Problem 56 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 53-56: Solve for the specified variable. \(\omega=\frac{1}{\sqrt{L C}}\) for \(C\) (alternating-current circuits)

Step-by-Step Solution

Answer

\( C = \frac{1}{L \omega^2} \)

Step 1: Understand the Formula

The given formula is \( \omega = \frac{1}{\sqrt{L C}} \). Here, \( \omega \) is a constant, \( L \) is the inductance, and \( C \) is the capacitance that we need to solve for.

Step 2: Square Both Sides

To isolate \( C \), first eliminate the square root by squaring both sides of the equation. This gives us: \[ \omega^2 = \frac{1}{L C} \]

Step 3: Multiply Both Sides by LC

To clear the fraction, multiply both sides of the equation by \( L C \):\[ \omega^2 \times L C = 1 \] This simplifies to: \[ L C \omega^2 = 1 \]

Step 4: Solve for C

To isolate \( C \), divide both sides of the equation by \( L \omega^2 \):\[ C = \frac{1}{L \omega^2} \]

Key Concepts

Variable Isolation

Variable isolation is a fundamental step in solving equations, particularly when one needs to solve for a specific variable in the equation. It involves a series of algebraic manipulations to express the desired variable on one side of the equation, while everything else is moved to the opposite side.

When isolating a variable, keep the following steps in mind:

• Identify the variable you need to solve for.
• Perform inverse operations to counteract the operations applied to the variable.
• Simplify each step; for example, clear fractions by multiplying every term by the denominator.
• Maintain balance by performing the same operation on both sides of the equation.

In the case of our given equation, we wanted to isolate the variable \( C \). This involved halting the components involved with \( L \) and \( \omega \) on the opposite side. Following the steps, such as squaring both sides and then multiplying through by the components involving \( C \), allows one to efficiently reach the solution.

Alternating-Current Circuits

An alternating-current (AC) circuit is characterized by the electric current changing direction periodically. These circuits are powered by alternating voltage sources, typically operating at a constant frequency like that in household or industrial power supplies.

Key features of AC circuits include:

• Alternation of current and voltage, typically alternating sinusoidally.
• Power can be efficiently transmitted over long distances due to the nature of current and voltage relationship.
• Components like resistors, inductors, and capacitors all respond differently to AC compared to direct current (DC).

Understanding how to solve equations within AC circuits is crucial because it reflects the real-world applications in electrical engineering, such as designing circuits that require specific responses at given frequencies. In dealing with AC circuits, it's important to have a grasp of how variables like inductance \( L \) and capacitance \( C \) affect the behavior or performance of the circuit.

Inductance and Capacitance Relationships

Inductance \( L \) and capacitance \( C \) are two fundamental properties in AC circuits. These properties significantly influence how circuits respond to changes in frequency and voltage. Understanding their relationship can facilitate a deeper insight into the dynamics of AC circuit behavior.

The relationship between inductance and capacitance is often governed by formulae like \( \omega = \frac{1}{\sqrt{L C}} \). This formula represents the resonant angular frequency \( \omega \), where the impedance of the circuit is minimized for oscillations. Here, inductance \( L \) tends to impede changes in current, while capacitance \( C \) stores and releases electrical energy.

• Inductance is measured in henrys (H) and represents a coil's tendency to resist changes in current.
• Capacitance is measured in farads (F) and indicates a capacitor's ability to store an electric charge.
• At resonant frequency, energy is ideally transferred back and forth between the inductor and the capacitor.

Manipulating these variables in formulas is essential for tasks like tuning circuits to achieve the desired frequency or performance criteria, critical in radio transmissions, signal processing, and various electronics.