Question

A 10.0 - \(\mu\) F capacitor is fully charged by a 12.0 - \(\mathrm{V}\) battery and is then disconnected from the battery and allowed to discharge through a 0.200 - \(\mathrm{H}\) inductor. Find the first three times when the charge on the capacitor is \(80.0-\mu \mathrm{C},\) taking \(t=0\) as the instant when the capacitor is connected to the inductor.

Short answer

Step 1: Identify the initial charge on the capacitor Using the formula Q₀ = CV, we have: Q₀ = (10.0 × 10⁻⁶ F)(12.0 V) = 1.20 × 10⁻⁴ C Step 2: Use the formula for the charge as a function of time in an LC circuit To find the angular frequency ω, we have: ω = 1/√(LC) ω = 1/ √(0.200 H × 10.0 × 10⁻⁶ F) ≈ 223.6 rad/s Step 3: Set the time-dependent charge expression equal to the desired charge 80.0 μC = Q₀ × cos(ωt) 80.0 × 10⁻⁶ C = (1.20 × 10⁻⁴ C) × cos(223.6t) Step 4: Solve the equation for the first three times To find the first three times, we need to find the smallest positive values of t: cos(223.6t) = 80.0 × 10⁻⁶ C / 1.20 × 10⁻⁴ C ≈ 0.6667 Angles whose cosine is 0.6667 correspond to π/3 and 2π/3. First time is when angle is π/3: 223.6t ≈ π/3 t₁ ≈ 0.00447 s Second and third times is when angle is 2π/3: 223.6t ≈ 2π/3 + 2πk, where k is an integer t₂ ≈ 0.00942 s (first period for k = 0) t₃ ≈ 0.02331 s (second period for k = 1) The first three times when the charge on the capacitor is 80.0 μC are approximately 0.00447 s, 0.00942 s, and 0.02331 s.

Step-by-step solution

Identify the initial charge on the capacitor

The capacitor is initially charged with a 12.0 V battery. To find the initial charge (Q₀), we can use the formula: $$ Q₀ = C V $$ Where C is the capacitance (10.0 μF) and V is the voltage (12.0 V). Plug in the values and calculate the initial charge.

Use the formula for the charge as a function of time in an LC circuit

In an LC circuit, the charge on the capacitor can be expressed in terms of time as: $$ Q(t) = Q₀ \cos(\omega t) $$ Where ω is the angular frequency, given by: $$ \omega = \frac{1}{\sqrt{LC}} $$ We need to plug the inductor, L = 0.200 H, and the capacitor, C = 10.0 μF, values to find the angular frequency ω.

Set the time-dependent charge expression equal to the desired charge

We want to find the times when the charge on the capacitor is 80.0 μC. So, we will set the expression for Q(t) equal to this value: $$ 80.0 \times 10^{-6} = Q₀ \cos(\omega t) $$ Solve this equation for t.

Solve the equation for the first three times

The equation from Step 3 has multiple solutions for t, as the cosine function oscillates with time. To find the first three times when the charge on the capacitor is 80.0 μC, we must solve the equation for the smallest positive values of t. Use the initial charge (Q₀) computed in Step 1 and the angular frequency (ω) computed in Step 2 and calculate the first three values of t.

Key concepts

Capacitor Discharge

When a capacitor discharges through an inductor, an LC circuit is formed. This circuit exhibits oscillatory behavior, much like a spring oscillates when compressed and released. In the context of a capacitor, the discharge involves the release of the stored energy in the capacitor's electric field.

As the capacitor discharges, the charge decreases over time from its maximum value to zero, then reverses polarity, reaching a negative peak, and continues this cycle, creating an alternating current. The discharge is not a simple linear decrease; it follows a sinusoidal pattern due to the energy exchange between the capacitor and the inductor. The capacitor releases energy to the inductor, which stores it as a magnetic field. When the capacitor's charge is depleted, the inductor's magnetic field collapses, recharging the capacitor with opposite polarity. This process repeats, leading to an oscillating charge and current, which can be described mathematically by specific equations.

Angular Frequency

Angular frequency, denoted by the symbol \(\omega\), is a key concept in understanding oscillations within an LC circuit. It represents the rate of change of the phase of the sinusoidal oscillation, or in simpler terms, it describes how quickly the system completes one full cycle of oscillation.

For an LC circuit, the angular frequency is determined by the formula:\[\omega = \frac{1}{\sqrt{LC}}\]Where \(L\) is the inductance in henrys (H) and \(C\) is the capacitance in farads (F). Angular frequency is related to the conventional frequency, \(f\), which counts the number of cycles per second, by the equation \(\omega = 2\pi f\). It’s crucial to understand that a higher angular frequency means that the circuit oscillates more rapidly. Therefore, by knowing the values of \(L\) and \(C\), we can calculate \(\omega\) and hence predict how the charge on the capacitor varies with time.

Time-dependent Charge Expression

The time-dependent charge expression describes how the charge on a capacitor changes over time in an LC circuit. The formula to determine the charge at any given moment is:\[Q(t) = Q_0 \cos(\omega t)\]Where \(Q(t)\) is the charge at time \(t\), \(Q_0\) is the maximum initial charge on the capacitor, and \(\omega\) is the angular frequency.

This expression is a result of the energy conservation in the LC circuit and the sinusoidal nature of the charge and current. The cosine function reflects the periodic oscillation of the charge, which goes through a full cycle between \(Q_0\) and \( -Q_0\) as time progresses. The duration of the cycle depends on the angular frequency \(\omega\), which is shaped by the circuit's inductance and capacitance. By using this expression, we can solve for specific instances in time when the charge on the capacitor reaches a certain value, such as the 80.0 µC in the given exercise.