The circuit shown in Figure P28.78 is set up in the laboratory to measure an unknown capacitance C in series with a resistance R = 10.0 MΩ powered by a battery whose emf is 6.19 V. The data given in the table are the measured voltages across the capacitor as a function of lime, where t = 0 represents the instant at which the switch is thrown to position b. (a) Construct a graph of In (ε/Δv) versus I and perform a linear least-squares fit to the data, (b) From the slope of your graph, obtain a value for the time constant of the circuit and a value for the capacitance.
Δv(V) | t(s) | In (ε/Δv) |
6.19 | 0 | |
5.56 | 4.87 | |
4.93 | 11.1 | |
4.34 | 19.4 | |
3.72 | 30.8 | |
3.09 | 46.6 | |
2.47 | 67.3 | |
1.83 | 102.2 |
(a)
To draw: The graph of
Answer to Problem 28.78AP
The graph of
And the equation of best fit is
Explanation of Solution
Introduction:
The linear least square is a technique of fitting a statistical model to a set of variables. This shows the data points linearly in terms of unknown parameters.
Given: The capacitance is
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Write the equation of line which gives a linear relationship.
Here,
Take
Substitute
Calculate the values of
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Write the formula of slope.
Here,
The number of data points
Calculate
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Substitute
Thus, the slope of line is
Write the formula of intercept.
Substitute
Thus, the value of intercept on
Substitute
Thus, the equation of best fit is
The graph of
Figure 1
(b)
The value of time constant of the circuit and the value of the capacitance.
Answer to Problem 28.78AP
The value of time constant of the circuit is
Explanation of Solution
Given: The capacitance is
Write the equation of time constant.
Here,
The time constant from the slope is,
Substitute
Thus, the time constant is
Substitute
The capacitance is
Conclusion:
Therefore, the value of time constant of the circuit is
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Chapter 28 Solutions
Physics for Scientists and Engineers, Technology Update (No access codes included)
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