EXPERIMENT 1
OHM'S LAW. RESISTORS IN SERIES AND IN PARALLEL.

OBJECT:

To verify Ohm's law, to study the voltage, current, and resistance relationships for circuits containing resistors in series and in parallel. 

METHOD:

The voltmeter-ammeter method of the measurement of resistance. 

APPARATUS:

Variable DC voltage supply, resistor board, connecting wires, and computer with current/voltage probes. 

THEORY:

Electrical current is the amount of charge passing by a given point in a conducting path (circuit) per unit time:

The unit of current is the Ampere, which is equal to a (Coulomb/second) and, although it is defined by other relations, a current of one ampere exists in a wire if approximately 6.21 x 1018 electrons (A charge of one Coulomb) flow through a given cross-section of wire in one second. It is agreed for convenience that the direction of the current is the same as the direction of movement of positive charges in electric field. In a metallic conductor, such as a wire, the only mobile particles are negatively charged electrons, which move in a direction opposite to that chosen for the conventional current.

The relationship between the voltage, current, and resistance in a metallic conductor is given by Ohm's law. It states as follows: If the temperature and other physical conditions of a metallic conductor are unchanged, the ratio of the potential difference across the conductor (V) to the current (I) is a constant. This constant ratio (R) is the resistance of the conductor.

If potential difference is measured in Volts and current is in Amperes, resistance will be in Ohms.

The resistance of a metallic conductor depends only on its length, the area of cross-section, the material of the conductor and its temperature. It does not depend on either V or I. Ohm's law may be applied either to the entire circuit or to any portion of the circuit in which there is no emf (source of voltage). Whenever this law is applied to the entire circuit, the voltage to be used is the net emf E and the resistance is the total resistance Rt of the circuit

The potential difference across a device may be measured by connecting a high resistance voltmeter V in parallel with it. Fig. 1 shows the conventional wiring diagram in which a voltmeter is used for the measurement of the terminal voltage of a battery. The current in a device is measured by inserting an ammeter in series in the circuit. All the electricity passing through the resistor thus has to pass through the ammeter.

  Fig. 1 Measurement of voltage battery by volt-meter

 Because most ammeters have a very small resistance, they do not greatly affect the current. In Fig. 2 an ammeter A is shown in series with a resistor R in a simple circuit.

   Fig. 2 Ammeter measuring of current in resistor

A straightforward technique for measuring the resistance of a resistor would be to measure V and I with a voltmeter and ammeter and find the ratio (Eq. 1). However, the introduction of meters into the circuit alters the circuit so that the measured values may not be the same as the actual values. Figures 3 and 4 show two methods of connecting a voltmeter and ammeter to determine the resistance R.

Figure 3.

 In the first method (Fig. 3) the ammeter A measures the total current I in the main circuit, but a portion of this current, Iv, shunted around R through the voltmeter. Hence, the ammeter reading does not give the exact value of the current I' in R, and, consequently, V/I does not give the exact value of R. The voltmeter reads the potential difference V between its own terminals as well as across the resistance R, so that here the voltmeter reading (V) is equal to V'.

If the resistance of the voltmeter Rv is known, the value of Iv is given by

Since the current I' through R is I - Iv we have for the corrected value of R.

The second method of connecting the meters is shown in Fig 4.

Figure 4.

In this arrangement, the ammeter reads the correct value of the current I' passing through the resistor, but the voltmeter V reads the potential drop across both the resistor R and the ammeter A. If Ra is the resistance of the ammeter, then
 

Note in Equation (3) that the larger the resistance of the voltmeter the less the effect on the measurement. Also, from Equation (4), we see that the smaller the resistance of the ammeter the less the effect on the measurement.

If we have a complicated circuit including resistors in series and in parallel the best way to analyze it is to break it down into equivalent simple resistances.

Resistors in Series. Conductors are said to be connected in series when they are joined as shown in Fig 5 so that electricity flows uniformly from one resistor into the next.

Fig. 5 Resistors in series

In this case:

1. The current in all of the resistors in a series circuit is the same

2. The voltage across a group of series resistors is the sum of the voltage across the individual resistors 3. The total resistance of a group of series resistors is the sum of the individual resistances Resistors in Parallel. When conductors are arranged as in Fig.6, so that the total current divides among the resistors with the respective ends of the resistors connected to common points, they are said to be in parallel.

Fig. 6 Resistors in parallel

In this case:

1. The currents in the various resistors are different and are inversely proportional to the respective resistances. The total current is the sum of the individual currents.

2. The voltage across each resistor is the same and is identical with the voltage across the whole group considered as a unit. 3. The reciprocal of the total resistance of the group is the sum of the reciprocals of the individual resistances. Connecting additional resistors in series increases the total resistance, but connecting additional resistors in parallel decreases the total resistance. The total resistance of a parallel group is always smaller than the resistance of the least resistor in the group. 

PROCEDURE:

Set up Procedure:

1.  Make sure all power supplies are OFF,  and that the voltage meter range is set to 10 V, and the voltage adjust knob is zeroed.  Set the Short Circuit Current  to 225 mA.

2. Plug in the Dual Channel Amplifier wires. Din 1 goes to Din 1 on the ULI box, and Din 2 goes to Din 2. Take a current box and plug it into Probe 2 on the Dual Channel Amplifier (this serves as the Ammeter in the circuits.) Plug the voltage probe (with the alligator clips) into Probe 1 on the Dual Channel Amplifier. This is your voltmeter.

3. Double click on the Logger Pro folder, open the UTC Physics folder, then open "Ohm's Law PT.MBL".  Click on the "zero" button at the top of the screen to zero all sensors. Click on the "Collect" icon and erase the most recent data and graph. In taking the following data, be careful to keep the voltage less than 5.0 V.


Data Collection

1. Connect the desired circuit shown in Figure 3. If you are using a resistor board with an even number on it, use resistor B.  If you are using an odd-numbered resistor board, use resistor E.

2. Click on the Start button. Turn on the voltage supply. Set the voltage reading on the power supply to 0. Gradually increase the voltage and press the "KEEP" button. Repeat this procedure until you have collected 8 points, making sure that you keep the voltage level beneath 5 V.

3. Now, click on Stop.

4. Go to Analyze and drag to Automatic Fit… A window appears.

7. Repeat the above procedure for two other single resistors. For even-numbered resistor boards, use resistors C and D.  For odd-numbered resistor boards, use resistors C and D as well.  To collect new data, simply click on the Start button again.

8. Connect the circuit for three resistors in series shown in Figure 5. Use the same three resistors. Repeat the Data Collection Procedure. Print the obtained graph.

9. Connect the circuit for three resistors in parallel shown in Figure 6. Use the same three resistors. Repeat the Data Collection Procedure. Print the obtained graph.

10.  Close the Logger Pro program (do not save anything). Shut down the computer by clicking on Start icon at the bottom left-hand corner of the screen, and choose "shut down".

11. Disconnect all circuits and leave the materials as they were on the lab bench. 

DATA ANALYSIS:

1. The slopes of the graphs obtained are the respective resistances for each circuit as can be seen in equation (1). Report the resistances in your results and conclusions as R ±DR W.

2. Using the experimentally measured resistances for the three individual resistors, compute a theoretical value for the total resistance of the series circuit. Compare this with the experimentally obtained value of resistance for the series circuit.  Compare the experimental and calculated values. Are they within the experimetal error?

3. Using the experimentally measured resistances for your three individual resistors, compute a theoretical value for the total resistance of the parallel circuit.  Compare this with the experimentally obtained value of resistance for the parallel circuit.  Compare the experimental and calculated values. Are they within the experimetal error?

You should know what was plotted on the graphs and how such a graph yields the resistance of the circuit.  Include all of your original data in your lab report and show your calculations for the data analysis questions.  In your results and conclusions, you should report the values of resistance obtained for each individual reistor.  Also include the answers to data analysis questions 2 and 3.