Use of a Potentiometer in Determining the Resistance of a Cell


Introduction
============

The experiment involves the determination, of the internal resistance
of a cell.

It is known that the internal resistance (r), of a cell (Eo) in a
potentiometer circuit as shown below can be found using the formula:

[IMAGE]
r = R( ( lo/l) -1)

Where R is the resistance of the resister in parrallel with the cell Eo.

In this experiment the same galvanometer, driver cell jockey and
potentiometer wire were used throughout. This was done to ensure that
the experiment was kept as fair as possible and was not biased due to
misreadings resulting from changing the apparatus. Also during the
experiment a variable resister was used to make sure the same current
flowed through the potentiometer. Whilst the experiment was being
conducted. So that the voltage gradient (k) across the wire was kept
constant during the experiment, as if k were to change the readings
would be changing throughout the experiment.

The things that were varied in the experiment were, the resistance in
parallel with the resister and the balance length (lo). The balance
length will be measured using a meter rule placed directly underneath
and in parallel with the potentiometer wire. The variable resister
was used to give as large a balance length as possible. To keep the
percentage error in readings, down to a minimum. After the variable
resister was set at the beginning of the experiment the current was
then kept the same by using the variable resister and ammeter. To
keep the number of oscillations, for every mass as similar to each
other as possible. To help keep the experiment fair.

So to find r the following experiment was devised and carried out:

A potentiometer circuit was devised (as shown above), this was then
used the value of R in the circuit was varied to see what effect it
has upon the balance length L to take trial readings of R and l these
trial results are shown in the table below:

R/W

L/m

1/L (m-1)

1/R (W-1)

0

0.1

0.4

0.5

0.8

1

2

4

6

8

10

0.890

0.098

0.014

0.034

0.075

0.099

0.213

0.357

0.456

0.526

0.573

1.124

10.204

71.429

29.412

13.333

10.101

4.695

2.801

2.193

1.901

1.745

¥

10.00

2.50

2.00

1.25

1.00

0.50

0.25

0.17

0.13

0.10

From the results taken in the trial of the experiment the following
table was devised:

R/W

l/m1

l/m2

l/m3

Average l

1/l

1/R W -1

Percentage Error in 1/R

Percentage Error in 1/l

Absolute Error in I/R

Absolute Error in 1/l

The quantities 1/R and 1/L were calculated as to find a good estimator
of r a straight-line graph would need to be plotted the graph would
come from the following rearrangements of the formulae and
substitutions we get the formula:

1/R = lo/r ´ 1/l – 1 /r

The following lines show how the formula is obtained:

For the cell,

Eo=klo (equation 1)

When R is in the circuit no current flows through the bottom cell so
that the voltage across its terminals drop to VAB,

VAB = kl (equation 2)

As the wire is made of the material throughout (r = constant) and the
same cross sectional area A and the current I has remained the voltage
gradient, k is the same in both cases,

\1/2 gives Eo = klo

VAB kl

Eo = lo

VAB IR (equation 5)

As the p.d. across the cell is equal to the p.d. across the resistor
you get,

VAB = IR (equation 3)

Applying the circuit equation to the bottom loop you get,

Eo = I(R+r) (equation 4)

On sub substituting in equation 5 you get,

I(R+r) = lo

IR l

As the current is the same you get,

(R+r) = lo

R l

Through further substitution,

(R+r) = R(lo/l)

r = R(lo/l) –R

r = R( ( lo/l) -1) )

From which r can be found as R, lo are known,

r = R( ( lo/l) -1)

r = lo – 1

R l

And after further substitution you get,

1 = lo ´ l – 1

R r l r

Which is in the form of y = mx + c

So then a straight-line graph of 1/R against 1/L could be plotted.
This graph would be used to calculate the gradient of the graph and
therefore the value of constant r. Which could be found by
rearranging the equation and using information from the graph as shown
above.

The intercept on the y-axis will be the value of 1/r so the value of r
can be easily found using this fact.

Apparatus

Potentiometer wire with meter rule, ammeter, resistance box, variable
resister, driver cell, cell of unknown internal resistance,
galvanometer, two switches, wires and a jockey, multimeter.


Circuit Diagram

[IMAGE]


[IMAGE]


The following diagram shows how the galvanometer was read from
directly above to avoid any parallax errors:

Results Section

Characteristics of Instruments

These show the characteristics of the instruments that were used in
the experiment they were taken before the experiment was conducted:

Instrument

Meter Rule

Galvanometer

Resistance Box

Ammeter

Multimeter

Range

0.001 to 1.000 m

+1 to –1 A

0.0 to 1000.0W

50.00 to –50.00 A

0.00 to 200 W

Resolution

0.001m

0.1 A

0.1W

0.01A

0.01W

Sensitivity

0.001m

0.1A

0.1W

0.01A

0.01W

Zero Error

0.000m

0.0A

0.0W

0.00A

0.00W

Uncertainty

0%

0.1A

5%

0.00A

0.1%

Trial Readings

R/W

L/m

1/L (m-1)

1/R (W-1)

0

0.1

0.4

0.5

0.8

1

2

4

6

8

10

0.890

0.098

0.014

0.034

0.075

0.099

0.213

0.357

0.456

0.526

0.573

1.124

10.204

71.429

29.412

13.333

10.101

4.695

2.801

2.193

1.901

1.745

¥

10.00

2.50

2.00

1.25

1.00

0.50

0.25

0.17

0.13

0.10

These were the trial readings taken before the experiment, they were
taken to help in the decision of the size of the limits. They were
used to make sure the results we were getting were reasonable and that
the lower and upper bounds produced comparable results that could be
used in the graph.


Main Readings

R/W

l/m1

l/m2

l/m3

Average l

1/l

1/R W -1

Percentage Error in 1/R

Percentage Error in 1/l

Absolute Error in I/R

Absolute Error in 1/l

0

0.9

0.9

0.900

0.900

1.111

0.00

5

0.1

0.00

0.001

0.7

0.107

0.1066

0.106

0.110

9.380

1.43

5

0.1

0.72

0.009

0.8

0.123

0.12

0.120

0.120

8.260

1.25

5

0.1

0.06

0.008

0.9

0.13

0.129

0.123

0.130

7.850

1.11

5

0.1

0.06

0.008

1

0.143

0.14

0.145

0.140

7.010

1.00

5

0.1

0.05

0.007

2

0.23

0.223

0.217

0.220

4.480

0.50

5

0.1

0.03

0.004

4

0.388

0.398

0.386

0.390

2.560

0.25

5

0.1

0.01

0.004

6

0.485

0.497

0.486

0.490

2.040

0.17

5

0.1

0.01

0.002

8

0.544

0.566

0.560

0.560

1.800

0.13

5

0.1

0.01

0.002

10

0.614

0.62

0.650

0.630

1.590

0.10

5

0.1

0.01

0.002

Procedure/Method

When planning the experiment, there were two methods of obtaining
information about the internal resistance of the cell, which were
discussed one being the one decided upon. The other being a circuit
based on the circuit equation. This experiment was chosen over the
other one, as there were practical problems as a voltmeter would have
to be in parallel with the cell and as it would not have ¥ resistance
it would over complicate the experiment and give rise to less accurate
results.

The procedure in the experiment was as follows:

1. The apparatus was set up as shown in the diagram.

2. A resistance was set on the resistance box the resistance was then
checked using the multimeter. Observers of the balance length were
then taken, the galvanometer was read from directly above to avoid any
parallax errors.

3. Once one observation was recorded another resistance was set on the
resistance box (observations were taken going up the resistance scale
and then on the way down again to make sure there was no bias to
results).

4. Then the apparatus was disassembled and put safely away.

Calculations/Derived Quantities

To find the value of r the following calculations were made:

Firstly the y intercept (which is equal to 1/r) was found from the
graph, it was found to be –0.1917 W-1.

This could then be used to find the value of r:

r =1¸ 0.1917

r = 5.2165 W

To find the uncertainty in r error bars were plotted on the graph and
were used to find the maximum and minimum values of r as follows:

For the minimum value, the maximum value of 1/r was used as follows:

r =1¸ 0.3073

r = 3.2541 W

For the maximum value, the minimum value of 1/r was used as follows:

r =1¸ 0.063

r = 15.8730 W

Using these results it is now possible to give the result of the
experiment giving an absolute error:

15.8730 + 3.2541= 19.127

19.127 ¸ 2 = 9.564 W

\ the result form the experiment is that r = 5 +/- 10W

[IMAGE]

Conclusion

The graphs show that the formula, 1 = lo ´ l – 1 does indeed give a
straight-line graph of form Y = mx + c.

R r l r

The graphs show that the value of r = 5 +/- 10 W.

The objective of this experiment was to determine r of the cell which
has been done within the limits, +15W and -5W.



Critical Analysis of Results
============================

The experiment could have been improved in the following ways:

1. More readings of l could have been taken to give a better average.

2. A greater range of resistance could have been used to get a more
accurate trend line for the graph.

3. There could have been more time for the experiment, so that the
readings etc would not be so rushed.

4. A longer potentiometer wire could have been used to give less
percentage error as the balance length would be longer.

5. The contact resistance in any connections could be reduced by using
soldered joints or cleaning contacts.