Question

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

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

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: \(A = \pi r^2.\)

Calculate the cross-sectional area

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

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, \(L.\)

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, \(X.\)

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.)