Problem 79 - Fundamentals of Physics

Question

(a) In an oscillating \(L C\) circuit, in terms of the maximum charge \(Q\) on the capacitor, what is the charge there when the energy in the electric field is \(50.0 \%\) of that in the magnetic field? (b) What fraction of a period must elapse following the time the capacitor is fully charged for this condition to occur?

Step-by-Step Solution

Answer

(a) \( q = \frac{Q}{\sqrt{3}} \); (b) \( t = \frac{T}{12} \).

Step 1: Understanding the Energy Distribution

In an LC circuit, the total energy is constant and is exchanged between the electric field of the capacitor and the magnetic field of the inductor. The total energy is given by \[ U = \frac{Q^2}{2C} \] where \( Q \) is the maximum charge. At any moment, the energy in the electric field \( U_E \) is \( \frac{q^2}{2C} \) and the energy in the magnetic field \( U_B \) is \( \frac{1}{2}L\left(\frac{dq}{dt}\right)^2 \). The total energy is the sum: \( U = U_E + U_B \).

Step 2: Setting Up the Energy Condition

According to the problem, the energy in the electric field \( U_E \) is 50% of that in the magnetic field \( U_B \). Thus, we have the equation:\[ \frac{q^2}{2C} = 0.5 \times \frac{L}{2}\left(\frac{dq}{dt}\right)^2 \].

Step 3: Express Current in Terms of Charge

The current \( I \) in the circuit can be expressed as \( I = \frac{dq}{dt} \). Thus, the energy stored in the magnetic field can be written as:\[ U_B = \frac{1}{2} L I^2 = \frac{L}{2}\left(\frac{dq}{dt}\right)^2 \] .

Step 4: Use Conservation of Energy

According to the conservation of energy, the total energy when the capacitor is fully charged is equal to the sum of energies in the electric and magnetic fields later:\[ \frac{Q^2}{2C} = \frac{q^2}{2C} + \frac{L}{2} \left(\frac{dq}{dt}\right)^2 \].

Step 5: Solve for Charge

From Steps 2 and 3, substitute the equivalent expressions and solve for \( q \):1. Substituting for \( \frac{dq}{dt} \) from the expression for the magnetic energy to the one derived from the condition given:\[ \frac{q^2}{2C} = 0.5 \left( \frac{Q^2}{2C} - \frac{q^2}{2C} \right) \] which simplifies to: \( q = \frac{Q}{\sqrt{3}} \).

Step 6: Determine the Fraction of a Period Elapsed

For a simple harmonic oscillator like an LC circuit, the charge oscillates as \( q = Q\cos(\omega t) \), where \( \omega = \frac{1}{\sqrt{LC}} \) and \( T = \frac{2\pi}{\omega} \) is the period. If \( q = \frac{Q}{\sqrt{3}} \), the cosine function reveals:\[ \cos(\omega t) = \frac{1}{\sqrt{3}} \] This gives us the equation: \( \omega t = \frac{\pi}{6} \) or \( \omega t = \frac{5\pi}{6} \). In terms of the period \( T \), \( t = \frac{T}{12} \).

Key Concepts

Energy Distribution in LC Circuits

When analyzing an LC circuit, understanding how energy is distributed is vital. In these circuits, energy constantly shifts between the capacitor and inductor. It swaps between the electric field in the capacitor and the magnetic field in the inductor. Essentially, the circuit forms a closed system where the total energy remains constant throughout the oscillation.
The total energy is represented as: \[ U = \frac{Q^2}{2C} \]where \( Q \) denotes the maximum charge on the capacitor. This energy doesn’t dissipate but is redistributed between:

• the electric field energy \( U_E = \frac{q^2}{2C} \)
• the magnetic field energy \( U_B = \frac{1}{2}L\left(\frac{dq}{dt}\right)^2 \)

Every moment the circuit oscillates, you're witnessing these constant shifts of energy back and forth.
Hence, grasping energy distribution is key to analyzing how these circuits operate so elegantly.

Harmonic Oscillator in LC Circuits

LC circuits can be seen as harmonic oscillators since they naturally undergo oscillations. The charge and current within this circuit oscillate similarly to a simple harmonic motion, governed by sinusoidal functions.
The charge \( q \) is described by:\[ q = Q\cos(\omega t) \]Here, \( \omega = \frac{1}{\sqrt{LC}} \), representing the angular frequency.
In simple terms, the charge oscillates in a predictable and cyclic pattern dictated by the time-varying cosine function.

• The periodic nature of the system ensures it returns to its starting state after a complete cycle.
• The angular frequency \( \omega \) depends on the values of the inductance \( L \) and capacitance \( C \).

Understanding the harmonic nature of LC circuits helps you predict the behavior of both voltage and current over time, revealing a resonant and cyclical sequence that defines the oscillation.

Conservation of Energy in LC Circuits

The principle of conservation of energy is fundamental in analyzing LC circuits. It states that energy in the circuit doesn’t disappear but shifts location. Initially, all energy resides in the capacitor when fully charged.
As the circuit oscillates, energy transfers between the capacitor and the inductor:

• When the capacitor is wholly charged, energy is primarily electrical.
• As the charge diminishes, energy converts into the magnetic form stored in the inductor.

Mathematically, this is captured by:\[ \frac{Q^2}{2C} = \frac{q^2}{2C} + \frac{L}{2} \left(\frac{dq}{dt}\right)^2 \] Thus, at any point, the total energy is simply the sum of energies in both electric and magnetic forms. This seamless exchange and balance of energies embodies the conservation law, emphasizing no net loss of energy within the ideal LC circuit.

Capacitor Charge in LC Circuit Analysis

The charge on the capacitor is a central aspect of LC circuit behavior. It's important to know how this charge changes as energy is being distributed within the circuit.
Initially, the capacitor begins fully charged, holding the maximum charge \( Q \). Over time, as the circuit oscillates, this charge fluctuates down, reaching different values.

• At distinctive points, like when \( q = \frac{Q}{\sqrt{3}} \), we gain insights into the relative amounts of energy in the electric and magnetic fields.
• These distinct charge levels correspond to specific fractions of the period, indicating how the circuit cycles through states.

In analyzing the charge, understanding the relationship \( q = Q\cos(\omega t) \) is crucial for predicting how the circuit behaves throughout its oscillation cycle. Each shift in charge heralds a shift in energy, guiding observers to the synchronized dance of electric and magnetic forces within the LC framework.