AP Physics C Electricity and Magnetism- 11.3 Resistance, Resistivity, and Ohm’s Law- Exam Style questions - FRQs- New Syllabus
Question
| Dough Cylinder | \(A\left(\text{m}^2\right)\) | \(\ell\left(\text{m}\right)\) | \(\Delta V\left(\text{V}\right)\) | \(R\left(\Omega\right)\) | Example: \(\dfrac{\ell}{A}\left(\text{m}^{-1}\right)\) |
|---|---|---|---|---|---|
| \(1\) | \(0.00049\) | \(0.030\) | \(1.02\) | \(23.6\) | \(61\) |
| \(2\) | \(0.00049\) | \(0.050\) | \(2.34\) | \(31.5\) | \(102\) |
| \(3\) | \(0.00053\) | \(0.080\) | \(3.58\) | \(61.2\) | \(151\) |
| \(4\) | \(0.00057\) | \(0.150\) | \(6.21\) | \(105\) | \(263\) |
Most-appropriate topic codes (AP Physics C: Electricity and Magnetism):
• Topic \(11.2\) — Simple Circuits (Part \( \mathrm{(a)} \), Part \( \mathrm{(c)} \))
• Topic \(11.3\) — Resistance, Resistivity, and Ohm’s Law (Part \( \mathrm{(a)} \), Part \( \mathrm{(b)} \), Part \( \mathrm{(c)} \))
▶️ Answer/Explanation
(a)(i)
For a uniform conductor, \(R=\rho\dfrac{\ell}{A}\).
Therefore, a useful linear graph is:
\(\boxed{\text{Vertical axis: }R\left(\Omega\right)}\)
\(\boxed{\text{Horizontal axis: }\dfrac{\ell}{A}\left(\text{m}^{-1}\right)}\)
The slope of the graph is the resistivity \(\rho\), because \(R=\rho\left(\dfrac{\ell}{A}\right)\).
(a)(ii)
Plot the points \(\left(61,23.6\right)\), \(\left(102,31.5\right)\), \(\left(151,61.2\right)\), and \(\left(263,105\right)\), where the horizontal quantity is \(\dfrac{\ell}{A}\left(\text{m}^{-1}\right)\) and the vertical quantity is \(R\left(\Omega\right)\). Draw a reasonable best-fit line through the data.
(a)(iii)
Use the slope of the best-fit line:
\(\rho=\text{slope}=\dfrac{\Delta R}{\Delta\left(\ell/A\right)}\)
Using two points from a reasonable best-fit line, for example approximately \(\left(60\,\text{m}^{-1},20\,\Omega\right)\) and \(\left(260\,\text{m}^{-1},105\,\Omega\right)\):
\(\rho=\dfrac{105\,\Omega-20\,\Omega}{260\,\text{m}^{-1}-60\,\text{m}^{-1}}\)
\(\rho=\dfrac{85\,\Omega}{200\,\text{m}^{-1}}\)
\(\rho\approx0.43\,\Omega\cdot\text{m}\)
\(\boxed{\rho\approx0.42\,\Omega\cdot\text{m}}\)
(b)
\(\boxed{\text{No}}\)
Resistivity is a property of the material, not the shape of the object. Changing the dough from cylinders to rectangular shapes changes the resistance because \(R=\rho\dfrac{\ell}{A}\), but it does not change \(\rho\) if the material and temperature remain the same.
(c)
Make several dough cylinders with the same length \(\ell\) and the same cross-sectional area \(A\). Keep the geometry of the samples fixed so that the only intended independent variable is temperature.
Place one dough cylinder in a temperature-controlled environment, such as a warm water bath or an ice-water bath inside a sealed bag so the dough does not get wet. Use a thermometer to measure the dough temperature \(T\). For each temperature, connect the dough cylinder to a DC power supply, ammeter, and voltmeter.
Apply the same potential difference \(\Delta V\) across the dough each time and measure the current \(I\). Then calculate the resistance using Ohm’s law:
\(R=\dfrac{\Delta V}{I}\)
Then calculate the resistivity:
\(\rho=R\dfrac{A}{\ell}\)
Repeat the measurement for several different temperatures while keeping \(A\), \(\ell\), and the dough material constant. A graph of \(\rho\) versus \(T\) can then be used to determine whether the resistivity depends on temperature.
\(\boxed{\text{If }\rho\text{ changes systematically as }T\text{ changes, then resistivity depends on temperature.}}\)