Problem

(a) Calculate the inductive reactance of a $3.00 \, \mathrm{mH}$ inductor when $60.0 \, \mathrm{Hz}$ and $10.0 \, \mathrm{kHz}$ AC voltage are applied.
(b) Determine the rms current at each frequency if the applied rms voltage is $120 \, \mathrm{V}$.

Data

• Inductance: $L = 3.00 \, \mathrm{mH} = 3.00 \times 10^{-3} \, \mathrm{H}$
• AC frequency 1: $f_1 = 60.0 \, \mathrm{Hz}$
• AC frequency 2: $f_2 = 10.0 \, \mathrm{kHz} = 10 \times 10^{3} \, \mathrm{Hz}$
• RMS voltage: $V_{\mathrm{rms}} = 120 \, \mathrm{V}$

Prerequisite Concepts

1. Inductive reactance is given by:

$$X_L = 2 \pi f L$$

2. RMS current is related to inductive reactance by Ohm’s law:

$$I_{\mathrm{rms}} = \frac{V_{\mathrm{rms}}}{X_L}$$

Solution

Part (a): Calculate Inductive Reactance at Each Frequency

For $f_1 = 60.0 \, \mathrm{Hz}$:

$$X_{L1} = 2 \pi f_1 L$$

Substitute the values:

$$X_{L1} = 2 \times 3.14 \times 60.0 \, \mathrm{Hz} \times 3.00 \times 10^{-3} \, \mathrm{H}$$ $$X_{L1} = 1.1304 \, \Omega$$

For $f_2 = 10.0 \, \mathrm{kHz}$:

$$X_{L2} = 2 \pi f_2 L$$

Substitute the values:

$$X_{L2} = 2 \times 3.14 \times 10 \times 10^{3} \, \mathrm{Hz} \times 3.00 \times 10^{-3} \, \mathrm{H}$$ $$X_{L2} = 188.4 \, \Omega$$

Part (b): Calculate RMS Current at Each Frequency

For $f_1 = 60.0 \, \mathrm{Hz}$:

$$I_{\mathrm{rms}1} = \frac{V_{\mathrm{rms}}}{X_{L1}}$$

Substitute the values:

$$I_{\mathrm{rms}1} = \frac{120 \, \mathrm{V}}{1.1304 \, \Omega}$$ $$I_{\mathrm{rms}1} = 106.16 \, \mathrm{A}$$

For $f_2 = 10.0 \, \mathrm{kHz}$:

$$I_{\mathrm{rms}2} = \frac{V_{\mathrm{rms}}}{X_{L2}}$$

Substitute the values:

$$I_{\mathrm{rms}2} = \frac{120 \, \mathrm{V}}{188.4 \, \Omega}$$ $$I_{\mathrm{rms}2} = 0.637 \, \mathrm{A}$$

Answer

• Inductive Reactance:
  • At $f_1 = 60.0 \, \mathrm{Hz}$: $X_{L1} = 1.1304 \, \Omega$
  • At $f_2 = 10.0 \, \mathrm{kHz}$: $X_{L2} = 188.4 \, \Omega$
• RMS Current:
  • At $f_1 = 60.0 \, \mathrm{Hz}$: $I_{\mathrm{rms}1} = 106.16 \, \mathrm{A}$
  • At $f_2 = 10.0 \, \mathrm{kHz}$: $I_{\mathrm{rms}2} = 0.637 \, \mathrm{A}$

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