Ch 32: AC Circuits

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 32: AC Circuits

Problem 26

Chapter 32, Problem 26

An inductor is connected to a 15 kHz oscillator. The peak current is 65 mA when the rms voltage is 6.0 V. What is the value of the inductance L?

Verified step by step guidance

Step 1: Understand the relationship between the inductive reactance (X_L), the frequency (f), and the inductance (L). The formula for inductive reactance is:

X L = 2 π f L

, where X_L is the inductive reactance, f is the frequency, and L is the inductance.

Step 2: Recall the relationship between the rms voltage (V_rms), the peak current (I_peak), and the inductive reactance (X_L). The peak current is related to the rms current (I_rms) by:

I rms = I peak / \sqrt{2}

. Then, the rms voltage and rms current are related by:

V rms = I rms X L

Step 3: Calculate the rms current (I_rms) using the given peak current (I_peak = 65 mA). Use the formula:

I rms = I peak / \sqrt{2}

. Substitute the value of I_peak into the equation.

Step 4: Solve for the inductive reactance (X_L) using the formula:

X L = V rms / I rms

. Substitute the values of V_rms (6.0 V) and I_rms (calculated in Step 3) into the equation.

Step 5: Finally, calculate the inductance (L) using the formula:

L = X L / 2 π f

. Substitute the values of X_L (calculated in Step 4) and f (15 kHz) into the equation to find L.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Inductance

Inductance is a property of an electrical component, typically a coil or inductor, that quantifies its ability to store energy in a magnetic field when an electric current flows through it. It is measured in henries (H) and is defined as the ratio of the induced electromotive force (emf) to the rate of change of current. Higher inductance values indicate a greater ability to store energy.

RMS Voltage and Peak Current

RMS (Root Mean Square) voltage is a way of expressing the effective value of an alternating current (AC) voltage, which is equivalent to a DC voltage that would deliver the same power. The peak current refers to the maximum instantaneous current flowing through the circuit. In AC circuits, these values are related through the impedance of the circuit, which includes resistance and reactance.

Impedance in AC Circuits

Impedance is the total opposition that a circuit offers to the flow of alternating current and is a combination of resistance and reactance. In inductive circuits, the reactance is frequency-dependent and increases with frequency. The relationship between voltage, current, and impedance is described by Ohm's law, adapted for AC as V = I * Z, where V is voltage, I is current, and Z is impedance.

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Related Practice

Textbook Question

FIGURE EX32.24 shows voltage and current graphs for an inductor. What is the value of the inductance L?

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The peak current through an inductor is 10 mA. What is the peak current if the emf peak voltage is doubled (at the original frequency)?

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A series RLC circuit has a 200 kHz resonance frequency. What is the resonance frequency if the capacitor value is doubled?

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A series RLC circuit has a 200 kHz resonance frequency. What is the resonance frequency if the capacitor value is doubled and, at the same time, the inductor value is halved?

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A series RLC circuit has a 200 kHz resonance frequency. What is the resonance frequency if the resistor value is doubled?

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Textbook Question

FIGURE EX32.24 shows voltage and current graphs for an inductor. What is the emf frequency f?

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