Resist! Resistance of a resistor equation
Uncovering the equation for resistance of a resistor using data.
Background
Circuits are now an imperative part of our world. Phones, electric scooters, light-up shoes, toys...as the modern world gets infused with technology, an understanding of circuits becomes more and more valuable. With nearly each new piece of electronic technology, a new custom circuit is created and tested to achieve the new task.
Intentional design of these circuits is important. When electricity is given little to no resistance to move, it will move with incredible speed (or current). This can become problematic as the speed of moving electrons creates heat and often destroys the very transport path it’s using. Scientists and engineers realized they need to slow that current down so it can be used more effectively and safely.
Enter: resistors! Resistors were created to resist electrical current. They are made out of many different materials and in different sizes to help craft the perfect flow of current for any circuit.
Task: You have been hired by a new toy designer, who has a very specific resistor value (2.2 ohms) they need to make their toy work correctly. They need the resistor to either be as slim as possible, or as short as possible for a space within the toy. Your job is to make a recommendation for two different ways a resistor could be built to meet their needs.
The space the resistor can fit within has a max length of 0.8 cm, and a max cross-sectional area of 0.4 cm². You have two big questions to address in your recommendation:
How short could you go? If you max out area (0.4 cm²), what length would you need to achieve 2.2 ohms of resistance?
How slim could you go? If you max out length (0.8 cm), what would your cross-sectional area need to be in order to achieve 2.2 ohms of resistance?
Use the following data set to explore how resistance changes and make your recommendation!
Dataset
This data set was gathered through a simulator. You can add your own data by going to a site like PHET’s Resistance in a Wire simulator. For any trial the variable that was not being manipulated was set to these values:
Length (cm): 0.8 (maximum for our scenario)
Area (cm^2): 0.4 (maximum for our scenario)
Resistivity (ohm/cm): 0.5
We assume the temperature of the wire remains constant, though we are not told its value. Material remains constant, and is represented by the resistivity value.
Variables
The variables should be explained and defined very clearly so that logical conclusions and interpretations can follow from analysis.
Trial # - This categorical variable simply references the trial for which these data were recorded.
Manipulated Variable - This categorical variable describes which variable was investigated during this trial. The variables that were not listed were held at the “constant” value.
Length - This numerical variable is the length of our tested wire in a particular trial. Measured in centimeters.
Area - This numerical value measures the cross-sectional area of our wire in a particular trial. Measured in centimeters^2.
Resistance - This numerical value measures how much current is opposed in a particular trial. Measured in ohms.
Activity
Use the Make a Graph tool to create a Trial # vs. Resistance graph by showing Trial # on X axis, and Resistance on Y axis. Differentiate between the data by showing Manipulated Variable on Z axis. Paste your graph below:
2. Write a sentence describing how changing length affects resistance.
3. Write a sentence describing how changing area affects resistance.
As these two variable patterns are not the same, we should examine them individually.
Part 1: Exploring LENGTH
4. Show Length on the x-axis. Keep Manipulated Variable as z-axis, and Resistance as y-axis. Exclude Area data. Paste your graph here:
5. Create a regression line, and Group by Z. Click Appearance, and select “include zero” for x- and y-axis range. Include a screenshot of your graph here:
6. Which regression line type seems to fit our length pattern (the non-vertical pattern) best? Include the equation below.
Extension A - Linearize? If it is not a linear fit, you should linearize to increase your confidence in the pattern (or type of relationship) you selected. Create a new column to linearize (transform with a function) your data and test your regression result. Include a screenshot and equation below as evidence of your new linearized graph.
7. Finalize your length equation by substituting your variables and units in the place of “x” and “y” in your equation.
8. Address the task: This equation assumes the maximum width of the resistor (Area = .4 cm^2) because that’s what was used in the experiment. Using your equation, what length is needed to create a resistor of 2.2 ohms for our toy company?
Extension B: How much material would this use? (Hint: the resistor is in the shape of a cylinder)
Part 2: Exploring AREA
9. Show Area on the x-axis. Keep Manipulated Variable as z-axis, and Resistance as y-axis. Exclude length from this graph. Paste your graph here (linear regression may already be present):
10. Does this line look like it fits well for your data points? If not, look for a better fit by changing the type of regression line. When r^2 = 1, that means it’s a perfect fit. Which line do you feel most confident about? Include the equation below, as well as pasting an image of the new fit.
Extension C - Linearize? if it is not a linear fit, you need to linearize to confirm you have selected the correct pattern. In table view, click the … at the column header. Go back into the data table and create a new column. Select the function you think will create a linear fit. Include a screenshot below as evidence of your linear graph. Include the new equation below the screenshot.
11. Finalize your length equation by substituting your variables and units in the place of “x” and “y” in your equation.
12. Address the task: This equation assumes the maximum length of the resistor (Length = .8 cm) because that’s what was used in the experiment. Using your equation, what length is needed to create a resistor of 2.2 ohms for our toy company?
Extension D: How much material would this use? (Hint: the resistor is in the shape of a cylinder)
13. Use the data to determine a solution: Look at your answers from #8 and #12. Here is our task briefing once again:
The space the resistor can fit within has a max length of .8 cm, and a max cross-sectional area of .4 cm^2. You have two big questions to address in your recommendation:
How slim could you go? If you max out length (.8 cm), what would your cross-sectional area be to achieve 2.2 ohms?
How short could you go? If you max out area (.4 cm^2), what length would you need to achieve 2.2 ohms?
Based on your findings, what would be the best solution to present to the toy company? Write your answer as a few short sentences in the Claim-Evidence-Reasoning format.
Extension E: Combine your equations for L vs. R and A vs. R into one general equation (do not stress about slope and units for slope) which reflects the relationship for all three.
*Teachers can request an answer key through the form below.