TI-Nspire™ Activities: Physics
March 11, 2008 through March 18, 2008
Capacitors
| Part 1 – Capacitors | |
| Step 1: Students should open the file PhysWeek07_capacitors_EN.tns and read the first two pages. Page 1.3 illustrates a charged capacitor. Students should examine the capacitor on page 1.3 and then answer questions 1–3. Q1. If the charge on the negative plate is –Q, what is the charge on the positive plate? A. +Q Q2. What is the total net charge of the capacitor? A. zero Q3. If the potential difference between the plates is increased from 10 V to 20 V, will Q increase, decrease, or be unchanged? A. The charge will increase. |
|
| Step 2: Next, students should read page 1.6 before examining the Lists & Spreadsheet application on page 1.7, which contains data on charge (q) and voltage (v) for a capacitor. On page 1.8, they should make a scatter plot of the data, using v as the x-variable and q as the y-variable. Students should then use the Linear Regression tool (Menu > Actions > Regression > Show Linear (mx + b)) to find an equation relating v and q. They should then answer questions 4–6. Note: Make sure students read the text on page 1.10 before answering question 6. Q4. Describe the relationship between charge (q) and potential (v). A. Charge increases linearly with electrical potential. Q5. Write an equation relating q and v. A. q = (0.000470)v Q6. What is the capacitance in μF of the capacitor graphed on page 1.8? A. 470 μF; you may need to review the conversion of farads to microfarads with students. |
|
| Part 2 – Physical characteristics of capacitors | |
| Step 1: Next, students should read the text on page 1.12 before moving to page 1.13, which shows a parallel-plate capacitor. Students can vary the separation and the area of the plates and observe the effects on capacitance. They can do this by dragging the black point (plate separation) up and down and the white point (plate area) left and right. After students have manipulated the plate area and separation, they should answer questions 7 and 8. Q7. How does increasing the plate area affect capacitance? A. Increasing plate area increases capacitance. Q8. How does decreasing plate separation affect capacitance? A. Decreasing separation increases capacitance. |
|
| Step 2: Next, students will attempt to find an equation relating C, A, and d for a capacitor. They should move to page 1.15, which shows data on area (ar) and plate separation (di) for a capacitor. They should use the simulation on page 1.13 to find and record the capacitance for the values of d and A in the Lists & Spreadsheet application. Students should record these values in the column labeled ca. |
|
Step 3: Next, students should use column D to calculate the value of  for the data in the spreadsheet. To do this, they should enter =a[]/b[] in the formula bar (light gray box) of column D. Then, students should use the Linear Regression tool (Menu > Statistics > Stat Calculations > Linear Regression (mx + b)) to find the equation relating ca and arbydi. Students should use arbydi in the X list and ca in the Y list for this calculation. Then, students should answer questions 9–12. Note: Make sure students read the text on page 1.18 before attempting to answer questions 11 and 12.Q9. Write a general equation for C as a function of A and d. A. The equation is  , where k is a constant. Students may struggle to relate the equation calculated by the Linear Regression tool to the equation above. Remind them that one of the variables they used for the linear regression, arbydi, is itself the ratio of area to plate separation. Q10. The constant of proportionality in the equation for C is the permittivity of a vacuum, ε0. What does your linear regression predict for the value of ε0 in units of farads per meter? A. ε0 = 8.85 x 10–12 F/m; remind students that the plate separation, di, was held constant for the data they used in the linear regression. Because the plate separation was constant, students can “remove” it from the value of k calculated in the linear regression. This yields the value of ε0. Remind students to check their units (the units for A, d, and C are given on page 1.13). Note that ε0 is also known as the permittivity of free space. Q11. A sliver of mica is placed between the plates of a capacitor with d = 1.2 x 10–8 m and A = 0.050 m2. The capacitance, C, is 260 µF. What is the dielectric constant for mica? A. The capacitance of a capacitor that contains a dielectric is given by the following equation:  Rearranging this equation to solve for εr, the dielectric constant, yields the following:  Substituting the given values yields the following:  Q12. What would be the capacitance without the mica? A. Without the mica, the capacitance decreases by a factor equal to mica's dielectric constant. In other words, the capacitance is equal to  . Students can confirm this calculation using the equation for the capacitance of a capacitor, as shown below:  Substituting the given values and solving yields the following:  You may wish to emphasize to students that calculating dielectric constants makes calculating the capacitance of the capacitor much simpler. |
|
| Part 3 – Rate of capacitor discharge | |
| Step 1: Next, students should read the information on page 1.20, which describes the discharge from an RC circuit. Then, students should move to page 1.21 and examine the diagram of the RC circuit shown there. They should read the information on page 1.22 and then move to the Lists & Spreadsheet application on page 1.23, which contains experimental data for the discharge of a 10 μF capacitor in a circuit with R = 22 kΩ. |
|
| Step 2: Students should use the data on page 1.23 to make a graph of volt1 vs. time1 in the Data & Statistics application on page 1.24. They should use this graph to answer questions 13–15. Q13. Describe the shape of the graph of volt1 vs. time1 for a discharging capacitor. A. Voltage decreases with time. The relationship is nonlinear and appears to follow an exponential curve. That is, the rate of change of voltage appears to decrease over time according to an exponential curve. Q14. How long does it take for the voltage to drop to one-half of the initial voltage? A. about 0.15 sec Q15. Show that the units of R • C (ohms • farads) are seconds. A.  |
|
Step 3: Next, students should read the text on page 1.27 and then move to page 1.28. Page 1.28 shows a graph of voltage as a function of time together with data on voltage vs. time. The equation for the change of voltage with time is  . Students should study the fit between the equation and the data. |
|
| Step 4: Next, students should examine the data on page 1.30 and the
graph on page 1.31. The data on page 1.30 make up another set of
experimental data for the discharge of a capacitor. Students should
change the value of c2 until the equation fits the data well. Then,
they should answer question 16. Q16. What is the capacitance of this second capacitor? A. 21 μF |
|