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Chapter 28: Problem 45

A 2.2 -nF capacitor and one of unknown capacitance are in parallel across a \(10-\mathrm{V}\) rms sine-wave generator. At \(1.0 \mathrm{kHz}\), the generator supplies a total current of \(3.4 \mathrm{mA}\) rms. The generator frequency is then decreased until the rms current drops to 1.2 mA. Find (a) the unknown capacitance and (b) the lower frequency.

Short Answer

Expert verified
The unknown capacitance value in this exercise can be found using given information and the equations for capacitive reactance and current in a capacitor. Subsequently, the lower frequency can be determined using these capacitance values and the current-voltage-frequency relationship in a capacitor.

Step by step solution

01

Determine the current through the known capacitor at the first frequency

We know the voltage, frequency, and capacitance for the 2.2-nF capacitor. We can use the formula for current in a capacitor: \(I=2πfCV\). Substituting the given values, the current \(I_1 = 2π * 1.0 kHz * 2.2 nF * 10 V\).
02

Calculate the current through the unknown capacitor at the first frequency

The total current supplied by the generator is the sum of the current through the known and unknown capacitors. Thus, the current through the unknown capacitor \(I_2 = I_{total} - I_1 = 3.4 mA - I_1\).
03

Determine the unknown capacitance

We can now use the formula for current in a capacitor for the unknown capacitor. Rearranging for C, we have \(C = I_2 / (2πfV)\). Substituting the values from the previous steps, we will get the value of the unknown capacitance.
04

Determine the current through the known capacitor at the second frequency

We can start by assuming that the frequency at which the rms current drops to 1.2mA is the frequency at which the current through the 2.2-nF capacitor alone is 1.2mA. Using the formula \(f = I / (2π * C * V)\), we can substitute the values and solve for the frequency.
05

Check the validity of the assumption

Now let's calculate the total current at this frequency using both capacitors' current equations. If the total current is more significant than 1.2mA, the assumption is valid; the frequency derived in Step 4 is correct. If not, the lower frequency at which the total current becomes 1.2mA needs to be calculated differently using trial and error.

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Most popular questions from this chapter

One-eighth of a cycle after the capacitor in an \(L C\) circuit is fully charged, what are the following as fractions of their peak values: (a) capacitor charge, (b) energy in the capacitor, (c) inductor current, (d) energy in the inductor?

Find the resonant frequency of an \(L C\) circuit consisting of a \(0.22-\mu \mathrm{F}\) capacitor and a 1.7 -mH inductor.

An inductor and capacitor are connected in series across an AC generator, and the voltage across the inductor is higher than across the capacitor. Is the generator frequency above or below resonance?

An industrial electric motor runs at \(208 \mathrm{V}\) rms and \(400 \mathrm{Hz}\). What are (a) the peak voltage and (b) the angular frequency?

Your professor tells you about the days before digital computers when engineers used electric circuits to model mechanical systems. Suppose a \(5.0-\mathrm{kg}\) mass is connected to a spring with \(k=1.44 \mathrm{kN} / \mathrm{m} .\) This is then modeled by an \(L C\) circuit with \(L=2.5 \mathrm{H} .\) What should \(C\) be in order for the \(L C\) circuit to have the same resonant frequency as the mass-spring system?
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