### Experiment 3

#### Series and Parallel Resistors

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 Vbat  divides amongst the three resistors R1, R2, and R3  proportional to their resistances such that

Vbat = Vab + Vbc + Vcd ,   (1)

Also, for series resistors:

Rad = Rab + Rbc + Rcd      (2)

Procedure (I):

If you have not done so, you may need to add the following Website to your Java exception list: http://www.walter-fendt.de/.   To do this, follow the path (Windows operating system),

Start → All Programs → Java → Configure Java → Security (use High) → Edit Site List … → Add → Type in the site URL (http://www.walter-fendt.de/).

Click on  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 first refer to Experiment 2 in order to learn how to use this applet.

1) Place 3 resistors: R1 = 100Ω,  R2 = 200Ω,  and R3 = 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 R1, R2, and R3, we keep in mind that the leftmost resistor that is 100.Ω is our R1.

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 V1, V2, and V3 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 V1 , V2 , and V3 (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 Vbat. (V) R1 (Ω) R2 (Ω) R3 (Ω) R Total (Ω) Series I Amps V/R V1 = R1I Volts V2 = R2I Volts V3 = R3I 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 Vbat , R1 , R2 , and R3, solve for I , V1 , V2 , and V3 , 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!

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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 Rab.  To find the imaginary Rab that can do the job of the three parallel resistors R1 , R2 ,and R3 , it is important to emphasize that current I at Point a divides into sub-currents I1 , I2 , and I3 .  At Point b, the sub-currents I1 , I2 , and I3 merge to form I again.  We may write I  = I1 + I2 + I3   (3)  in which

I = Vab /Rab ,   I1 = Vab /R1 ,    I2 = Vab /R2 ,  and     I3 = Vab /R3 .

Substitution in (3) results in:

Dividing through by Vab , we get:

Rab = 1 / (1/R1 + 1/R2 + 1/R3 ).   (4)

R1 , R2 , and R3 all together draw the same current from the battery that Rab would draw if replaced them.  Rab is the "equivalent resistance" between a & b.

Figure 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 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 R1 , R2 , and R3 .

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 12.0-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.  You will then have a circuit that has a module of 3 parallel resistors connected to a battery.

2) Name the top one 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.

3) Calculate the equivalent resistance Rab by using the parallel-resistors 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 (V1 = V2 = V3 = Vbat).  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 Vbat to the overall resistance Rab.   This means:  I = Vbat /Rab.

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 I1 , I2 ,and I3 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 R1 , R2 , and R3 .  Resistors in a parallel module experience the same voltage across.  This will be the measured voltage V, V1 , V2 , or V3 .  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 ( I1 , I2 , and I3 ). These are you measured values for the currents.

9) As a test, add  I1 , I2 , and I3 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 Rab 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 Vbat (V) R1(Ω) R2(Ω) R3(Ω) Rab(Ω) V1 = V2 = V3 = V = Vbat ( Volts ) I (main) V/Rab (Amps) I1 V/R1 (Amps) I2 V/R2 (Amps) I3 V/R3 (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 Vbat , R1 , R2 , and R3 , solve for Rab , I , I1 , I2 , I3 , V1 , V2 , and V3  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:

Vbat = Vab + Vbc .       (5)

The parallel module R1 , R2 , and R3 may be replaced by Rab  given by

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      or     Rac =  Rab + R4 .

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 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 Rab to calculate Vab.  It can also be multiplied by Rbc to calculate Vbc .

Vab = Rab      and      Vbc = Rbc I.

When Vab is calculated, 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 (III):

1) Refresh the applet to again 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 a 4th 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.  This time 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 .  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 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 each resistor.  Calculate currents I1 , I2 , I3 , and I4  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 V1 , V2 , V3 , and V4 .

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

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:

 Ca-se Vbat.(V) R1(Ω) R2(Ω) R3(Ω) R123 (Ω) R4 (Ω) I Vbat/R (A) V123 (V) V4 (V) I1 V123/R1  (A) I2 V123/R2 (A) I3 V123/R3 (A) I4 V4/R4 (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 Vbat , R1 , R2 , R3 and R4 , solve for I, I1 , I2 , I3 , I4 , V1 , V2 , V3 and V4 , 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.