Experiment 3

Series and Parallel Resistors

 

Objectives:

 

The objective of this experiment is to verify Ohm's law applied to

 

1.      series resistors (experimentally verifying that they experience the same current),

2.      parallel resistors (experimentally verifying that they experience the same voltage), and

3.      a module of parallel resistors collectively in series with another resistor.

 Note: Such observations based on your measurements must be reflected in your conclusion.

 

Equipment:

 

A computer with the Internet connection, a calculator (The built-in calculator of the computer may be used.), paper, and pencil

 

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:

 

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 Vbat will be divided between the three resistors R1, R2, and R3, proportional to their resistances, such that

    Vbat = Vab + Vbc + Vcd, or                                                       (1)

 

    Vad = Vab + Vbc + Vcd.                                                            (2)

 

Also, for series resistors:                   

   Rad = Rab + Rbc + Rcd                                                                                  (3)

 

 

Procedure:

 

    Click on the following link:   http://www.walter-fendt.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 refer to Experiment 2 first, in order to learn how to use this applet.

  1. Place 3 resistors: R1 = 100.Ω,  R2 = 200.Ω,  and R3 = 300.Ω,  from left to right, respectively.  Let the battery voltage be 12.0 volts.

  2. Although, the resistors in the applet are not labeled as R1, R2, and R3, we keep in mind that the leftmost one that is 100.Ω is our R1.

  3. Do not make any measurements at this point.

  4. Calculate the total resistance R and record it in the appropriate box in Table 1.

  5. Based on this total resistance that the battery faces, calculate the current that it can push or flow in the circuit. V=RI  from which  I=V/R.  Record I  in the appropriate place in Table 1.

  6. 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.

  7. Knowing the current through each resistor, apply V=RI to each resistor to calculate the voltage across that resistor.  Record your calculated value of V1, V3, and V3 in Table 1You have found all calculated (accepted) values.

  8. 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 the Table.

  9. Remove the ammeter by deselecting the "Amperage."  Select each resistor one at a time, click on the "Voltage",  read and record the voltage across that resistor, and deselect the "Voltage" before selecting the next resistor.  Record these under V1, V2, and V3 (measured).

  10. Calculate a % error for each measured value using the %error formula:

  11. Repeat the experiment for trial 2 as well.

    The %error formula to be used is

Data:

Given and Measured:

Trial Vbat.

Volts

R1

Ω

R2

Ω

R3

Ω

R

Total (Ω)

Series (S)

  I

Amps

V/R

V1

Volts

R1I

V2

Volts

R2I

V3

Volts

R3I

1 12.0 100. 200. 300.            Calculated                         
Measured        
% error        
2 13.0 500. 250. 250.   Calculated        
Measured        
% error        

 

Table 1

Calculations:

 

Knowing the given values for Vbat, R1, R2, and R3, solve for I, V1, V2, and V3, 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.

 

ii)  Parallel Resistors

 Theory:

 

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 ( Vab ) 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 (V1=V2=V3=Vab), the larger the resistance R of a particular resistor, the smaller the current through that resistor.   Charge or flow conservation requires that

I = I1 + I2 + I3.

 

Point a is a dividing point.  Current I divides at point a into I1, I2, and I3.  At point bI1, I2, and I3 merge to form I again.

Fig. 2

Note that the voltage across each of the resistors is Vab.  If we neglect the small voltage-drop across the ammeter, the voltage Vab is equal to the battery voltage Vbat.  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, Vab is not equal to Vbat.  This is going to be case (iii) in this experiment.  Here, for parallel resistors:

Procedure:

  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 12.0-V battery connected to a 100.-Ω resistor.   On the applet, select the resistor and click on the "Parallel Connection" 2 times to put 2 more 100.-Ω resistors in parallel with the existing one.  Now you have a circuit that has a module of 3 parallel resistors connected to a battery.  Name the top one be R1, the middle one R2, and the bottom one R3.   Change R2 and R3 to 200 and 300 Ohms, respectively.  Now you have a circuit that is similar to Fig. 2.

  2. Calculate the overall (the equivalent) or total resistance R by using the parallel-resistors formula shown above, and record its value in Table 2.

  3. 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.

  5. Now that you know the voltage across each resistor, go ahead and calculate the current through each as well as the total current.

  6. Calculate the total current I by applying the battery voltage Vbat to the overall resistance R..   This means:  I = Vbat / R..

  7. 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 I1, I2, and I3 in Table 2 in the line for accepted values.

  8. 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 R1, R2, and R3.  Resistors in a parallel module have the same voltage.  This will be the measured voltage V, V1, V2, or V3.

  9. 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 ammeterEach current you read is the current through that particular resistor, the currents are different ( I1, I2, and I3). These ar you measured values for the currents.

  10. As a test, add  I1, I2, and I3 to see if the add up to I.

  11. 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 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.

  12. Calculate all %errors and record them.

  13. Repeat the above procedure for line 2 of Table 2.

 

Data:

 Given and Measured:

Trial Vbat.

Volts

R1

Ω

R2

Ω

R3

Ω

Total

R (Ω)

Para (P)

  V1=V2=V3

=Vbat

Volts

I

Amps

V/R

I1

Amps

V/R1

I2

Amps

V/R2

I3

Amps

V/R3

1 12.0 100. 200. 300.  

 

Calculated                 
Measured          
% error          
2 16.0 400. 250. 250.  

 

Calculated          
Measured          
% error          

 

Table 2

Calculations:

 

Knowing the given values for Vbat, R1, R2, and R3, solve for R, I, I1, I2, I3, V1, V2, and V3, 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.

 

III)     Parallel Resistors in Series with Another Resistor:

 

Theory:

 

Fig. 3 below, 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 the rest across the bc portion.  It is logical to write:

 

Vbat = Vab + Vbc

 

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 = I1 + I2 + I3.  The same voltage Vab drives currents I1, I2, and I3 into resistors 1, 2, and 3.)

 

This balance-of-currents equation may be written in terms of voltages and resistances as:

 

I = I1 + I2 + I3, or

Dividing through by Vab yields:    1 / Rab = 1 /R1  +  1 /R2   +  1 /R3

 

Rab = 1 / (1 / R1 + 1/ R2 + 1/ R3 )

This equation is good only for the parallel portion.  To find the total resistance of the circuit, one may write:

Rac = Rab + Rbc      where Rbc = R4         (Rab and Rbc are in series)

 

Once the total resistance Rac is determined, the main current I can be calculated by using Ohm's law:

 

Vbat = Rac I.

 

The total resistance Rac determines how much current the battery can push through.  (The ammeter (A) is a low-resistance device, and its resistance may be ignored.)

Fig. 3

When I is determined, it can be multiplied by Rab to calculate Vab.  It can also be multiplied by Rac to calculate Vac as shown:

 

Vab = Rab I      and        Vac = Rac I

Now that Vab is known, Ohm's law may be used again to calculate I1, I2, and I3, as follows:

I1 = Vab / R1

I2 = Vab / R2

I3 = Vab / R3

Procedure:

 

  1. Refresh the applet to get the default case of a 12.0-V battery in series with a 100.-Ω 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 another resistor.  Change the module to 100, 200, 300 ohms from top to bottom.  Name these as R1, R2, and R3.   Name the right resistor R4, and leave it at 100 ohms.

  4. Calculate the equivalent resistance of the parallel module as you did in Part 2.  Name it R123, and record its value in Table 3.

  5. Calculate the overall resistance of the circuit R by considering R123 and R4 as series resistors. and record its value.   R = R123 + R4 .

  6. Calculate the main current I  that the battery can send out (push through) by applying its voltage Vbat  to the overall resistance R.   [ I = Vbat / R ].  Record its value.

  7. Calculate the voltage V123 across the parallel module. V123  = R123 I Record its value.

  8. Calculate V4 ,the voltage across R4.    V4  = R4 I .   Record its value.

  9. Now, you have the voltage across and the resistance of each resistor.  Calculate the currents I1, I2, I3, and I4. and record them in Table 3.

  10. The above calculated values are the accepted values, and are what we expect to get when doing the measurements.

  11. Measurements: Select each resistor and measure the voltage across it and record its measured value.  Record them as V1, V2, V3, and V4.

  12. Select each resistor and measure the current through it and record its measured value.  Record them as I1, I2, I3, and I4.

  13. 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.  Measure I and record its value.

  14. Calculate a %error for each measured value and record.

  15. Repeat the experiment for the values given in line 2 of Table 3.

Data:    Given and Measured:

 

Trial Vbat.

Volts

R1

Ω

R2

Ω

R3

Ω

R123

(Ω)

R

(Ω)

  I

(A)

V/R =

 

V123

Volts

 

V4

Volts

I1

(A)

V123/R1 =

I2

(A)

V123/R2 =

I3

(A)

V123/R3 =

I4

(A)

V4/R4=

1 12.0 100 200 300     Calc.                            
Meas.              
Error              
2 16.0 400 250 250     Calc.              
Meas.              
Error              

 

Table 3

Calculations:

 

Knowing the given values for Vbat, R1, R2, R3and R4, solve for I, I1, I2, I3, I4, V1, V2, V3and V4, 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