V = IR
By examining the previous formula, this is less than the e.m.f by Ir, referred to as lost volts, and by combining the last two equations I get the following:
V = IR – Ir or V = E – Ir
For graphical method, we can plot a graph of Voltage against Current. The graph should result in a straight line because the graph of V against I is proportional. The equation for a straight line is Y = MX + C, which can be applied to the e.m.f formula V = E-Ir, given that the voltage is on the Y-axis, the intercept (c) equals the electromotive force and the gradient (m) is equal to the internal resistance r.
Procedures
- The switch was unplugged.
- The rheostat was turned to the maximum value.
- All the apparatus was connected as the above electric diagram.
- The switch was plugged.
- The readings of the voltmeter and the ammeter were recoded.
- The rheostat was changed into different values.
- The corresponding readings of ammeter and voltmeter were recorded.
- The data obtained was tabulated.
Precautions
To ensure the accurate and precise data is obtained, we should be aware of the following precautions:
Firstly, the internal resistance of the dry cell is not constant; it increases with the time of the cell used. Therefore, the longer time is duration of the experiment, the larger is the internal resistance of the dry cell. As a result, we should not leave the circuit connected longer than necessary to take the readings.
Secondly, the rheostat should be set to its maximum value in the beginning of the experiment, so that the current is the lowest at first, then increase gradually. As high current produces heat, it would increase the resistance of the connecting wires and the internal resistance. The current through the wire will heat up the wire and lead to increase resistance of the apparatus. It may lead to the inaccurate and imprecise data obtained and hence the inaccurate calculated value of the e.m.f and internal resistance in the dry cell. Therefore, we should not leave the circuit connected longer than necessary to take the readings.
In addition, we should ensure the ammeter and voltmeter are connected to the cell with suitable way (positive terminal to the direction of positive terminal and negative terminal to the direction of negative terminal). Hence the ways of connection of ammeter and voltmeter also should be in correct ways (voltmeter in parallel while ammeter in series). Otherwise, the pointer will deflect to the opposite direction. The ammeter and voltmeter may be damaged.
Results & Calculations
The values of e.m.f and the internal resistance of the dry cell used in the experiment are calculated in both mathematical and graphical method:
For mathematical method,
Two dry cells are used:
V represents the e.m.f. of the dry cell.
With the key plug opened, the voltmeter reading (V ) is 2.6 V ± 0.1 V.
E.m.f of one cell = 2.6 / 2 = 1.3 V ± 0.1 V.
Total resistance: V = – Ir
1.40 = 2.60 – 0.16 r
r = 7.50 Ω
The internal resistance of the dry cells: 7.50 Ω – resistance of the resistor
= 7.50 Ω – 5.60 Ω
= 1.90 Ω
Maximum error in ( – V) = (0.10 + 0.10) V
= 0.20 V
Maximum error in total internal resistance:
= ± [(0.20 / 1.2) + (0.01 / 0.16)] Ω
= ± 0.23 Ω
The uncertainty in the total internal resistance:
= ± 0.23 x 7.50
= ± 1.73 Ω
Total internal resistance of the dry cell = 7.50 Ω ± 1.73 Ω
Since the resistance of the resistor was assumed to be 5.6 Ω with no error,
The maximum error in the internal resistance of the dry cell = (1.73 / 2) Ω= 0.87Ω
The internal resistance of one dry cell is 0.95 Ω ± 0.87 Ω.
Maximum percentage error in the total internal resistance of the dry cell:
= 0.23 x 100%
= 23.0 %
Maximum percentage error in the internal resistance of the dry cell:
=(0.87 / 0.95) x 100%
= 0.92 %
For graphical method,
From the graph,
e.m.f of the dry cells= y-intercept = 2.58 V
e.m.f of one dry cell= y-intercept = 1.29 V
The total resistance = -slope = (1.14 – 1.21) / (0.20 – 0.19) = 7Ω
The total internal resistance of two cells = 7 –5.6 = 0.14Ω
The internal resistance of one cell = 0.14/2 = 0.7Ω
The coordinate of the centroid C: (0.24, 0.87)
Let the y- intercepts of the two good- fit lines be c1 and c2.
c1 = 2.84
c2 = 2.40
Δ c = [(2.84-2.58) + (2.40- 2.58)] / 2
= 0.04
Let the slopes of the two good- fit lines be m1 and m2.
m1= - (0.87 - 2.84) / (0.24 - 0.00) = 8.21
m2= - (0.87 – 2.40) / (0.24 - 0.00) = 6.38
Δm = [(8.21-7) + (6.38-7)] /2
= 0.30
The e.m.f. of the dry cell: 1.24 V ± 0.04 V
Total internal resistance of the dry cell: 0.7Ω ± 0.3 Ω
The e.m.f. of one cell = 1.29 V± 0.04V
Since the resistance of the resistor was assumed to be 5.6 Ω with no error,
The maximum error in the internal resistance of the dry cell = 0.3 Ω
The internal resistance of the dry cell is 0.7Ω ± 0.3 Ω.
Maximum percentage error in the e.m.f. of the dry cell:
= (0.04/1.29) x 100%
= 3.1 %
Maximum percentage error in the internal resistance of the dry cell:
= ± (0.3 / 0.7) x 100%
= 42.9 %
Discussion
Accuracy & improvements
In this experiment, we had made several experimental errors but we can improve the experiment by the following improvements.
For a start, the change in the rheostat only leads to a very small variation of current and voltage, the scale of the ammeter and voltmeter cannot show that tiny variation in current and voltage. As the ammeter and voltmeter cannot show the more accurate and precise readings, the data used for calculation also will be inaccurate and hence the calculated value of the internal resistance will be inaccurate. To improve this error, we should choose other ammeter and voltmeter with smaller scale which can enhance the accuracy of the results.
Secondly, the set-up of the experiment is used for long time or the switch is plugged on for a long time, it will produce heat to the set-up and then the resistance of the set-up will increase. As a result, the calculated internal resistance of the dry cell will be greater than the expected as we have not minus the total resistance by the resistance of resistor and the extra resistance produced by the heat. To get rid of the error, we should open the circuit after taking each set of readings by unplugging the switch.
Besides, the connecting wires used in the experiment are ideal wires with no resistance; it will cause inaccurate readings of both ammeter and voltmeter. Each wire has its own resistance and then most of them are connected in series circuit. The resistance of the whole set-up will increase as the sum of the resistances of each wires have been added to the total resistance of the set-up. The calculated internal resistance will be greater than expected. To do away with such error, we should use the thicker wire, which is made up of copper to minimize resistance of the wires.
On top of that, the connecting joints of the cells had rusts on it, it may affect the conductivity of the current and the resistance of whole set-up will increase. As the conductivity is decreased, the e.m.f measured by the voltmeter is not accurate and may be smaller than the expected. As the resistance of the set-up increases, the calculated internal resistance will be greater than expected. To avoid such error, we should check the connecting joints of all the apparatus have no rust. If there is rust, we should another one to replace to enhance the accuracy and the precise of the meters.
Finally, the ammeter and voltmeter are not ideal and have zero error. It leads to the inaccurate values of e.m.f and internal resistance of the dry cell. In the reality, there is also a totally ideal ammeter and voltmeter, we only can choose the meters, which is more ideal than others. So we carry out some experiments to find out which ammeter and voltmeter is more ideal and smaller zero errors.
In this experiment, there are some assumptions we had made. Firstly, we assume that he d.c. voltmeter has infinite internal resistance and no current would be drawn from the circuit by the voltmeter. In the reality, there is no ideal voltmeter and it causes inaccurate readings of the voltmeter. So we only can assume it is ideal to calculate the internal resistance and e.m.f.of the dry cell. Secondly, we also had assumed the d.c. ammeter has zero internal resistance. Just like the assumption in voltmeter, there is no an ideal ammeter without internal resistance. The only thin we can do is to assume the ammeter is ideal. Besides, we also had made the assumption that the connecting wires and the key plug have zero resistance. Otherwise, the resistances of them are needed to measure that is very difficult to find out and the calculated internal resistance is inaccurate. Finally, we assume that the resistance of the 5.6 Ω resistor is ideally 5.6 Ω. The resistance of the resistor is easily affected by several factors such as temperature and magnetic fields.
The experimental errors can be divided into two errors, systematic errors and random errors. The zero errors of the ammeter and voltmeter, the heating effect on resistance, the internal resistance of the connecting wires and unclear scales of the ammeter and voltmeter are belonged to the systematic errors. Rusting on the connecting joints is the random errors. An experiment with small systematic and random errors is more precise.
Questions
6. determined graphically is 0.7Ω ± 0.3Ω while the value of V0 determined in step 2 is 0.95 V ± 0.01 V. So, the graphical measurement is more reliable.
- It is a bad method because it cannot show any errors which are random and mistakes either cannot easily be seen and not included. So that the more inaccurate results is obtained.
Conclusion
By both mathematical and graphical methods, we can find out the e.m.f and the internal resistance of one dry cell.
For mathematical method, the e.m.f and internal resistance of the cell are 1.3 V ± 0.1 V and 0.95 Ω ± 0.87 Ω. respectively. For graphical method, the e.m.f and internal resistance of the cell are 1.29 V ± 0.04 V and 0.7 Ω ± 0.3 Ω respectively. Either the e.m.f from the mathematical method or the one from the graphical method are not equal to the value stated on the experiment menu, 1.5V. It may result from the consumption of chemical energy in the dry cell as the dry cell may be used for a period of time. In addition, they are closed to each other; it means that the results calculated are precise.
However, internal resistances calculated from both methods are not closed to each other and there are many experimental errors in the experiment. To obtain a more accurate and precise result, we should have more preparations and minimize the experimental errors as much as possible. For further investigation, we can find out the effect of heating on the internal resistance of the apparatus especially the dry cell.
References
Wikipedia (internal resistance)
http://en.wikipedia.org/wiki/internal_resistance
New Way Physics for advanced level Fields, Electricity and Electromagnetism P.125 – 126 for better understanding of internal resistance.
