A capacitor stores electrical energy that can be discharged quickly or slowly. For example, energy from a battery is stored in a camera's capacitor and then the voltage quickly discharges to create the bright flash of light created by a flash bulb.

In this lesson you will examine data produced by a capacitor as it discharges. The discharge discussed is much slower than that of a flash bulb because an electronic resistor, which retards the discharge, was placed in the circuit with the capacitor.

Finding the Voltage Equation

When a capacitor discharges through a resistor, the voltage across the capacitor decreases at a rate proportional to the amount of remaining voltage. This is described by the differential equation

V' = kV and V(0) = V₀

where V represents the voltage across the capacitor at any time t, k is a negative parameter that depends on the physical characteristics of the capacitor and resistor, and V₀ represents the capacitor's initial voltage.

A Texas Instruments CBL™ and voltage probe were used to measure the discharge of energy from a capacitor. The design for this experiment is described in Real-World Math with the CBL-2 and LabPro, Activity 18, which is published by Texas Instruments and can be found in the Online Store at http://education.ti.com/us/product/book/rwmwb.html.

The data from the capacitor experiment are stored in lists l1 and l2, which should be downloaded to your computer and calculator.

Downloading the Data to Your Computer

• Click on l1 and then l2 to download the data to your computer.
• Choose to save the files
• Save the files on your local hard disk in a folder that you can access later

Transferring the Data to the TI-89

Click here to get information about how to obtain the needed cable and to review the procedure to transfer the program from your computer to your calculator.

• Send the lists l1 and l2 from your computer to your TI-89

Time, measured in seconds, is in l1 and voltage, measured in volts, is in l2.

Plotting the Data and Solving the Differential Equation

• Set the TI-89 to Graph Function mode
• Perform NewProb
• Define a scatter plot with l1 in the x-list and l2 in the y-list

Each list can be found by pressing after it is downloaded to the TI-89.

• Select "Dot" as the Mark type

• Use a Zoom Data window and display the graph by pressing

The graph of the data in l1 and l2 shows a particular solution to the differential equation

V' = kV and V(0) = V₀.

• Press to open the Trace feature

The graph indicates that the initial voltage was V₀ 7.93407 volts.

• Return to the Home Screen

Solve the initial-value problem V' = kV and V(0) = 7.93407

• deSolve(v' = k*v and v(0) = 7.93407,t,v)

Use another point on the graph to find the value of k.

• Return to the graph and trace to t = 30

The value of k can be found by solving V = 7.93407e^kt with the values t = 30 and V = 2.0928.

• solve(v = 7.93407*e^(k*t), k) | t = 30 and v = 2.0928

Graph the function to see that the equation fits the original data well.

• Set y1 = 7.93407e^(-0.044422x)

19.3.1 Trace on the scatter plot of the data to t = 50 and write down the corresponding value of v. Use this value of v to find a new value for k. How well does the solution to the differential equation using this value of k fit the original data?
Click here for the answer.

Finding Half-Life

19.3.2 Find the half-life of the capacitor by solving the equation for t.
Click here for the answer.