Module 21  Exponential Growth and Decay  
Introduction  Lesson 1  Lesson 2  Lesson 3  SelfTest  
Lesson 21.3: Charge on a Capacitor  
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 from the flash bulb. In this lesson you will examine data produced by a capacitor as it discharges. The discharge discussed here is much slower than that of a flash bulb because an electronic resistor, which retards the discharge, is placed in the circuit with the capacitor. 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. The situation is described by the same basic differential equation that was explored in the previous two lessons. In the equations above, V represents the voltage across the capacitor at any time t, k represents a negative parameter that depends on the physical characteristics of the capacitor and resistor, and V_{0} represents the capacitor's initial voltage. The TI83 can accept realworld data, plot the data, and display a graph of the voltage across the capacitor. The data can be used to find values for V_{0} and k, leading to a solution of the initial value problem given above. The particular solution can then be used to calculate the voltage across the capacitor at any given time. Getting the Data The data from the capacitor experiment are stored in lists L_{1} and L_{2}, which should be downloaded to your computer and calculator. Downloading the Data to Your Computer
Click here to get information about how to obtain the needed cable and to review the procedure to transfer the lists from your computer to your calculator.
Time, measured in seconds, is in L_{1} and voltage, measured in volts, is in L_{2}. Plotting the Data and Solving the Differential Equation
Writing the Equation The graph of the data shows the particular solution to the differential equation That is, the equation that corresponds to the graph has the form V = V_{0}e^{k · t}. The equation that models the scatter plot can be found by using two points of the data, the initial value and another point. Specific values in the data set can be found by using the Trace feature of the Graph screen.
The graph indicates that the initial voltage was V_{0} 7.93407 volts. Other values may be found by using the arrow keys. We will use this initial voltage in the model of the voltage. The resulting differential equation is given in Question 21.3.1. 21.3.1 Solve the initialvalue problem and V(0) = 7.93407. Click here for the answer. To find the value of k, use another point on the graph.
Substituting these values into the equation gives:
Now graph the function to see that the equation fits the original data.
21.3.2 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 HalfLife 21.3.3 Find the halflife of the capacitor by solving the equation for t. Click here for the answer. 

< Back  Next >  
©Copyright
2007 All rights reserved. 
Trademarks

Privacy Policy

Link Policy
