Objectives:
The objective of this experiment is to verify Ohm's law applied to
1. series resistors,
2. parallel resistors, and
3. a module of parallel resistors collectively in series with another resistor.
Equipment:
A few ceramic resistors (100 to 700 ohms), a dc-power source, 2 multi-meters, a calculator, and a few connecting wires with alligator clips
Theory:
Ohm's law simply states that the ratio of voltage across an electric device to the current through that device is a constant called the electric resistance of that device.
If V is in volts and I is in amperes, R will be in ohms. A more often used version of this formula is
V = R I
i) Series Resistors:
Theory and Procedure:
Series resistors experience the same current, but (possibly) different voltages. A typical pure series circuit is shown below. On one hand, the current I has to be the same everywhere.
Fig. 1
On the other hand, the voltage will be different across each resistor (if their resistances are different). The battery voltage V_{bat} will be divided between the three resistors R1, R2, and R3, proportional to their resistances, such that
V_{bat} = V_{ab} + V_{bc} + V_{cd}, or (1)
V_{ad} = V_{ab} + V_{bc} + V_{cd}. (2)
Also, for series resistors:
R_{ad} = R_{ab} + R_{bc} + R_{cd }(3)
Measure the four ceramic resistors you are provided with a multi-meter (set on ohm setting) and name them R1, R2, R3, and R4. Name the smallest R4.
Arrange a circuit as shown in Fig.1. To measure the voltages, there is no need for three voltmeters. One voltmeter is sufficient. The voltages across R1, R2, and R3 can be measured one at a time by placing the voltmeter terminals across each resistor.
To measure the current, the ammeter must be placed in the circuit in series. Note that the circuit must be opened at one of the connections and the ammeter be placed in between the open ends. Note that the approximate current must be calculated first and the correct and safe mA range be selected; otherwise, the ammeter will be damaged.
Read the current, I, through the circuit and the voltage across each resistor and record them.
Turn the circuit off or disconnect the battery.
Data:
Given:
Use an ohmmeter to measure more-accurate values for the following resistances.
R_{1} = (100 - 700) Ω
R_{2} = (100 - 700) Ω
R_{3} = (100 - 700) Ω
R_{4} = (100 - 700) Ω
V_{bat} = 9 V (When the circuit is on, use the voltmeter to measure the exact voltage supplied by the battery (V_{bat } ). This is the voltage that must be used in your calculations).
Measured:
When the circuit is closed, measure the following values:
V_{1} = ? V_{2} = ? V_{3} = ? I = ? (These are measured values. Do not use these in calculations).
Calculations:
Knowing the given values for V_{bat}, R_{1}, R_{2}, and R_{3}, solve for I, V_{1}, V_{2}, and V_{3}, and use these calculated values as accepted values. Note that the exact values of R_{1}, R_{2}, and R_{3} measured by the ohmmeter must be used in your calculations.
Comparison of the Results:
Corresponding to every measured value, there is an accepted value. Calculate a percent error for each, using the percent error formula you used in Experiment 2.
ii) Parallel Resistors:
Theory and Procedure:
Parallel resistors experience the same voltage, but (possibly) different currents. This is in contrast to series resistors. A typical pure parallel circuit is shown below. The voltage ( V_{ab} ) has to be the same for the three resistors, because they are all between the same two points of the circuit (a and b). The current in each resistor is determined by the equation I=V/R. Since the voltage V is the same for each resistor (V_{1}=V_{2}=V_{3}=V_{ab}), the larger the resistance R of a particular resistor, the smaller the current through that resistor. Charge or flow conservation requires that
I = I_{1} + I_{2} + I_{3}.
Fig. 2
Note that the voltage across each of the resistors is V_{ab}. If we neglect the small voltage-drop across the ammeter, the voltage V_{ab } is equal to the battery voltage V_{bat}. This is because there is no circuit element (other than the ammeter) between point a and the battery, or between point b and the battery. The ammeter (A) does not contribute to any significant voltage drop; the voltage drop across an ammeter can usually be neglected.
For a different case, where there is a fourth resistor between point a and the battery, or between point b and the battery, V_{ab} is not equal to V_{bat}. This is going to be case (iii) in this experiment. Here, for parallel resistors:
Use the same three resistors R1, R2, and R3.
Arrange a circuit as shown in Fig.2. To measure the voltages, place the voltmeter across the junction points a and b. The voltage you read is the voltage across each of R1, R2, and R3. Parallel resistors share the same voltage.
To measure the current, the ammeter must be placed in the line of each resistor once to read the current in that branch. Note that each branch must be opened at one of the connections and the ammeter be placed in between the open ends. Note that the approximate current must be calculated first and the correct and safe mA range be selected; otherwise, the ammeter will be damaged. Do this for each branch and record the values in (mA).
Read the current, I, through the main branch (between the battery and a or between b and the battery). That gives the current in the main branch. It is the current that leaves the battery from one terminal and finally goes into the battery at its other terminal. Record its value.
Turn the circuit off or disconnect the battery.
Data:
Given:
R_{1} = (100 - 700) Ω
R_{2} = (100 - 700) Ω
R_{3} = (100 - 700) Ω
Measured:
When the circuit is closed, measure the following values:
V_{1} = V_{2} = V_{3} = ? I = ? I_{1} = ? I_{2} = ? I_{3} = ?
(These are measured values. Do not use these in calculations).
Calculations:
Knowing the given values for V_{bat}, R_{1}, R_{2}, and R_{3}, solve for R, I, I_{1}, I_{2}, I_{3}, V_{1}, V_{2}, and V_{3}, and use these calculated values as accepted values.
Comparison of the Results:
Corresponding to every measured value, there is an accepted value. Calculate a percent error for each, using the appropriate formula as in Experiment 2.
iii) Parallel Resistors in Series with Another Resistor:
Theory and Procedure:
Fig. 3 shows a parallel portion between points a and b that is in series with another resistor between points b and c. Here, the battery voltage partially drops across the ab portion and partially across the bc portion. It is logical to write:
V_{bat} = V_{ab} + V_{bc}
The voltage across resistors 1, 2, and 3 is the same because they are in parallel. (The parallelism is not because they are drawn parallel to each other on paper; it is because the current between points a and b must divide into three branches such that I = I_{1} + I_{2} + I_{3}. The same voltage V_{ab} drives currents I_{1}, I_{2}, and I_{3} into resistors 1, 2, and 3.)
This balance-of-currents equation may be written in terms of voltages and resistances as:
Dividing through by V_{ab} yields: 1 / R_{ab} = 1 /R_{1} + 1 /R_{2} + 1 /R_{3}
R_{ab} = 1 / (1 / R_{1} + 1/ R_{2} + 1/ R_{3} )
This equation is good only for the parallel portion. To find the total resistance of the circuit, one may write:
R_{ac} = R_{ab} + R_{bc } where R_{bc} = R_{4} (R_{ab} and R_{bc} are in series)
Once the total resistance Rac is determined, the main current I can be calculated by using Ohm's law:
V_{bat} = R_{ac} I.
The total resistance R_{ac} determines how much current the battery can deliver. (The ammeter (A) is a low-resistance device, and its resistance may be ignored.) The reason that R_{ac} is the determining factor is that there are no other significant resistors between point a and the battery or point c and the battery.
Fig. 3
When I is determined, it can be multiplied by R_{ab} to calculate V_{ab}. It can also be multiplied by R_{ac} to calculate V_{ac} as shown:
V_{ab} = R_{ab} I and V_{ac} = R_{ac} I
Now that V_{ab} is known, Ohm's law may be used again to calculate I_{1}, I_{2}, and I_{3}, as follows:
I_{1} = V_{ab} / R_{1}
I_{2} = V_{ab} / R_{2}
I_{3} = V_{ab} / R_{3}
Use the four resistors R_{1}, R_{2}, R_{3} and R_{4}.
Arrange a circuit as shown in Fig. 3. To measure the voltages, place the voltmeter across points a and b to read V_{ab} and then across points b and c to read V_{bc}. Of course, V_{ab} is the voltage across each of R1, R2, and R3, and V_{bc} is the voltage across R_{4} .
To measure the current, the ammeter must be placed in the line of each resistor once to read the current in that branch. Note that each branch must be opened at one of the connections and the ammeter be placed in between the open ends. Note that the approximate current must be calculated first and the correct and safe mA range be selected; otherwise, the ammeter will be damaged. Do this for each branch in the parallel section and record its value in (mA). Record the
Read the current, I, through the main branch (between the battery and a or between c and the battery). That gives the current in the main branch. It is the current that leaves the battery from one terminal and finally goes into the battery at its other terminal. Record its value.
Turn the circuit off or disconnect the battery.
Data:
Given:
R_{1} = (100 - 700) Ω
R_{2} = (100 - 700) Ω
R_{3} = (100 - 700) Ω
R_{4} = (100 - 700) Ω
Measured:
When the circuit is closed, measure the following values:
V_{1} = V_{2} = V_{3} = ? V_{4} = ? I = ? I_{1} = ? I_{2} = ? I_{3} = ?
(These are measured values. Do not use these in calculations).
Calculations:
Knowing the given values for V_{bat}, R_{1}, R_{2}, R_{3}and R_{4}, solve for I, I_{1}, I_{2}, I_{3}, V_{1}, V_{2}, V_{3}and V_{4}, and use these calculated values as accepted values.
Comparison of the Results:
Corresponding to every measured value, there is an accepted value. Calculate a percent error for each.
Conclusion: To be explained by students
Discussion: To be explained by students