The Wheatstone Bridge - Ryerson Department of Physics

OBJECTIVE
E9 - 1
MEASUREMENT OF RESISTANCE USING A WHEATSTONE BRIDGE
Various forms of the Wheatstone bridge are used to accurately determine the value of an
unknown resistance, capacitance or inductance. In this experiment, a Wheatstone bridge will be
used for the following purposes:
1. To check the specified tolerance of five ± 5% carbon resistors
2. To check the series and parallel formulas
and
THEORY
R S = R 1 + R 2 + R 3 (a)
for any three of the five resistors used for part 1.
!
1
R P
= 1
+
R1 1
+
R2 1
R3 In the circuit in figure 1, the unknown resistor is RX, and the resistor RB is known; the two
resistors RA and RD have a known ratio RA/ RD, although their individual values need not be
necessarily known. The galvanometer G measures the current flowing between the points B and
D. Either the known resistor RB or the ratio RA/ RD is adjusted until the voltage difference VBD is
zero and no current flows through G. At this point, when VBD = 0, the bridge is said to be
“balanced”.
_
+
S
B
RA
RB
Figure 1. The Wheatstone Bridge Circuit
IB
IA
I
I
IG
G
A
C
IX
RD
RX
ID
D
(b)

E9 - 2
Therefore, the following relationships will hold true for the electric potentials VA, VB, VC and
VD and the voltages (potential differences) VCB, VCD, VBA and VDA in the four branches.
First, for the branches marked CB and CD:
V CB = V C – V B = I B R B (1)
V CD = V C – V D = I X R X (2)
And since the bridge is balanced:
V BD = V B – V D = 0 (3)
From (1), (2) and (3) we obtain:
V BD = V B – V D = (I B R B - V C) - (I X R x - V C) = 0 (4)
Therefore:
I B R B = I X R x (5)
Since the current through the galvanometer IG = 0, according toKirchhoff’s first law for the
junctions denoted B and D, we have:
I A = I B (6)
I D = I X (7)
Similar to equations (1) to (3), for the branches marked BA and DA, we have:
VBA = V B – V A = I A R A (8)
V DA = V D – V A = I D R D (9)
And since the bridge is balanced:
V BD = V B – V D = 0 (10)
Now, from equations (8), (9) and (10):
Or:
and using (6) and (7), we obtain
V BD = V B – V D = (I A R A + V A) - (I D R D + V A) = 0 (11)
I A R A = I D R D (12)
I B R A = I X R D (13)
Dividing equations (13) by equation (5) we get: I B R B = I X R x
and solving for R X :
!
!
RX RD = R B
R A
R X = R B " R D
R A
(14)
(15)

PROCEDURE
E9 - 3
Set up the Wheatstone bridge circuit as in Figure 2.
Figure 2: Actual setup of the Wheatstone bridge.
Use the precision 500 Ω for R A and the decade resistance box for R B • R C is a variable resistor,
continuously adjustable from 0 to 1 kΩ. R X is the resistor whose resistance is to be measured. The
galvanometer G is a sensitive current sensing meter.
Precaution:
If the galvanometer swings rapidly off scale when the switch S is closed, then release the
switch quickly. Check and/or adjust your circuit before closing the switch again.
To determine the value of R X:
a. set the resistance box at about double the expected nominal value of the resistor R X
(use the colour code on page 5).
b. Close the switch S and adjust R D until the galvanometer reads zero. The bridge is
now balanced.
DATA ANALYSIS
Part A: Use the table below to record your data, and apply the method described in the theory
section above to measure the resistance of six resistors of different values. Determine the
percentage difference between the calculated and expected resistance values (use the colour coding
table at the end of this set of instructions to determine the manufacturer predicted resistance).
R B
(kΩ)
_
+
R D
(kΩ)
S
R A
(kΩ)
0.5
0.5
!
0.5
0.5
0.5
0.5
B
RA
RB
- decade box
G
A
C
R X1 = R B " R D
R A
(kΩ)
RD = Rpot
0 → 1 kΩ
D
R1, R2, R3, . . . R6
Expected Value
(Colour Code)
(kΩ)
Percentage
Difference (%)

E9 - 4
Part B: Use the table below to record your data, and apply the method described in the theory
section above to measure the resistance of any three of the previous six resistors connected in
series. Determine the percentage difference between the calculated and expected value.
R B
(kΩ)
R B
(kΩ)
R D
(kΩ)
R A
(kΩ)
!
R S = R B " R D
R A
(kΩ)
Expected:
R S= R 1 + R 2+ R 3
(kΩ)
Percentage
Difference (%)
Part C: Use the table below to record your data, and apply the method described in the theory
section above to measure the resistance of the same three resistors used in part B, connected in
parallel this time. Determine the percentage difference between the calculated and expected value.
CONCLUSIONS
R D
(kΩ)
R A
(kΩ)
!
R P = R B " R D
R A
(kΩ)
(1) Are all six resistors within the expected tolerance?
(2) Do the series and parallel equivalent resistance formulas:
!
Expected:
RP = 1
+
R1 1
+
R2 1
"
%
$
'
# R3 &
(kΩ)
RS = R1 + R2 + R3 (a)
1
RP = 1
+
R1 1
R2 + 1
R3 (b)
hold true within experimental error?
(1
Percentage
Difference (%)
(3) In ! case the expected and measured values are a lot more different, explain the possible
causes for this difference.
(4) How reliable does the Wheatstone bridge method seem to be in determining precisely
the resistance of an electric circuit?

COLOUR CODES FOR RESISTORS.
E9 - 5
Resistors are usually marked with coloured bands or rings to allow for identification. Referred to as
colour codes, these markings are indicative of their resistance, and tolerance.
A: first significant figure
B: second significant figure
C: third significant figure (in case of only 4 coloured bands, the third significant digit is not given!)
D: multiplier (factor by which the significant figures are multiplied to yield the nominal value)
E: tolerance (%)
Examples:
Colour Colour
Name
Digit
"A"
Digit
"B"
Digit
"C"
Black 0 0 0 1
Multiplier
"D"
Brown 1 1 1 10 ±1%
Red 2 2 2 100 ±2%
Orange 3 3 3 1k
Yellow 4 4 4 10k
Tolerance
"E"
Green 5 5 5 100k ±0.5%
Blue 6 6 6 1M ±0.25%
Violet 7 7 7 10M ±0.1%
Gray 8 8 8
White 9 9 9
Gold 0.1 ±5%
Silver 0.01 ±10%
Colour Code Resistance
A=Brown, B=Red, D=Orange, E=Gold. 12 KOhm ±5%
A=Brown, B=Red, D=Orange, E=Gold. 12 KOhm ±5%
A=Red, B=Violet, D=Green, E=Yellow 2.7 MOhm ±5%
A=Gray, B=Red, D=Orange, E=Gold. 82 KOhm ±5%
A=Yellow, B=Violet, C=Black, D=Red, E=Brown. 47 KOhm ±1%
A=Orange, B=White, C=Black, D=Orange, E=Brown 390 KOhm ±1%