PHYS272_Lab3

Physics for Scientists and Engineers: Lab 3 Prelab
 Assignment: The following uncertainty propagations rely on using the following: If a quantity //y// is a function of multiple measurements (x 1, x 2 , ..., x n ) with uncertainty (Δx 1 , Δx 2 , ..., Δx n ) so y=f(x 1 , x 2 , ..., x n ) then the uncertainty in y is math \displaystyle \Delta y = \sqrt{\left( \frac{\partial f}{\partial x_1} \Delta x_1 \right)^2 + \left( \frac{\partial f}{\partial x_2} \Delta x_2 \right)^2 + \ldots \left( \frac{\partial f}{\partial x_3} \Delta x_3 \right)^2} math This assignment is due at the start of the laboratory period.

1. Three resistors R 1, R 2 , R 3 are connected in series. The net resistance can be found via R net = R 1 + R 2 + R 3. If we measure the resistance of the three resistors individually and find that R 1 = 2000 ± 10 Ω, R 2 = 400 ± 5 Ω and R 3 = 1200 ± 8 Ω , what is the net resistance and the uncertainty in the net resistance?

2. Now the three resistors are connected in parallel. The net resistance can be found via 1/R net = 1/R 1 + 1/R 2 + 1/R 3. What is the net resistance and the uncertainty in the net resistance?

3. A Kirchhoff's laws-style circuit is set up. The currents, resistances, and voltages in various branches of the circuit obey the following relationships: I 1 + I 2 + I 3 = 0, R 1 I 1 - R 2 I 2 = V 1, R 2 I 2 - R 3 I 3 = V 2. Solve this system of equations analytically for each of the 3 currents in terms of R 1, R 2 , R 3 , V 1 , and V 2. Do not plug in any numbers.


 * Extra Credit **

Calculate an analytical expression for the uncertainty of each of the three currents from question 3 based on the resistance and voltage measurements and their uncertainties. This is sufficiently mathematically tedious that you will likely want to use a tool like [] to help you. Write a computer program or set up a spreadsheet to perform the calculations of the currents and their uncertainties. Here is some data you can use to test your calculations: if R 1 = 1200 ± 30 Ω, R 2 = 3000 ± 50 Ω , R 3 = 5300 ± 70 Ω , V 1 = 3.0 ± 0.1 V, V 2 = 7.0 ± 0.2 V then I 1 = 1.77 ± 0.04 mA, I 2 = -0.29 ± 0.03 mA, I 3 = -1.48 ± 0.04 mA

=Physics for Scientists and Engineers – Lab 3 =

Objectives:

 * To study the relationship between current and voltage in DC circuits.
 * To observe resistance, current, and voltage relationships in series and parallel combinations.

Equipment:

 * Digital multimeters, 2
 * DC power supplies, 0-15 V, 2
 * Resistors, 100 Ω, 1000 Ω, 2200 Ω, 3300 Ω, 4700 Ω
 * Circuit boards, circuit board leads
 * Graphical Analysis and mathcad software

Physical Principles:
 math \displaystyle R=\frac{V}{I} math where R is called the resistance and is measured in ohms (Ω).
 * Ohm's Law** For many materials the current resulting when a voltage is applied is directly proportional to the voltage. The proportionality constant is defined by

The effective resistance of the three resistors in figure 1 connected in series is equal to the sum of the resistances of each resistor. math R_{eq}=R_1+R_2+R_3 math
 * Resistors in Series**

The effective resistance of three resistors connected in parallel (see figure 2) is given by the following formula. math \displaystyle \frac{1}{R_{eq}}=\frac{1}{R_{1}}+\frac{1}{R_{2}}+\frac{1}{R_{3}} math According to this law, the sum of the currents leaving a node is zero. A node is a point where more than two conducting paths connect. math I_1+I_2+I_3=0 math
 * Resistors in Parallel**
 * [[image:https://www.andrews.edu/phys/wiki/PhysLab/lib/exe/fetch.php?media=fig.1.jpg link="https://www.andrews.edu/phys/wiki/PhysLab/lib/exe/detail.php?id=272s11l3&media=fig.1.jpg"]] || [[image:https://www.andrews.edu/phys/wiki/PhysLab/lib/exe/fetch.php?media=fig.2.jpg link="https://www.andrews.edu/phys/wiki/PhysLab/lib/exe/detail.php?id=272s11l3&media=fig.2.jpg"]] ||
 * Kirchoff's Current Law**

The sum of all the voltage drops within a loop always adds up to zero. math V+V_1+V_2+V_3=0 math
 * Kirchoff's Voltage Law**

1. Resistance in series
 Connect the three resistors with values of about 2200 Ω, 3300 Ω and 4700 Ω in series as shown in figure 1. Record in your journal a table of nominal (from the color code bands) and measured resistance values. The uncertainty in readings taken with your multimeter can be found [|here] Record the uncertainty in the resistance based on the presence of the fourth gold or silver band and the uncertainty in your measured resistance. For each resistor, do these uncertainties overlap? (Note: For all following calculations, use the measured resistance values of each resistor, not their nominal values.) Measure and record resistance values of R 1 + R 2, R 2 + R 3 , and R 1 + R 2 + R 3. For each combination, calculate the net resistance and the uncertainty in the net resistance. Do the measured values of the resistor combinations overlap your calculated intervals?

2. Voltages for resistors in series
 Plug the banana leads of two breadboard leads into the positive and negative receptacles on the power supply. Insert the wires of these leads into the circuit board across the three resistors in series with an ammeter as shown in the figure 3. Set the voltage to about 15 volts. Draw a schematic diagram of the circuit in your journal. Read the voltage and the current from the digital meter displays. Disconnect the voltmeter from its original position and measure and record in a table the voltages across each of the three resistors, as well as across the resistance combinations R 1 + R 2, R 2 + R 3 , and R 1 + R 2 + R 3. Compare these values with the products of the current and the corresponding resistance values by calculating the uncertainty in the current times resistance.

3. Resistance in parallel
 Connect the three resistors in parallel. Measure the resistance of the parallel combination of three resistors. Find the measured value of R eq and the calculated R eq obtained from: math \displaystyle \frac {1}{R_{eq}}=\frac {1}{R_1}+\frac {1}{R_2}+\frac {1}{R_3} math Also calculate the uncertainty in R eq and show that it overlaps your direct measurement.

4. Currents for resistors in parallel
<span style="color: #333333; display: block; font-family: Arial,sans-serif; font-size: 14px;"> Connect the power supply to the parallel combination of the three resistors (similar to figure 4). Set the voltage to approximately 15 volts. Connect the meters in such a way that you can measure the supply voltage and the currents I, I 1, I 2 and I 3. Complete a table of resistance, voltage, and current along with their uncertainties in your journal. Compare the measured currents with the corresponding V/R values the uncertainty in the V/R values. State whether the calculated and measured intervals overlap.

<span style="color: #333333; font-family: Arial,sans-serif; font-size: 1.125em;">5. Complex Circuit
<span style="color: #333333; display: block; font-family: Arial,sans-serif; font-size: 14px;"> Set up a complex resistor network of three or more resistors that is not a simple series or parallel network. Draw a schematic diagram of the network in your journal. Compare the calculated and measured equivalent resistance of the resistor network. Connect the battery to the resistor network and measure the voltage drop and current through each resistor. Compare the measured and calculated values in a table in your journal. Calculate the power dissipated by each resistor.

<span style="color: #333333; font-family: Arial,sans-serif; font-size: 1.125em;">6. Kirchhoff’s Law Circuit
<span style="color: #333333; display: block; font-family: Arial,sans-serif; font-size: 14px;"> Using the three resistors with values of about 2200 Ω, 3300 Ω and 4700 Ω. Build the circuit in figure 5, and then record the values of V 1, V 2 and I 1 , I 2 , I 3 and compare with the theoretical values that you can find by using Ohm’s Law and Kirchoff’s Laws. The values of I 1, I 2 and I 3 are determined theoretically from the three equations

math I_1+I_2+I_3=0 math

math R_1I_1-R_2I_2+0I_3=V_1 math

math 0I_1+R_2I_2-R_3I_3=V_2 math

The first equation is the current law, the second is the voltage law,applied to the left side loop and the last equation is the voltage law applied to the right side loop of the circuit shown in figure 5. Use your solution for the currents from the prelab to calculate values for the currents. If you did the extra credit portion of the prelab, use your program to calculate uncertainties for the currents and see if the calculated intervals overlap the measured ones. If you did not do the extra credit portion, calculate percent error between measured and calculated currents.