How to calculate internal resistance of a cell

Experiment 5

The Slide Wire Potentiometer

Abstract:

In this experiment, we are going to calculate the Electromotive force and the internal resistance for the three unknown batteries. Also, our goal is to calculate the average and Standard deviation of these three unknown batteries and do a comparison for the E.M.F and Internal Resistance for the three unknown batteries which are Old Battery, New battery, and Daniel Cell. Afterwards, we will calculate the uncertainty for the two cases (E.M.F. & Internal Resistance) for the unknowns. We expect our results for E.M.F for the three unknowns to be somewhat close to each other. For calculating E.M.F (and internal resistance “r” for these three battery cells, we used equation that are explained in the introduction.

Introduction:

We are going to observe how much or the maximum Voltage the three unknown batteries can have which is the Electromotive Force of a battery. As well we are going to observe and calculate the Internal Resistance for these three unknowns by changing the current in the battery and resistance. Refer to the equations for better understanding:

Electromotive force >> or 2)

Either one can be used to calculate the Electromotive force. Basically in equation 1 stands for Electromotive force for the standard cell, “x” stands for the length we measure for our unknowns (either New, Old, or Daniel cell) and “S” stands for the length for the standard cell. On the other hand, in equation 2 the “X” stands for length we measure for the three unknown, and stands for the voltage we set obtained from the Power supply.

Internal Resistance >>

“r” is the internal resistance for any of the three unknowns, stands for Electromotive force for our unknown batteries stands for terminal voltage which we obtain using the following equation: (x) and is the current (Amps) we calculate using this equation: I= Ω. In our Daniel cell, we used 50 Ω as our standard resistor because we were have trouble using 20 Ω.

Procedure:

Step 1: We assembled our circuit according to figure 1, and we measured “S” length (cm) for the standard cell using a slide wire and get the reading for the length when the galvanometer reads zero current (just like we did in experiment 4). We did it 4 times, or 4 runs.

Step 2: We connected our new battery to slide wire potentiometer, then we turned our switch towards the New battery instead of the standard cell and we did the same Procedure as step 1 to the length “X” using uniform slide wire so we can calculate the Electromotive force for these unknowns. In addition we prepared the solutions for the Daniel cell which is copper sulfate and zinc sulfate. We did the same procedure for the Old battery and Daniel cell.

Step 3: We got our calibrated standard resistor of 20Ohms, and we connected it to our circuit instead of so we can calculate the internal resistance for these three unknowns. We got our length “X” for our new battery using the uniform slide wire. We did the same procedure for Old battery, and Daniel Cell.

Apparatus:

Slide wire potentiometer, galvanometer (protective resistance), knife switch DPDT, standard cell, new 1.5 volt dry cell, old 1.5 dry cell, decade resistor box, Daniel cell, copper sulfate, and zinc sulfate.

Data:

· The e.m.f listed on the standard cell E = 1.01912 V

· The voltage of the power supply VAB = 2.0 V

a. New Battery:

· The balance position of the standard cell s = 55.0 cm

Length X (Centimeters) for Dry Cell

Length X(cm) for Loaded Dry Cell

Run 1

74.2

73.3

Run 2

77.2

75.0

Run 3

73.9

75.9

Run 4

76.1

73.6

Run 5

74.4

74.4

b. Old Battery:

· The balance position of the standard cell s = 53.3 cm

Length X (Centimeters) for Old Dry Cell

Length X (cm) for Loaded Old Dry Cell

Run 1

77.6

5.4

Run 2

77.4

6.6

Run 3

75.2

6.3

Run 4

77.7

6.2

Run 5

77.3

5.3

c. Daniel cell:

· The balance position of the standard cell s = 54.2 cm

Length X (Centimeters)for Daniel Cell

Length X(cm) for Loaded Daniel Cell

Run 1

54.5

33.5

Run 2

57.2

34.3

Run 3

56.4

34.9

Run 4

55.6

34.2

Run 5

55.5

35.2

Calculations:

· New Dry Cell:

x (cm)

xt (cm)

I (A)

r (Ω)

74.2

73.3

1.37

1.36

0.0679

0.246

77.2

75.0

1.43

1.39

0.0695

0.587

73.9

75.9

1.37

1.41

0.0703

-0.527

76.1

73.6

1.41

1.36

0.0682

0.679

74.6

74.4

1.38

1.38

0.0689

0.0538

· Old Dry Cell:

x (cm)

xt (cm)

I (A)

r (Ω)

77.6

5.4

1.48

0.103

0.00516

267

77.4

6.6

1.48

0.126

0.00631

215

75.2

6.3

1.44

0.120

0.00602

219

77.7

6.2

1.49

0.119

0.00593

231

77.3

5.3

1.48

0.101

0.00507

272

· Daniell Cell:

x (cm)

xt (cm)

I (A)

r (Ω)

54.5

33.5

1.02

0.630

0.0315

12.5

57.2

34.3

1.08

0.645

0.0322

13.4

56.4

34.9

1.06

0.656

0.0328

12.3

55.6

34.3

1.05

0.645

0.0322

12.4

55.5

35.2

1.04

0.662

0.0331

11.5

Conclusion:

The Experiment went as expected. We had a little disagreement, we expected the Electromotive force on the New battery to be higher than the E.M.F of the Old battery, although they are pretty much close (not that big of a difference). The reason for that is common sense, a “New battery” will have higher E.M.F. (max volts) in a circuit, than an old battery which is constantly used. Also, for our internal resistance we were satisfied since our internal resistance for our Old battery was much higher than the internal resistance for the new battery; in which we expected that since Internal resistance for a dry cell rises as the battery ages. While E.M.F pretty much stays the same.

Questions:

1, If we use a voltmeter we would have a different measurement for our voltage, because when using voltmeter we would have current flowing through the battery. While in slide wire potentiometer is adjusted in which no current flow through (terminal voltage) or Vt would give more accuracy when calculating the electromotive force. Therefore, the measured potential would be higher that the potential goes through the potentiometer.

2, Yes, we experienced quite a shift for the internal resistance in the New battery and Daniel cell, especially Old battery.

3, It is advisable to recheck the potentiometer because the standard cell has an average e.m.f. Es at 200C marked on each cell. The variation of Es with temperature in the 200C to 300C range is negligible compared to the errors inherent in the used of the slide wire used in this laboratory.

4, Yes, the measurements would get effected if the leads were long or short. We would have a different values for the internal resistance for the unknowns. Because to calculate internal resistance we depended on Vt and I in which I = Vt / 20 according to our lab manual. So if you change the 20 Ohms to different value, you would obtain different values for the internal resistance for the three unknowns.

5, Yes, the values are almost the same for E.M.F using equation 3 and using equation 5. However, equation 5 is probably more accurate since you are using the length “S” and E.M.F for the standard battery also.

For the New battery using equation 3: 74.2cm () = 1.48 Volts

For the New battery using equation 5: = 1.01912V( ) = 1.37 Volts

6, Comparing the values for and “r” from our analysis, we see that “r” (internal resistance) varies more as the battery ages. The reason is because the internal resistance increases as the battery becomes old because as battery gets old, chemicals inside the battery resist any current flowing inside of the battery resulting in higher Internal resistance.

I = 1.36 20

= 0.0679A

I=

1.36

20

=0.0679A

r = 1.37−1.36 0.0679

= 0.147Ω

r=

1.37-1.36

0.0679

=0.147W

)

(

V

X

e

)

(

V

V

T

εx = 1.37+1.43+1.37+1.41+1.38

5 =1.39V

e

x

=

1.37+1.43+1.37+1.41+1.38

5

=1.39V

σεX = (1.36−1.39)2 +...+ (1.38−1.39)2

5−1 = 0.03V

se

X

=

(1.36-1.39)

2

+...+(1.38-1.39)

2

5-1

=0.03V

r = 0.246+...+ 0.0538 5

= 0.208Ω

r=

0.246+...+0.0538

5

=0.208W

σ r = (0.246− 0.208) 2 +...+ (0.0538− 0.208)2

5−1 = 0.5

sr=

(0.246-0.208)

2

+...+(0.0538-0.208)

2

5-1

=0.5

Ω

W

)

(

V

X

e

)

(

V

V

T

εx =1.47V

e

x

=1.47V

σεX = 0.02V

se

X

=0.02V

r = 241Ω

r=241W

σ r = 27V

sr=27V

)

(

V

X

e

)

(

V

V

T

εx =1.05V

e

x

=1.05V

σεX = 0.02V

se

X

=0.02V

r =12.4Ω

r=12.4W

σ r = 0.6V

sr=0.6V

εX =1.01912× 74.2 55.0

=1.37V

e

X

=1.01912´

74.2

55.0

=1.37V

VT =1.01912× 73.3 55.0

=1.36V

V

T

=1.01912´

73.3

55.0

=1.36V