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 computer with the Internet connection, a calculator, paper, and pencil
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):
Click on http://www.walterfendt.de/ph14e/combres.htm . This is the same link you used for Experiment 2. If you have not done Experiment 2 online, you need to first refer to Experiment 2 in order to learn how to use this applet.
1) Place 3 resistors: R_{1} = 100Ω, R_{2} = 200Ω, and R_{3} = 300Ω from left to right, respectively. Let the battery voltage be 12.0 volts.
2) Although, the resistors in the applet are not labeled as R_{1}, R_{2}, and R_{3}, we keep in mind that the leftmost resistor that is 100.Ω is our R_{1}.
Do not take any measurements at this point.
3) Calculate the total resistance R and record it in the appropriate box in Table 1.
4) Based on this total resistance that the battery faces, calculate the current that it can push or flow in the circuit. Of course, we use Ohm's law: V = RI from which I=V/R. Record I in the appropriate place in Table 1.
5) Since the resistors are in series, the same current I must be flowing through each of them; therefore, you know the current through each resistor.
6) Knowing the current through each resistor, apply V = RI to each resistor to calculate the voltage across that resistor. Record your calculated values of V_{1}, V_{2}, and V_{3} in Table 1. You have found all calculated (accepted) values.
7) Now, place an ammeter in the circuit by clicking on the "Amperage." The applet quickly measures the current and shows it by the ammeter. Record this under I ( measured ) in Table 1.
8) Remove the ammeter by deselecting the "Amperage." Select each resistor one at a time, click on the "Voltage". Read and record the voltage across each resistor, and deselect the "Voltage" before selecting the next resistor. Record these under V_{1 }, V_{2 }, and V_{3} (measured).
9) Calculate a % error for each measured value using the % error formula:
(10) Repeat the experiment for trial 2 as well.
Data (I):
Given and Measured:
Table 1
Trial  V_{bat.} _{(V)} 
R_{1} (Ω) 
R_{2} (Ω) 
R_{3} (Ω) 
R Total (Ω) Series 
I Amps V/R 
V_{1 }
= R_{1}I Volts 
V_{2}
= R_{2}I Volts 
V_{3}
= R_{3}I Volts 

1  12.0  100  200  300  Calculated  
Measured  
% error  
2  13.0  500  250  250  Calculated  
Measured  
% error 
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 }, and use these calculated values as accepted values. You might have already done these in steps 1 through 6 above!
Comparison of the Results (I):
Corresponding to every measured value, there is an accepted value. Calculate a percent error on each. You might have already done these!

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 subcurrents I_{1 }, I_{2 }, and I_{3 }. At Point b, the subcurrents 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 voltagedrop 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.
Equation (4) calculates the equivalent resistance R of the three parallel resistors R_{1 }, R_{2 }, and R_{3 }.
Procedure (II):
1) Refresh the screen of the applet to the get the default circuit. This means that you will start with a circuit that has a 12.0V battery connected to a 100Ω resistor. On the applet, select the resistor and click on the "Parallel Connection" 2 times to put 2 more 100Ω resistors in parallel with the existing one. You will then have a circuit that has a module of 3 parallel resistors connected to a battery.
2) Name the top one R_{1}, the middle one R_{2}, and the bottom one R_{3}. Change R_{2} and R_{3} to 200 and 300 Ohms, respectively. Now you have a circuit that is similar to Fig. 2.
3) Calculate the equivalent resistance R_{ab} by using the parallelresistors formula, Equation (4) above, and record its value in Table 2.
Do not take any measurements at this point.
4) What is the voltage across each resistor or what do you expect it to be? Write your answer in Table 2 in the space provided (V_{1} = V_{2} = V_{3} = V_{bat}). The battery voltage is given, but you write it down where it says "Accepted."
5) Calculate the main current I by applying the battery voltage V_{bat} to the overall resistance R_{ab}. This means: I = V_{bat }/R_{ab}.
6) To calculate the current through each resistor, the voltage across it must be divided by its resistance. Since V = RI ; therefore, I = V/R. Record your calculated values (Accepted values) of I_{1 }, I_{2 }, and I_{3} in Table 2 in the line for accepted values.
Measurements:
7) To measure the voltages, select each resistor by clicking on it, then click on the "Voltage", read the voltage, record it, and then deselect the "Voltage" to remove the voltmeter. The voltage you read is the voltage across each of R_{1 }, R_{2 } , and R_{3 }. Resistors in a parallel module experience the same voltage across. This will be the measured voltage V, V_{1 }, V_{2 }, or V_{3 }. Record the value of voltage you read in Table 2.
8) To measure the currents, select each resistor by clicking on it, then click on the "Amperage", read the current, record it, and then deselect the "Amperage" to remove the ammeter. Each current you read is the current through that particular resistor, the currents are different ( I_{1 }, I_{2 }, and I_{3 }). These are you measured values for the currents.
9) As a test, add I_{1 }, I_{2 }, and I_{3 }to see if the add up to I .
10) To read the current, I, through the main branch (up to the dividing point), remove the ammeter first if it is already in one of the branches. Next, click on one of the resistors and then click somewhere near the battery (out of the area of the resistors). The three resistors will be selected and the overall resistance R_{ab} will be shown in a white box at the top left corner. Now, if you click on the "Amperage", an ammeter will be placed in the main branch to read the overall current I that comes out of the battery. Record the overall current I in Table 2. This is your measured value for the main current I.
11) Calculate all % errors and record them.
12) Repeat the above procedure for line 2 of Table 2.
Data (II):
Given and Measured:
Table 2
Trial  V_{bat} _{(V)} 
R_{1} (Ω) 
R_{2} (Ω) 
R_{3} (Ω) 
R_{ab} (Ω) 
V_{1 }=
V_{2} = V_{3} = V = V_{bat } ( Volts ) 
I
(main) V/R_{ab} (Amps) 
I_{1} V/R_{1} (Amps) 
I_{2} V/R_{2} (Amps) 
I_{3} V/R_{3} (Amps) 

1  12.0  100  200  300 

Accepted  
Measured  
% error  
2  16.0  400  250  250 

Accepted  
Measured  
% error 
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 }, I_{3 }, V_{1 }, V_{2 }, and V_{3 } and use these calculated values as accepted values. You might have already done all of these!
Comparison of the Results (II):
Corresponding to every measured value, there is an accepted value. Calculate a percent error for each. You might have already done all of these!
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 lowresistance 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_{ac} to calculate V_{ac} .
V_{ab} = R_{ab} I and V_{ac} = R_{ac} 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):
1) Refresh the applet to again get the default case of a 12.0V battery in series with a 100Ω resistor.
2) Select the resistor and click on "Series Connection." Now you have 2 resistors in series.
3) Click on the left resistor to select it. Then click on the "Parallel connection" 2 times to make the left resistor a module of 3 parallel resistors. Now you have a module of 3 parallel resistors in series with a 4th resistor. Change the module to 100, 200, 300 ohms from top to bottom. Name these as R_{1 }, R_{2 }, and R_{3}_{ }. Name the right resistor R_{4} and leave it at 100 ohms.
4) Calculate the equivalent resistance of the parallel module as you did in Part 2. This time name it R_{123} , and record its value in Table 3.
5) Calculate the overall resistance of the circuit R by considering R_{123} and R_{4} as series resistors and record its value. R = R_{123} + R_{4} . The value of R will be used in the next step.
6) Calculate the main current I that the battery can push out by applying its voltage V_{bat} to the overall resistance R. [ I = V_{bat }/ R ]. Record its value.
7) Calculate the voltage V_{123} across the parallel module. V_{123} = R_{123} I . Record its value.
8) Calculate V_{4 },the voltage across R_{4 }. V_{4} = R_{4} I . Record its value.
9) Now, you have the voltage across each resistor. Calculate currents I_{1 }, I_{2 }, I_{3 }, and I_{4} and record their values in Table 3. These calculated values are the accepted values, and are what we expect to get when doing the measurements.
Measurements:
10) Select each resistor and measure the voltage across it and record its measured value. Record them as V_{1 }, V_{2 } , V_{3 }, and V_{4 }.
11) Select each resistor and measure the current through it and record its measured value. Record them as I_{1 }, I_{2 } , I_{3 }, and I_{4}_{ }.
To measure the main current I, all resistors must be selected at once and then the ammeter added. If you click near the battery (away from all resistors), then all resistors will be selected.
12) Measure I and record its value.
13) Calculate a % error for each measured value and record.
14) Repeat the experiment for the values given in line 2 of Table 3.
Data (III): Given and Measured:
Case  V_{bat.} _{(V)} 
R_{1} (Ω) 
R_{2} (Ω) 
R_{3} (Ω) 
R_{123} (Ω) 
R_{4} (Ω) 
I V_{bat}/R (A) 
V_{123} _{(V)} 
V_{4} _{(V)} 
I_{1} V_{123}/R_{1} (A) 
I_{2} V_{123}/R_{2} (A) 
I_{3} V_{123}/R_{3} (A) 
I_{4} V_{4}/R_{4} (A) 

1  12.0  100  200  300  Accpt.  
Meas.  
Error  
2  16.0  400  250  250  Accpt.  
Meas.  
Error 
Table 3
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 }, I_{4 }, V_{1 }, V_{2 }, V_{3 } and V_{4 }, and use these calculated values as accepted values. You might have already done all of these!
Comparison of the Results (III):
Corresponding to every measured value, there is an accepted value. Calculate a percent error for each. You might have already done all of these!
Conclusion: To be explained by students.
Discussion: To be explained by students.