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In the realm of LC circuit analysis, understanding the concept of resonant frequency is crucial. The resonant frequency is the frequency at which an LC circuit naturally oscillates. It depends on the values of both the inductor (\(L\)) and the capacitor (\(C\)) within the circuit. For a standard LC circuit, the resonant frequency (\(f\)) can be calculated using the formula:

• \( f = \frac{1}{2\pi\sqrt{LC}} \)

Here, \(L\) is the inductance in henrys (H), and \(C\) is the capacitance in farads (F). The resonant frequency is measured in hertz (Hz). In this particular exercise, the given resonant frequency is 17 kHz (or 17,000 Hz). Converting capacitive values from picofarads (pF) to farads (by multiplying by \(10^{-12}\)) helps in accurately solving these problems. An essential part of working with an LC circuit is manipulating this formula to find unknown variables when some parameters are given.

Calculating inductance is an integral part of analyzing LC circuits, especially when the resonant frequency and capacitance are known. Using the resonant frequency formula, the inductance \(L\) is derived by rearranging it to:

• \(L = \frac{1}{(2\pi f)^2 C}\)

This formula enables us to calculate the inductance based on the known values of frequency \(f\) and capacitance \(C\). For instance, using the given resonant frequency of 17 kHz and a capacitance of 2200 pF, converting to appropriate units gives:

• \(L = \frac{1}{(2\pi \times 17 \times 10^3)^2 \times 2200 \times 10^{-12}} \approx 4.0 \times 10^{-3} \, \mathrm{H}\)

Understanding how to manipulate these mathematical relationships is key to solving many practical problems in electronics.

The peak current in an LC circuit reveals the maximum current that can flow through the circuit at resonance. To find this, a different formula is utilized:

• \(I_{max} = V_0 \times \omega \times C \)

where \(I_{max}\) is the peak current, \(V_0\) is the initial voltage across the capacitor, \(\omega = 2\pi f\) is the angular frequency, and \(C\) is the capacitance. When the voltage is 120 V, frequency is 17 kHz with capacitance of 2200 pF, substituting these into the formula yields:

• \(I_{max} = 120 \times 2\pi \times 17 \times 10^3 \times 2200 \times 10^{-12} \approx 26 \, \mathrm{mA}\)

This calculation is straightforward once all parameters are in their correct units.

An important feature of inductors in LC circuits is their ability to store energy in a magnetic field. This stored energy reaches its maximum at resonance. The formula for calculating the energy stored in a magnetic field of an inductor is:

• \( U = \frac{1}{2} L I_{max}^2 \)

where \(U\) is the energy stored, \(L\) is the inductance, and \(I_{max}\) is the peak current. Using the inductance of 4.0 mH and the peak current of 26 mA calculated earlier, the stored energy computes to:

• \( U = \frac{1}{2} \times 4.0 \times 10^{-3} \times (26 \times 10^{-3})^2 \approx 1.40 \times 10^{-6} \mathrm{J} \)

This value, in microjoules, illustrates the capability of inductors in storing significant energy even within small-scale circuits.