Question

For which values of \(f\) is \(X_{L}>X_{C} ?\) a) \(f>2 \pi(L C)^{1 / 2}\) b) \(f>(2 \pi L C)^{-1}\) c) \(f>\left(2 \pi(L C)^{1 / 2}\right)^{-1}\) d) \(f>2 \pi L C\)

Short answer

Question: Find the range of values of 'f' for which the inductive reactance, \(X_{L}\), is greater than the capacitive reactance, \(X_{C}\), given the following options: a) \(f>2 \pi(L C)^{1 / 2}\) b) \(f>(2 \pi L C)^{-1}\) c) \(f>\left(2 \pi(L C)^{1 / 2}\right)^{-1}\) d) \(f>2 \pi L C\) Answer: c) \(f>\left(2 \pi(L C)^{1 / 2}\right)^{-1}\)

Step-by-step solution

Write the condition for \(X_L > X_C\)#

Considering the expressions for \(X_L\) and \(X_C\), we will write the inequality \(X_L > X_C\): \(2\pi f L > \frac{1}{2\pi f C}.\)

Solve the inequality for 'f'#

To solve the inequality, let's first multiply both sides by \(2\pi f C\) to eliminate the denominator: \((2\pi f L)(2\pi f C) > 1\) Simplify this expression to get: \((2\pi)^2 f^2 LC > 1\)

Determine 'f' from the expression#

To solve for 'f', we can first divide by \((2\pi)^2 LC\), then take the square root on both sides: \(f^2 > \frac{1}{(2\pi)^2 LC}\) \(f > \frac{1}{(2\pi L C)^{1/2}}\) Now, let's compare this result to the given options: a) \(f>2 \pi(L C)^{1 / 2}\) b) \(f>(2 \pi L C)^{-1}\) c) \(f>\left(2 \pi(L C)^{1 / 2}\right)^{-1}\) d) \(f>2 \pi L C\) The expression found for 'f', \(\frac{1}{(2\pi L C)^{1/2}}\), matches with option (c). So, the correct answer is: c) \(f>\left(2 \pi(L C)^{1 / 2}\right)^{-1}\)

Key concepts

Electrical Reactance

Electrical reactance is a fundamental concept in circuit analysis, referring to the opposition that inductors and capacitors offer to alternating current (AC). Reactance varies with the frequency of the AC signal and the circuit elements involved. Specifically, inductors have an inductive reactance (\(X_L\)) that increases with frequency, formulated as \(X_L = 2\pi f L\), where \(f\) is the frequency and \(L\) is the inductance. On the other hand, capacitors have a capacitive reactance (\(X_C\t)) that decreases with frequency, given by the equation \(X_C = \dfrac{1}{2\pi f C}\), where \(C\) denotes capacitance. These two types of reactance are critical in the analysis of AC circuits, and their relationship determines the overall behavior of the circuit, especially in a resonance scenario.

When comparing inductive and capacitive reactance, we often deal with the condition \(X_L > X_C\), which signifies that the inductive reactance dominates over the capacitive reactance. To find the frequency \(f\) that satisfies this condition, we must handle inequalities involving these reactances carefully, a process demonstrated in the textbook solution.

LC Circuit Frequency

LC circuits, composed of an inductor (\(L\)) and a capacitor (\(C\)), play a vital role in electronics, particularly in the tuning and filtering of signals. The frequency of such a circuit is an essential characteristic that affects its reactance and behavior. The resonant frequency, or natural frequency, is given by the equation \(f_0 = \dfrac{1}{2\pi\sqrt{LC}}\).

Understanding this frequency is key to analyzing and designing LC circuits because it indicates the frequency at which the circuit will naturally oscillate when not driven by an external force. It's also important for determining conditions like the one found in our exercise: when the inductive reactance is greater than the capacitive reactance. By manipulating the frequency \(f\), we can understand under what conditions the reactances differ in an LC circuit.

Solving Inequalities

Solving inequalities is a mathematical skill used to find the range of values that satisfy an inequality expression. When dealing with inequalities, the same basic principles that apply to equations are used, but with added attention to the direction of the inequality sign. As seen in the provided solution, to solve the inequality \(2\pi f L > \frac{1}{2\pi f C}\), we must perform operations cautiously to maintain the proper inequality relationship between the two sides.

Multiplication or division by a negative number would reverse the inequality sign, a situation that was not present in this particular problem. However, understanding such nuances is essential for correctly solving inequalities. As we move through mathematical transformations, keeping track of these signs is crucial to arriving at an accurate solution, and helps to determine the correct conditions for a given scenario.

Resonance Condition

The resonance condition in an electrical circuit is a unique state where the inductive and capacitive reactances are equal and cancel each other out. This phenomenon occurs at the resonant frequency \(f_0 = \dfrac{1}{2\pi\sqrt{LC}}\), which we discussed earlier in the context of LC circuit frequency. At resonance, the impedance of the circuit is purely resistive, and the circuit naturally oscillates at its maximum amplitude.

In practical applications, achieving resonance can be useful for tuning a circuit to select or reject specific frequencies. In the context of the exercise, knowing the resonance frequency helps us establish a benchmark for comparisons such as determining when the inductive reactance exceeds the capacitive reactance, which is critical for understanding the behavior of LC circuits at different frequencies.