Problem 33 - University Physics with Modern Physics

Question

An \(L C\) cirruit containing an \(80.0-\mathrm{mH}\) inductor and a 1.25-nF capacitor oscillates with a maximum current of 0.750 \(\mathrm{A}\) . Calculate: (a) the maximum charge on the capacitor and (b) the oscillation frequency of the circuit. (c) Assuming the capacitor had its maximum charge at time \(t=0\) , calculate the energy stored in the inductor after 2.50 \(\mathrm{ms}\) of oscillation.

Step-by-Step Solution

Answer

(a) 7.50×10⁻⁵ C; (b) 1592.5 Hz; (c) 3.45×10⁻⁹ J.

Step 1: Find the maximum charge on the capacitor

To find the maximum charge on the capacitor, we can use the relationship between the maximum current and maximum charge in an LC circuit:\[ Q_{ ext{max}} = I_{ ext{max}} \times rac{1}{ ext{angular frequency}} \]The angular frequency, \( \omega \), can be calculated with:\[ \omega = \frac{1}{\sqrt{LC}} \]Where:\( L = 80.0 \times 10^{-3} \, \text{H} \) (inductance of the inductor)\( C = 1.25 \times 10^{-9} \, \text{F} \) (capacitance of the capacitor)First, calculate the angular frequency:\[ \omega = \frac{1}{\sqrt{(80.0 \times 10^{-3})(1.25 \times 10^{-9})}} \approx \frac{1}{\sqrt{1.00 \times 10^{-10}}} \approx 10000 \, \text{rad/s} \]Now, calculate the maximum charge:\[ Q_{ ext{max}} = 0.750 \, \text{A} \times \frac{1}{10000} = 7.50 \times 10^{-5} \, \text{C} \]

Step 2: Compute the oscillation frequency

The oscillation frequency \( f \) of an \( LC \) circuit is related to the angular frequency by:\[ f = \frac{\omega}{2\pi} \]Substituting the calculated angular frequency:\[ f = \frac{10000}{2\pi} \approx 1592.5 \, \text{Hz} \]

Step 3: Calculate the energy stored in the inductor after 2.50 ms

To calculate the energy stored in the inductor at any time during the oscillation, we start with the total energy in the circuit, which is the energy of the capacitor when it is fully charged:\[ E_0 = \frac{1}{2}C Q_{ ext{max}}^2 \]Substitute the known values:\[ E_0 = \frac{1}{2}(1.25 \times 10^{-9})(7.50 \times 10^{-5})^2 \approx 3.5156 \times 10^{-9} \, \text{J} \]At any time \( t \), the energy in the inductor is given by:\[ E_L = E_0 \cos^2(\omega t) \]Given \( t = 2.50 \, \text{ms} = 2.50 \times 10^{-3} \, \text{s} \), calculate \( E_L \):\[ E_L = 3.5156 \times 10^{-9} \cos^2(10000 \times 2.50 \times 10^{-3}) \approx 3.5156 \times 10^{-9} \times \cos^2(25) \]Since \( \cos(25) \approx -0.991 \),\[ E_L \approx 3.5156 \times 10^{-9} \times (0.991)^2 \approx 3.45 \times 10^{-9} \, \text{J} \]

Key Concepts

Inductance

Inductance is a fundamental property of inductors in an electrical circuit. It defines an inductor's ability to store energy in the form of a magnetic field. One common symbol for inductance is the letter "L", and its unit of measurement is the henry (H). In an LC circuit, the inductor works hand in hand with the capacitor to enable oscillations. This occurs because the energy is alternatively stored and released between the magnetic field of the inductor and the electric field of the capacitor. Understanding inductance is crucial because it determines the manner in which the circuit reacts to changes in electric current. The inductance value in the exercise was given as 80.0 millihenries (mH), which equals 80.0 x 10^-3 H. In practical terms, high inductance would mean that the circuit resists changes in current instantaneously.

Capacitance

Capacitance is an essential characteristic of capacitors, representing the capacity to store electric charge. It is symbolized by the letter "C," and measured in farads (F), though much smaller units like microfarads (μF) or nanofarads (nF) are often used. Capacitance occurs in a capacitor due to its structure, which consists of two conductive plates separated by an insulating material. In the exercise, the capacitance is specified as 1.25 nanofarads (nF), or 1.25 x 10^-9 F. Like inductance, capacitance plays a critical role in the operation of an LC circuit, as it helps determine the circuit's capacity to sustain oscillations. The capacitor temporarily stores electrical energy, which is then released as the circuit oscillates.

Oscillation Frequency

The oscillation frequency of an LC circuit represents how often the circuit completes one full cycle of charging and discharging per second. It is denoted by "f" and is typically measured in hertz (Hz). The frequency is inversely proportional to the period of oscillation, indicating how quickly or slowly the oscillations occur. In the provided solution, the frequency was derived from the angular frequency by dividing it by 2π. This results in an oscillation frequency of approximately 1592.5 Hz, meaning the LC circuit oscillates roughly 1592.5 times per second. A higher frequency would correspond to faster oscillations, while a lower frequency indicates slower oscillations.

Angular Frequency

Angular frequency, represented by the symbol \( \omega \), is another key concept within LC circuits. It measures how fast the circuit oscillates in radians per second. A complete oscillation is equivalent to \( 2\pi \) radians, which means angular frequency plays a critical role in determining both the oscillation frequency and the characteristics of the oscillating circuit. The equation for calculating angular frequency in an LC circuit is \( \omega = \frac{1}{\sqrt{LC}} \). This formula shows that angular frequency is dependent on both inductance (L) and capacitance (C). In the exercise, \( \omega \) was found to be approximately 10000 rad/s. This value helps in determining not only how the circuit oscillates but also the maximum charge and energy within the components over time.

Energy Storage in Inductor

An inductor stores energy in its magnetic field when current flows through it. The energy stored is a result of the inductor's resistance to changes in current, which creates and maintains the magnetic field. The magnitude of energy stored can fluctuate over time, especially in an oscillating circuit such as the LC circuit from the exercise.At the beginning of the oscillation cycle, when the capacitor is fully charged, the energy is primarily stored in the capacitor and gradually transferred to the inductor. The energy stored in the inductor at any given time \( t \) can be calculated using the formula: \[ E_L = E_0 \cos^2(\omega t) \]where \( E_0 \) is the initial energy provided by the fully charged capacitor. While at \( t = 2.50 \text{ ms} \), the value of \( E_L \) is used to determine how much energy the inductor is containing at that specific time, as calculated in the exercise to be approximately \( 3.45 \times 10^{-9} \text{ J} \). This understanding helps in managing and predicting the behavior of the circuit under various conditions.