Objectives:
The objective of this experiment is to verify Ohm's law applied to (a) series resistors, (b) parallel resistors, and (c) a module of parallel resistors in series with another resistor.
Equipment:
A few ceramic resistors (100 to 500 ohms), a dc-power source, 2 multi-meters, a calculator, and a few connecting wires with alligator clips
I) Series Resistors:
Theory (I):
Ohm's law simply states that the ratio of voltage across an electric resistor to the current through that resistor is a constant called the electric resistance of that resistor.
If V is in volts and I in amps, R will be in ohms. The preferred version of this formula is V = R I.
Series resistors experience the same current, but generally different voltages. A typical series circuit is shown below. Current I is the same everywhere.
Figure 1
Voltage across different resistors are different. The battery voltage V_{bat} divides amongst the three resistors R_{1}, R_{2}, and R_{3 } proportional to their resistances such that
V_{bat} = V_{ab} + V_{bc} + V_{cd } , (1)
Also, for series resistors:
R_{ad} = R_{ab} + R_{bc} + R_{cd }(2)
Procedure (I):
Select 4 ceramic resistors between 100Ω to 500Ω. Measure their resistances with a multi-meter (set on ohm setting) and name them R_{1 }, R_{2 }, R_{3 }, and R_{4} in increasing order.
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 V_{1 }, V_{2 }, and V_{3} 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 opened ends. Also, the approximate current may be calculated first and the correct and safe mA range be selected; otherwise, the ammeter will get damaged.
Read the current, I, through the circuit and the voltage across each resistor and record them.
Turn the circuit off.
Data: (I):
Given: The measured resistances of the selected resistors are to be treated as "given."
R_{1} = ........ Ω, R_{2} = ........ Ω, R_{3} = ........ Ω, and R_{4} = ........ Ω.
V_{bat} = ........... = whatever you read for the voltage between Points a and d when the circuit is on and stabilized. Doing this, you eliminate the small voltage drop by the ammeter. This is like bypassing the ammeter.
Measured:
When the circuit is on and stabilized, measure the following values:
V_{1} = .......... V_{2} = ......... V_{3} = ........., and I = ........... .
(These are measured values. Do not use them in calculations).
Calculations (I):
Knowing the given values for V_{bat }, R_{1 }, R_{2 }, and R_{3}, solve for I , V_{1 } , V_{2 }, and V_{3 }. Use these calculated values as accepted values.
Comparison of the Results (I):
Corresponding to every measured value, there is an accepted value. Calculate a percent error for each, using the % error formula.
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II) Parallel Resistors:
Theory (II):
Parallel resistors experience the same voltage, but generally different currents. A typical parallel module is Fig. 2.
We want to develop a formula that calculates the equivalent resistance between Points a and b. See Fig. 2. Let's name this equivalent resistance R_{ab}. To find the imaginary R_{ab} that can do the job of the three parallel resistors R_{1 }, R_{2 } , and R_{3 }, it is important to emphasize that current I at Point a divides into sub-currents I_{1 }, I_{2 }, and I_{3 }. At Point b, the sub-currents I_{1 }, I_{2 }, and I_{3} merge to form I again. We may write: I = I_{1} + I_{2} + I_{3 } (3) in which
I = V_{ab } /R_{ab} , I_{1} = V_{ab }/R_{1} , I_{2} = V_{ab }/R_{2} , and I_{3} = V_{ab }/R_{3} .
Substitution in (3) results in:
Dividing through by V_{ab} , we get:
R_{ab} = 1 / (1/R_{1} + 1/R_{2} + 1/R_{3} ). (4)
R_{1 } , R_{2 }, and R_{3 }all together_{ }draw the same current from the battery that R_{ab} would draw if replaced them. R_{ab} is the "equivalent resistance" between a & b.
Figure 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 any other circuit element between Point a and the battery, or between Point b and the battery to cause a voltage drop. The ammeter A is just a measuring device and does not contribute to significant voltage drop. The voltage drop across an ammeter can usually be neglected. In this part, it is better to measure V_{ab} and use it as V_{bat} in your calculations. This is like bypassing the ammeter.
Equation (4) calculates the equivalent resistance R of the three parallel resistors R_{1 }, R_{2 }, and R_{3 }.
Procedure (II):
Use the same three resistors R_{1 }, R_{2 }, and R_{3 } as in Part I.
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 R_{1 }, R_{2 }, and R_{3 } . 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 opened ends. Also, the approximate current may be calculated first and the correct and safe mA range be selected; otherwise, the ammeter can get 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.
Data (II):
Given: The measured resistances of the selected resistors are to be treated as "given."
R_{1} = ........ Ω, R_{2} = ........ Ω, R_{3} = ........ Ω, and R_{4} = ........ Ω.
V_{bat} = ........... = whatever you read for the voltage between Points a and b when the circuit is on and stabilized. Doing this, you eliminate the small voltage drop by the ammeter. This is like bypassing the ammeter.
Measured:
When the circuit is on and stabilized, measure the following values:
I_{1} = .......... I_{2} = ......... I_{3} = ........., and I = ........... .
(These are measured values. Do not use them in calculations).
Calculations (II):
Knowing the given values for V_{bat }, R_{1 }, R_{2 }, and R_{3}, solve for R_{ab} , I , I_{1 }, I_{2 }, and I_{3 }. Use these calculated values as accepted values.
Comparison of the Results (II):
Corresponding to every measured value, there is an accepted value. Calculate a percent error for each, using the % error formula.
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III) A Parallel Module in Series with Another Resistor:
Theory (III):
Fig. 3 below, shows a parallel module between Points a and b that is in series with another resistor between Points b and c. Here, the battery voltage partially drops across ab and partially across bc. It is logical to write:
V_{bat} = V_{ab} + V_{bc } ._{ } _{(5)}
The parallel module R_{1 }, R_{2 }, and R_{3} may be replaced by R_{ab } given by
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 } or R_{ac} = R_{ab} + R_{4} .
Once the total resistance R_{ac} 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 I , the main current, the battery can push through. Ammeter A is a low-resistance device, and its resistance may be ignored.
Figure 3
When I is determined, it can be multiplied by R_{ab} to calculate V_{ab}. It can also be multiplied by R_{bc} to calculate V_{bc} .
V_{ab} = R_{ab} I and V_{bc} = R_{bc} I.
When V_{ab} is calculated, 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 } .
Procedure (III):
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 R_{1 }, R_{2 }, and R_{3 }, 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 opened ends. Also, the approximate current may be calculated first and the correct and safe mA range be selected; otherwise, the ammeter may get damaged. Do this for each branch in the parallel section and record its value in mA.
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.
Data (III):
Given: The measured resistances of the selected resistors are to be treated as "given."
R_{1} = ........ Ω, R_{2} = ........ Ω, R_{3} = ........ Ω, and R_{4} = ........ Ω.
V_{bat} = ........... = whatever you read for the voltage between Points a and c when the circuit is on and stabilized. Doing this, you eliminate the small voltage drop by the ammeter. This is like bypassing the ammeter.
Measured:
When the circuit is on and stabilized, measure the following values:
V_{1} = ........ V_{2} = ....... V_{3} = ......., V_{4} = ........ and I = ......... .
I_{1} = ........ I_{2} = ........ I_{3} = .......... .
(These are measured values. Do not use these in calculations).
Calculations (III):
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 (III):
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.