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Chapter 21: Problem 25

What is the reactance of an air core solenoid of length \(8.0 \mathrm{cm},\) radius \(1.0 \mathrm{cm},\) and 240 turns at a frequency of $15.0 \mathrm{kHz} ?$

Short Answer

Expert verified
Short answer: To find the reactance of the air core solenoid, first calculate the cross-sectional area, \(A = \pi r^2\) where \(r = 1.0\ \mathrm{cm}\). Then, find the inductance using the formula \(L = \frac{\mu_0 N^2 A}{l}\) with given values \(\mu_0 = 4\pi \times 10^{-7}\ \mathrm{T\cdot m/A}, N = 240, A = \pi\ \mathrm{cm^2},\) and \(l = 8.0\ \mathrm{cm}\). Finally, use the formula \(X = 2 \pi f L\) with the given frequency \(f = 15.0 \mathrm{kHz}\) to find the reactance, \(X\ \Omega\).

Step by step solution

01

Calculate the solenoid's inductance

To find the inductance of an air core solenoid, we can use the formula \(L = \frac{\mu_0 N^2 A}{l}\), where \(L\) is the inductance, \(\mu_0\) is the permeability of free space \((4\pi \times 10^{-7} \mathrm{T\cdot m/A})\), \(N\) is the number of turns, \(A\) is the cross-sectional area, and \(l\) is the solenoid's length. First, we need to calculate the cross-sectional area using the given radius: \(A = \pi r^2.\)
02

Calculate the cross-sectional area

Find the cross-sectional area by substituting the given radius \((r = 1.0\ \mathrm{cm})\) into the formula: \(A = \pi r^2 = \pi (1.0 \mathrm{cm})^2 = \pi \mathrm{cm^2}.\)
03

Calculate the inductance

Now, use the inductance formula to find the value of inductance, with \(\mu_0 = 4\pi \times 10^{-7}\ \mathrm{T\cdot m/A}\), \(N = 240\), \(A = \pi\ \mathrm{cm^2} = 0.0001\ \mathrm{m^2}\), and \(l = 8.0\ \mathrm{cm} = 0.08\ \mathrm{m}\): \( L = \frac{\mu_0 N^2 A}{l} = \frac{(4\pi \times 10^{-7}\ \mathrm{T\cdot m/A})(240)^2 (0.0001\ \mathrm{m^2})}{0.08\ \mathrm{m}}.\) Calculate the inductance, \(L.\)
04

Find the reactance

Using the inductance value \(L\) and the given frequency \((f = 15.0 \mathrm{kHz} = 15,000\ \mathrm{Hz})\), find the reactance \((X)\) of the solenoid using the formula: \(X = 2 \pi f L.\) Calculate the reactance, \(X.\)
05

Present the result

The reactance of the air core solenoid at a frequency of \(15.0\ \mathrm{kHz}\) is \(X\ \Omega.\) (Replace \(X\) with the value calculated in Step 4.)

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Most popular questions from this chapter

A \(150-\Omega\) resistor is in series with a \(0.75-\mathrm{H}\) inductor in an ac circuit. The rms voltages across the two are the same. (a) What is the frequency? (b) Would each of the rms voltages be half of the rms voltage of the source? If not, what fraction of the source voltage are they? (In other words, \(V_{R} / \ell_{m}=V_{L} / \mathcal{E}_{m}=?\) ) (c) What is the phase angle between the source voltage and the current? Which leads? (d) What is the impedance of the circuit?

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