THE POTENTIOMETER:
Internal Resistance of a Test Cell
OBJECT
To calibrate a one meter slide wire potentiometer using a standard cell and then to use this potentiometer to measure the emf of a test cell. The terminal voltage of the same test cell is then measured as different load resistors are connected across the test cell and these data are used to determine the internal resistance of the test cell.
THEORY
The electromotive force (emf) of a cell is its terminal voltage when no current is flowing through it. The terminal voltage of a cell is the potential difference between its electrodes. A voltmeter cannot be used to measure the emf of a cell because a voltmeter draws some current from the cell. To measure a cell's emf a potentiometer is used since in a potentiometer measurement no current is flowing. It employs a null method of measuring potential difference, so that when a balance is reached and the reading is being taken, no current is drawn from the source to be measured.
Figure 1.
This is the basic circuit diagram for a potentiometer. Point
C is the sliding contact which can be adjusted for zero current deflection
through the galvanometer.
In this method (refer to Figure 1) a uniform, bare slide wire AB is connected across the power supply. If you were to connect a voltmeter between the + power supply terminal and point A you would measure essentially zero volts. If you were to now connect the voltmeter between the + power supply and point B you would measure a voltage equal to the terminal voltage of the power supply which is approximately 2.5 volts. The potential relative to point A then varies from zero at A to approximately 2.5 volts at B.
The cell whose emf is to be determined is then connected so that its emf opposes the potential along the wire. At some point C the potential difference between A and C is exactly equal to the emf of the cell so that if the other terminal of the cell is connected to the point C, no current will flow. The calibration procedure is to locate this point C using a standard cell whose emf is accurately known (emf = 1.0186 volts). You then know that at this point C the potential difference relative to point A is exactly 1.0186 volts.
Since the wire is uniform, the length of wire spanned is proportional to the potential drop and the wire can now be calibrated in volts per cm. The emf of an unknown cell is then found by finding a new point C whose potential is exactly equal to the emf of the unknown cell and multiplying this new distance AC times the calibration factor determined using the standard cell.
It is crucial in this experiment that the current flowing through wire AB remain constant throughout the experiment. If the current varies then the potential at all points along the wire will vary and you cannot trust your calibration. An ammeter is included in series with wire AB so that you can monitor this current. (See Figure 2.) The circuits used in this experiment are shown below in Figures 2 and 3.
In this
photograph, the apparatus used for the potentiometer is shown:
an adjustable resistor, test cell, standard cell,
galvanometer, ammeter, direct-current source, and the bare wire above
a meter stick along which the potential drop is measured.
Figure 2.
This circuit diagram
shows the location of the standard cell for calibrating the potentiometer.
Figure 3.
In this circuit diagram, the location of the test cell is shown for measuring the
voltage across the load resistor R.
Here e _{s} is the standard cell (emf = 1.0186 volts), and e _{x} is the unknown cell whose emf is to be measured. G is the galvanometer which has an internal resistor R_{1} in series with the meter to decrease its sensitivity. Once the potentiometer is balanced by adjusting point C until there is no deflection of G, switch K1 (a pushbutton on top of the galvanometer) is closed to increase the sensitivity of G by shorting out R_{1}. Point C is then further adjusted with K1 closed until there is no deflection of G.
Since the electromotive force of the standard cell is equal to the potential drop in the length of wire spanned (measured from A) for a condition of balance and the same is true for the unknown cell, the emf of each cell is proportional to the lengths of wire spanned. Thus
and the unknown emf is given by
where e _{x} is the unknown emf and, e _{s} is the emf of the standard cell, L_{x} is the length of wire (AC) used for balancing the unknown cell, and L_{s} is the length of wire used for balancing with the standard cell.
If we have a test cell of emf, e and internal resistance r supplying current to a variable load resistor R (see figure 4), then we will measure a terminal voltage V which is a function of the load resistance R.
Figure 4.
The load resistance is not external to the test cell, but it is shown in this
circuit diagram for the purpose of determing the voltage across the load
resistor R and finding the internal resistance r.
Since V = e - Ir, if you plot V versus I the negative of the slope of the graph will be the internal resistance of the cell r.
PROCEDURE
Use the experimental arrangement shown in Figure 2 for the calibration of the potentiometer wire, using the standard cell e _{s}. Start with your sliding contact C near the center of the bridge. Press the contact C. The galvanometer will probably deflect. Find a point C where there is no deflection. Now close switch K1 and again adjust C for no deflection. Pushbutton switch K1 (Figure 2) shorts out the protective resistance R1 and gives the galvanometer maximum sensitivity. Record the final setting of the contact point C and known value of the emf of the standard cell. Compute the calibration factor f in volts/cm.
Connect the test cell e _{x} into the circuit as shown in Figure 3. Determine the emf of this cell by again locating a point C where no galvanometer deflection occurs when contact C is pressed. Remember to close switch K1 for a finer adjustment of C. When no galvanometer deflection occurs with the switch K1 closed the potential drop along the wire from A to C exactly equals the emf of the test cell. Record the final balance position of the contact C and the emf of the test cell.
Now adjust the load resistor R to 150 ohms. You must hold switch K2 down in order for the circuit connecting R across the test cell to be complete. While holding K2 down again balance the bridge as described above. When balanced, again record the distance AC and compute the terminal voltage of the test cell. Repeat this procedure for R = 100, 60, 30, 15, 10, 8, 6, and 4 ohms. Using your measured value for the terminal voltage V and the resistance R, compute the current I being supplied by the test cell for each value of R used.
Plot a graph with terminal voltage V on the vertical axis and current on the horizontal axis. Draw the best straight line through your data points. Determine the value of the internal resistance of the test cell from this graph.
AN EXAMPLE USING THE PROGRAM: p23best1.m
THE ASSIGNMENT FOR PHYSICS 2322
Use the program p23best1.m to determine the best fit line among three sets of data. For load resistors of 150, 100, 60, 30, 15, 10, 8, 6, and 4 ohm, find the best fit parameters and the internal resistance of each cell. In the first set, the measured voltages were: 1.467, 1.466, 1.465, 1.457, 1.442, 1.428, 1.416, 1.401, and 1.369 V. In the second set of data, the voltages were: 1.261, 1.273, 1.255, 1.251, 1.228, 1.203, 1.183, 1.150, and 1.105. In the third set of data, the measured voltages were: 1.525, 1.523, 1.517, 1.498, 1.466, 1.446, 1.418, 1.403, and 1.340. For which set of data does one find the best fit to the measured values? Why? Explain and justify your answer. Examples of best fit and least squares analysis have been given by Etters (1996) and Pratap (1999).
REFERENCES