Problem 116 - A Graphical Approach to Precalculus with Limits

Question

Let the energy stored in the inductor be $$L(t)=3 \cos ^{2} 6,000,000 t$$ and the energy in the capacitor be $$C(t)=3 \sin ^{2} 6,000,000 t$$ where \(t\) is time in seconds. The total energy \(E\) in the circuit is given by \(E(t)=L(t)+C(t)\) (a) Graph \(L, C,\) and \(E\) in the window \(\left[0,10^{-6}\right]\) by \([-1,4],\) with \(\mathrm{Xscl}=10^{-7}\) and \(\mathrm{Yscl}=1 .\) Interpret the graph. (b) Make a table of \(L, C,\) and \(E,\) starting at \(t=0\) and incrementing by \(10^{-7}\). Interpret your results. (c) Use a fundamental identity to derive a simplified expression for \(E(t)\)

Step-by-Step Solution

Answer

E(t) = 3 remains constant; oscillations in L(t) and C(t) conserve total energy.

Step 1: Analyze the given functions

The energy stored in the inductor is given by \( L(t) = 3 \cos^2(6,000,000t) \) and for the capacitor by \( C(t) = 3 \sin^2(6,000,000t) \). The total energy in the circuit \( E(t) \) is calculated as \( E(t) = L(t) + C(t) \). This means \( E(t) = 3 \cos^2(6,000,000t) + 3 \sin^2(6,000,000t) \).

Step 2: Simplify E(t) using trigonometric identity

Use the Pythagorean identity \( \cos^2(x) + \sin^2(x) = 1 \) to simplify the expression for the total energy. Thus, \( 3(\cos^2(6,000,000t) + \sin^2(6,000,000t)) = 3 \times 1 \). Therefore, \( E(t) = 3 \).

Step 3: Interpret the simplified form of E(t)

The simplified form \( E(t) = 3 \) indicates that the total energy in the circuit remains constant over time, regardless of the variations in \( L(t) \) and \( C(t) \). This behavior is typical for LC circuits under ideal conditions.

Step 4: Graph L(t), C(t), and E(t)

To graph \( L(t) \), \( C(t) \), and \( E(t) \) in the specified window, note that both \( L(t) \) and \( C(t) \) are sinusoidal functions with high frequency oscillation, while \( E(t) \) is a constant line at \( y = 3 \). In the window \([0, 10^{-6}]\), the graph for \( L(t) \) and \( C(t) \) will oscillate rapidly between 0 and 3, while \( E(t) \) will remain a horizontal line.

Step 5: Create a table for L(t), C(t), and E(t)

Set up a table starting from \( t=0 \) and incrementing by \( 10^{-7} \). Calculate \( L(t) \), \( C(t) \), and \( E(t) = 3 \). Each pair \( L(t) \) and \( C(t) \) will oscillate rapidly between 0 and 3, since they're based on \( \cos^2 \) and \( \sin^2 \); however, \( E(t) \) remains consistently 3 across the table.

Step 6: Interpret results from the table

As you observe the table, \( L(t) \) and \( C(t) \) will show alternating values due to their sinusoidal nature, yet \( E(t) \) remains constant at 3. This illustrates energy conservation, where energy oscillates between the inductor and capacitor but the total energy is unchanged.

Key Concepts

Trigonometric Identity

In LC circuits, trigonometric identities play a critical role in simplifying the mathematical expressions that describe the behavior of the system. A key identity used here is the Pythagorean identity. It states that for any angle \( x \), the following is true:

• \( \cos^2(x) + \sin^2(x) = 1 \)

This identity helps in transforming complex expressions into simpler forms, easing analysis. In the given problem, the energy stored in both the inductor and capacitor is represented by trigonometric functions:

• \( L(t) = 3 \cos^2(6,000,000t) \)
• \( C(t) = 3 \sin^2(6,000,000t) \)

By substituting into the Pythagorean identity, the total energy expression \( E(t) = L(t) + C(t) \) simplifies to a constant:

• \( E(t) = 3 \cos^2(6,000,000t) + 3 \sin^2(6,000,000t) = 3 \times 1 = 3 \)

Therefore, the trigonometric identity shows that the total energy within the circuit remains constant over time, which is a crucial observation in studying LC circuits.

Oscillation

Oscillation in the LC circuit describes how energy transfers back and forth between the inductor and the capacitor. This back-and-forth movement is analogous to a perpetual energy dance within the circuit. Due to this process, the forms of energy involved are time-dependent. The functions \( L(t) = 3 \cos^2(6,000,000t) \) and \( C(t) = 3 \sin^2(6,000,000t) \) capture this oscillation. Here’s how it happens:

• Both \( \cos^2 \) and \( \sin^2 \) describe cyclical behaviors, depicting the energy stored in the inductor and capacitor respectively.
• As time progresses, the squared cosine and sine functions oscillate between the values 0 and 3.

These rapid changes convey that while the individual components (inductor, capacitor) experience fluctuating energy levels, the total energy continues balancing out to a constant value—highlighting the seamlessly coordinated oscillation in LC circuits.

Energy Conservation

Energy conservation is a foundational principle in physics, ensuring that energy cannot be created or destroyed, only converted from one form to another. In the context of an LC circuit, this principle is vividly illustrated. The given functions, \( L(t) \) for the inductor and \( C(t) \) for the capacitor, demonstrate how energy shifts between these two components over time. The total energy within the circuit is expressed by:

• \( E(t) = L(t) + C(t) = 3 \)

This equation highlights energy conservation effectively. As the inductor's energy drops, the capacitor's rises, and vice versa, maintaining a constant total energy level. In simpler terms, energy appears to "bounce" between the inductor and capacitor, yet never leaves or diminishes within the ideal circuit. This ensures that the behavior over time is a harmonized transfer—a core hallmark of many physical systems respecting energy conservation.