things so that the change in R_{x}, ,
is the entire signal. The bridge serves to "balance out" the
signal due R_{x}, leaving only the signal due to .

To show that ,we
consider Fig.1, and note that .
We assume that the galvanometer is a perfect voltmeter, so that no current
flows through it, even when the bridge is not balanced. We also assume
that the bridge has been balanced with the sample resistance at an initial
value of R_{xo }, so that .
Then we consider what happens to V_{AB} when the sample resistor
is changed by a small amount to a new value .

Applying Kirchhoff's Voltage Law and Ohm's Law to the upper and lower arms of the bridge, we have

(4) .

We are trying to find V_{AB}, which we can relate to I_{a}
and I_{b}.

(5) .

We can use (4) to solve for I_{a} and I_{b} and then
substitute for I_{a} and I_{b} in (5),

(6) .

This equation shows how V_{AB} depends on R_{x}. Notice
that eqn.(6) yields
when . To see
how much V_{AB} changes when R_{x} changes from R_{xo}
to , we write

(7) .

We wrote eqn.(7) by regarding V_{AB} as a function of R_{x},
and remembering (from Calculus) that if
then .

Substituting into (7) yields

(8)

Finally, we remember that V_{AB,initial} = 0, so the change
in V_{AB }is the same as V_{AB}, and we have

(9) .

** **

**Experiment**

We will use a slide-wire Wheatstone bridge, in which the two resistors
R_{1} and R_{2} are two portions of a single, uniform Ni-Cr
wire. Electrical contact is made at some point along the wire by a sliding
contact (this contact corresponds to point A). The two portions of the
wire on either side of the contact have resistances R_{1} and R_{2},
and the ratio is
the same as the ratio of the lengths of the two portions of wire, .
The lengths are readily measured with a meter stick which the wire rests
upon.

Fig. 2. Physical Layout of the Wheatstone Bridge.

The 10W resistor in series with the 6V battery
serves to limit the current through the bridge to less than 1A. (Higher
currents might over-heat components of the bridge.) The rechargeable battery
lasts only 6 hours when connected to the bridge, so please remember to
disconnect it when done. The known resistor R_{k} is adjustable
and can be set to any value from 1W to 999W
in 1W steps with a decade resistance box, which
is accurate to 0.02%. The Ni-Cr wire has a total resistance of about 2W
. The sliding contact is spring loaded; you have to push it down to make
contact to the wire at point A. There are two buttons on the sliding contact;
push one or the other, but not both, to contact the wire.

The galvanometer has three spring-loaded buttons, labeled 1, 2, and
3. The buttons close internal switches which connect the galvanometer to
the external voltage V_{AB }. The 3 buttons correspond to 3 different
sensitivities with button 1 the least sensitive and 3 the most sensitive.
*Always *begin your measurements with button 1 depressed, and adjust
the position of the sliding contact to balance the bridge (zero deflection
on the galvanometer). Then re-balance the bridge with button 2 and then
finally with button 3. *Never *begin with button 3. The galvanometer
is quite sensitive and can be easily damaged if a large voltage is applied
with button 3 depressed.

For Part 1 of the lab, the unknown resistor R_{x }is one of
5 coils of wire, numbered 1 through 5, mounted on a board. The lengths
of the wires, their composition, and their *gauge number* are printed
on the board. The gauge number is a measure of the thickness of the wire.
22 gauge wire has a diameter of 0.644mm; 28 gauge wire has a diameter of
0.321mm. For part 2 of the lab, R_{x} is a stand-alone coil of
copper.

The resistance R of a wire is related to its length L, its cross-sectional
area A, and its *resistivity* r by the
formula

(10) .

The resistivity r of a material depends on
composition, on defect structure, and on the temperature. For metals, r
is approximately constant at very low temperatures and
increases approximately linearly with temperature (measured in ^{0}K)
at high temperatures.

At T = 20C, the resistivity of pure, defect-free, copper is . (RT stands for room temperature.)

** **

**Procedure**

**Part 1: Resistivity of Copper**

In this section, you will use the bridge to make precise measurements of the resistance of each of the 5 coils of wire on the coil board. First, however, use the digital multimeter (DMM) to make an approximate measurement of the resistance of each coil. Also use the DMM to see how the resistance of the decade box changes when you turn the knobs. You must temporarily disconnect the coil board and the decade box from the bridge when testing them with the DMM. With the DMM, measure the resistance of each of the five coils to the nearest 0.1W . The resistance of the wire leads used to connect the DMM to the coils is not negligible, so first use the DMM to measure the resistance of the two wire leads in series, and then subtract this lead resistance from your measurements.

Now use the bridge to measure the resistance of each of the 5 coils.
Before connecting the battery to the bridge, carefully check that all the
connections are correct. Select a coil, attach it to the bridge, and set
the decade box resistance R_{k} to be as near to R_{x}
as possible(you know R_{x }roughly from your DMM measurements).
Balance the bridge by moving the sliding contact along the wire while watching
the galvanometer. Remember that when using the galvanometer you should
always begin with button 1, then button 2, and finally button 3. With the
bridge balanced, measure L_{1} and L_{2}, and compute

(11) .

Repeat this procedure for the other 4 coils.

For each of these coils, compute the resistivity, using your measured resistances, the data printed on the coil board, and eq'n (10). Make a table with the headings: coil #, R from DMM, R from bridge, and computed resistivity. Four of the 5 sample coils are made of copper (Cu). For these four coils, compute the average resistivity, and the uncertainty of the average (). Compare your average value with the known value. Coil #5 is made of a copper-nickel alloy. What is the ratio ?

**Part 2. Temperature dependence of resistivity.**

For this part, the unknown R_{x} is a copper wire coil (the
one which is not attached to the coil board). We will use eq'n(9) to measure
the change in resistance of the coil when its temperature is changed by
plunging it into an ice bath. Note that eq'n(9) can be written as

(9')

Prepare an ice bath by filling a beaker with ice from the icemaker (next to the video area) and adding some water. Measure room temperature and the ice bath temperature with the digital thermometer.

Measure the coil resistance R_{x} with the DMM and set R_{k}
equal to R_{x}, as nearly as possible. With R_{x} and R_{k}
connected in the bridge, measure with
the DMM. See Fig. 2 to locate V_{CD}. Note that V_{CD }is
__not__ the battery voltage.

For this bridge measurement, we must replace the galvanometer in the bridge circuit with the DMM, set to DC volts with the most sensitive range(400mV). We need to do this because the galvanometer does not behave as an ideal voltmeter, as assumed in the derivation of eq'n (9).

With the sample coil at room temperature, balance the bridge and compute
the coil's resistance at room temperature. Now without changing the position
of the slide wire contact, place the coil into the ice bath and watch as
the voltage V_{AB} on the DMM changes. Record V_{AB
}after it reaches equilibrium. Use eq'n(9) to compute ,
the change in resistance of the coil.

As a measure of how sensitive the resistivity is to temperature changes, we can compute the fractional change in the resistivity divided by the change in temperature,

(12) .

This quantity, multiplied by 100%, is the % change per degree. From your measurements, compute the % change in resistivity per degree for copper, and compare this value with the value you expect if . [Hint: If , then , where C is some constant, and .]

** **

**Questions**

**1.** Two copper wires, labeled A and B, are at the same temperature,
but the temperature is unknown. Wire B is twice as long and has half the
diameter of wire A. Compute the ratio R_{B}/R_{A}.

**2.** A 28 gauge copper wire is 30 meters long. What is its resistance
at room temperature?

**3.** Name two advantages of a Wheatstone bridge over an ordinary
ohmmeter.

**4. **What is the formula for the total resistance of two resistors
in parallel? Consider two resistors and
. What is the
total resistance of these two resistors in parallel. Give your answer to
the nearest ohm.

** **

**5.** Consider Fig.2 and recall that the total resistance of the
Ni-Cr wire is 2W . If R_{k} and R_{x}
are both very, very large compared to 2W , how
much current flows through the Ni-Cr wire?

**6.** Suppose an unknown resistance R_{x} is measured with
the bridge circuit shown in Fig. 2 and the result is R_{x} = 8.65W
. The 6V battery is then replaced with an 10V battery and R_{x}
is re-measured. What is the new measured value of R_{x}?

**7.** A sample of copper wire is 100m long and has a diameter of
0.250 mm. It's resistance is determined to be 38.0W
. What is the resistivity of the copper in this wire?

**8. (Counts as 3 questions) **A Wheatstone Bridge such as in Fig.
1 has a . With
R_{k} = 7.00W and the unknown resistor
R_{x} at room temperature, the bridge is balanced with L_{1}
= 44.0cm and .
What is the value of R_{x}? While still attached to the bridge,
the unknown R_{x} is placed in an oven, which raises its temperature
by 23.0^{o} C. The value of V_{AB} is then found to be
0.087V. What is the resistance change of the unknown, D
R_{x}? What is the % change per degree of the resistivity of this
sample?