Question

A certain circuit has a \(25-\Omega\) resistor and one other component in series with a \(12-\mathrm{V}\) (rms) sinusoidal ac source. The rms current in the circuit is 0.317 A when the frequency is \(150 \mathrm{Hz}\) and increases by \(25.0 \%\) when the frequency increases to \(250 \mathrm{Hz}\). (a) What is the second component in the circuit? (b) What is the current at $250 \mathrm{Hz} ?$ (c) What is the numerical value of the second component?

Short answer

Answer: The second component in the series circuit is a capacitor with a value of 37.79 μF.

Step-by-step solution

Find Impedance at Given Frequencies

We can calculate the impedance of the circuit at two given frequencies using Ohm's law. At 150 Hz, the rms current is 0.317 A, and the rms voltage is 12 V. The resistance R is given as 25 Ω. The impedance Z can be found using Ohm's law: \(Z = \frac{V}{I} = \frac{12 \mathrm{V}}{0.317 \mathrm{A}} \approx 37.87 \Omega\) At 250 Hz, the current increases by 25%, so the new current I' is: \(I'= 0.317 \mathrm{A} \times 1.25 = 0.39625\, \mathrm{A}\) Now let's find the new impedance Z': \(Z' = \frac{V'}{I'} = \frac{12 \mathrm{V}}{0.39625 \mathrm{A}} \approx 30.28 \Omega\)

Determine the Type of the Second Component

As the frequency increases, the impedance decreases, meaning that the unknown component is capacitive. The capacitive reactance \(X_C\) can be calculated as: \(X_C = Z^2 - R^2 \Rightarrow X_C = \sqrt{Z^2 - R^2}\) And, \(X'_C = Z'^2 - R^2 \Rightarrow X'_C = \sqrt{Z'^2 - R^2}\) Now, let's calculate \(X_C\) and \(X'_C\): \(X_C = \sqrt{37.87^2 - 25^2} \approx 28.11 \Omega\) \(X'_C = \sqrt{30.28^2 - 25^2} \approx 18.39 \Omega\)

Calculate Current at 250 Hz and Numerical Value of the Second Component

(b) We already found the current at 250 Hz in Step 1, which equals to \(I' = 0.39625 \, \mathrm{A}\). (c) Now, we will determine the capacitance of the second component. The capacitive reactance is given by: \(X_C = \frac{1}{2 \pi f C}\) Solving for C: \(C = \frac{1}{2 \pi f X_C}\) By plugging in the values at 150 Hz: \(C = \frac{1}{2 \pi \times 150 \mathrm{Hz} \times 28.11 \Omega} \approx 37.79 \, \mu\mathrm{F}\) Thus, the second component is a capacitor with a value of 37.79 μF.