
Concept explainers
For the circuit in Fig. 7.88, if
(a) Find R and C.
(b) Determine the time constant.
(c) Calculate the initial energy in the capacitor.
(d) Obtain the time it takes to dissipate 50 percent of the initial energy.
Figure 7.88
For Prob. 7.8.
(a)

Find the value of resistance R and capacitance C in the circuit.
Answer to Problem 8P
The value of resistance R in the circuit is
Explanation of Solution
Given data:
The voltage across the capacitor
The current flows through the circuit
Formula used:
Write the expression to find the voltage across the capacitor for a source-free RC circuit.
Here,
Write the expression to find the time constant for RC circuit.
Here,
R is the resistance of the resistor, and
C is the capacitance of the capacitor.
Write the expression to find the current through the capacitor.
Here,
Calculation:
Refer to Figure 7.88 in the textbook. In the circuit, the direction of current is given as leaving form the positive terminal the capacitor, therefore in equation (3), the current direction is taken as negative. The equation (3) becomes,
Substitute
Rearrange the equation to find capacitance C.
Substitute the unit F for
Compare the given voltage across the capacitor
Substitute
Substitute
Rearrange the equation to find resistance R.
Substitute the unit
Conclusion:
Thus, the value of resistance R in the circuit is
(b)

Calculate the value of time constant for the RC circuit.
Answer to Problem 8P
The value of time constant for the RC circuit is
Explanation of Solution
Given data:
Refer part (a),
The value of resistance is
The value of capacitance
Calculation:
Substitute
Substitute the units
Conclusion:
Thus, the value of time constant for the RC circuit is
(c)

Find the initial energy in the capacitor
Answer to Problem 8P
The initial energy in the capacitor
Explanation of Solution
Given data:
Refer part (a),
The value of capacitance
The initial voltage
Formula used:
Write the expression to find the initial energy in the capacitor.
Calculation:
The voltage across the capacitor,
Compare the equation (10) and (1) for the initial voltage
Substitute
Substitute the unit
Convert the unit J to mJ.
Conclusion:
Thus, the initial energy in the capacitor
(d)

Find the time taken to dissipate 50 percent of the initial energy.
Answer to Problem 8P
The time taken to dissipate 50 percent of the initial energy is
Explanation of Solution
The energy in the capacitor to dissipate 50 percent of the initial energy is,
Substitute
Substitute the unit
Rearrange the equation as follows.
Take ln on both sides of the equation (14) to find the time t in seconds.
Rearrange the equation as follows.
Convert the unit s to ms.
Conclusion:
Thus, the time taken to dissipate 50 percent of the initial energy is
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