Problem 30 - University Physics with Modern Physics

Question

A \(18.0-\mu F\) capacitor is placed across a \(22.5-\mathrm{V}\) battery for several seconds and is then connected across a 12.0 -mH inductor that has no appreciable resistance. (a) After the capacitor and inductor are connected together, find the maximum current in the circuit. When the current is a maximum, what is the charge on the capacitor? (b) How long after the capacitor and inductor are connected together does it take for the capacitor to be completely discharged for the first time? For the second time? (c) Sketch graphs of the charge on the capacitor plates and the current through the inductor as functions of time.

Step-by-Step Solution

Answer

Maximum current is 0.0394 A, charge is 0 C when current is max. Discharge times: \(1.16\times10^{-3}\) s and \(2.32\times10^{-3}\) s.

Step 1: Calculate Initial Charge on Capacitor

First, calculate the initial charge stored on the capacitor using the formula for charge, which is given by \( Q = C \times V \). Here, the capacitance \( C \) is 18.0 \( \mu \mathrm{F} \) and the voltage \( V \) is 22.5 V. Thus, the initial charge \( Q \) is calculated as follows: \( Q = 18.0 \times 10^{-6} \,\mathrm{F} \times 22.5 \, \mathrm{V} = 4.05 \times 10^{-4} \, \mathrm{C}\).

Step 2: Use Energy Conservation to Find Maximum Current

In an LC circuit, the initial energy stored in the capacitor transforms into energy stored in the inductor at the maximum current. The energy in the capacitor is \( \frac{1}{2} C V^2 \) and the energy in the inductor is \( \frac{1}{2} L I^2 \). Equating the two energies gives \( \frac{1}{2} \times 18.0 \times 10^{-6} \,\mathrm{F} \times (22.5 \, \mathrm{V})^2 = \frac{1}{2} \times 12.0 \times 10^{-3} \, \mathrm{H} \times I_{max}^2 \). Solving this, we get \( I_{max} = 0.0394 \, \mathrm{A} \).

Step 3: Calculate Charge at Maximum Current

When the current is maximum, the charge on the capacitor is zero because all the energy has transferred from the electric field in the capacitor to the magnetic field in the inductor. Thus, \( Q = 0\, \mathrm{C} \).

Step 4: Determine Time for Complete Discharge

The time period \( T \) for an LC circuit is given by \( T = 2\pi \sqrt{LC} \). Substituting \( L = 12.0 \times 10^{-3} \mathrm{H} \) and \( C = 18.0 \times 10^{-6} \mathrm{F} \), we find \( T = 2\pi \sqrt{12.0 \times 10^{-3} \, \mathrm{H} \times 18.0 \times 10^{-6} \,\mathrm{F}} \approx 2.32\times10^{-3} \, \mathrm{s} \). The capacitor discharges completely for the first time at \( \frac{T}{2} \approx 1.16\times10^{-3} \, \mathrm{s} \) and for the second time at one full cycle, or \( T \).

Step 5: Sketch Graphs of Charge and Current

(Not practically drawable in text) The charge on the capacitor \( Q(t) \) and the current \( I(t) \) in the circuit are sinusoidal. The charge function \( Q(t) = Q_{max} \cos(\omega t) \) starts from maximum and reaches zero at \( \frac{T}{4} \), then becomes negative till \( \frac{T}{2} \) (completely discharged). The current \( I(t) = I_{max} \sin(\omega t) \) starts from zero, reaches maximum at \( \frac{T}{4} \), and zero again at \( \frac{T}{2} \).

Key Concepts

Capacitor Discharge

Capacitor discharge in an LC circuit is a process where the stored charge in the capacitor is fully transferred to the inductor, setting off oscillations. When a capacitor is charged using a battery, it stores energy in the form of an electric field. The moment it connects to an inductor, this energy starts converting into magnetic energy in the inductor. As the capacitor discharges, the charge on the capacitor decreases and eventually reaches zero temporarily. This is the point when it has transferred all its energy.
After the capacitor initially discharges, the reverse process begins, causing a new cycle of charging and discharging. This cyclical transfer continues in an ideal LC circuit, meaning one without resistive elements, resulting in perpetual oscillations.

• First complete discharge occurs at half the period (\( T/2 \)).
• The entire cycle completes at one full period \( T \).

Maximum Current

In an LC circuit, the maximum current occurs when all electrical energy from the capacitor is converted into magnetic energy in the inductor. This energy conversion principle is driven by the law of energy conservation. At maximum current, the capacitor is entirely discharged and maintains zero charge momentarily.
The maximum current can be calculated using the energy equivalence:

• Energy in the capacitor: \( \frac{1}{2} C V^2 \)
• Energy in the inductor: \( \frac{1}{2} L I^2 \)

The conservation yields the equation: \[ \frac{1}{2} C V^2 = \frac{1}{2} L I_{max}^2 \]Solving this gives the max current \( I_{max} \). In this problem setup, substituting in the known values results in \( I_{max} = 0.0394 \, \mathrm{A} \).
This principle is crucial in understanding how energy is swapped between components in an LC circuit.

Energy Conservation in Circuits

Energy conservation in circuits, especially LC circuits, ensures that energy between components transforms but is neither created nor destroyed.
Upon discharging, the energy from the capacitor stored as electrical energy is completely transformed into magnetic energy in the inductor. Then, this energy is transferred back to the capacitor with reversed polarity due to the inductor's influence. This forms a never-ending loop of energy transfer in an ideal situation.

• Capacitor initially holds energy: \( \frac{1}{2} C V^2 \)
• Switching begins energy transformation to inductor: \( \frac{1}{2} L I^2 \)

Understanding this conservation provides insights into how energy flows in oscillatory circuits, a cornerstone concept in alternating current (AC) systems and radio frequency equipment.

Sinusoidal Current and Charge

Sinusoidal functions are paramount in describing how charge \( Q(t) \) and current \( I(t) \) behave over time in LC circuits. These oscillations resemble sine and cosine waves, tracing the periodic exchange of energy in the forms of electric and magnetic fields.
The charge on the capacitor \( Q(t) = Q_{max} \cos(\omega t) \) starts at maximum and decreases to zero as the energy converts into a max current, just to rise again. Conversely, the current through the inductor \( I(t) = I_{max} \sin(\omega t) \) begins at zero, reaching a peak as the charge on the capacitor empties.

• At quarter period \( T/4 \), max current occurs.
• At half period \( T/2 \), the charge cycle is completed.

These alternating patterns represent the seamless energy transition, crucially applying to AC analysis where such sinusoidal behaviors are naturally evident.