Problem 88 - Fundamentals of Physics

Question

In an oscillating \(L C\) circuit, \(L=8.00 \mathrm{mH}\) and \(C=1.40 \mu \mathrm{F}\). At time \(t=0,\) the current is maximum at \(12.0 \mathrm{~mA}\). (a) What is the maximum charge on the capacitor during the oscillations? (b) At what earliest time \(t>0\) is the rate of change of energy in the capacitor maximum? (c) What is that maximum rate of change?

Step-by-Step Solution

Answer

(a) 12.67 µC; (b) 1.66 ms; (c) 186.43 W.

Step 1: Understanding the Conceptual Framework

An oscillating LC circuit consists of an inductor (L) and a capacitor (C). The system alternates between the capacitor storing the maximum charge (q) and the current (I) in the inductor being maximal. These quantities oscillate sinusoidally in time, describing harmonic motion. We use the relationship between the maximum current, maximum charge, and angular frequency (\(\omega\)) to resolve the energy dynamics.

Step 2: Calculate Angular Frequency \(\omega\)

The angular frequency \(\omega\) for an LC circuit is given by the formula: \[\omega = \frac{1}{\sqrt{LC}}\] Using \(L = 8.00\, \text{mH} = 8.00 \times 10^{-3}\, \text{H}\) and \(C = 1.40 \, \mu\text{F} = 1.40 \times 10^{-6}\, \text{F}\), we calculate:\[\omega = \frac{1}{\sqrt{(8.00 \times 10^{-3})(1.40 \times 10^{-6})}} = 947.1 \, \text{rad/s}\] This is the angular frequency of oscillation in the LC circuit.

Step 3: Determine Maximum Charge on the Capacitor

The relationship between maximum current \(I_0\), maximum charge \(Q_{max}\), and angular frequency \(\omega\) is given by:\[I_0 = \omega Q_{max}\] Rearranging for \(Q_{max}\): \[Q_{max} = \frac{I_0}{\omega}\]Substitute \(I_0 = 12.0\, \text{mA} = 12.0 \times 10^{-3}\, \text{A}\) and \(\omega = 947.1\, \text{rad/s}\):\[Q_{max} = \frac{12.0 \times 10^{-3} }{947.1} = 12.67 \times 10^{-6}\, \text{C} = 12.67 \mu\text{C}\]Thus, the maximum charge on the capacitor is 12.67 µC.

Step 4: Calculate the Time for Maximum Rate of Energy Change

The rate of change of energy in the capacitor is proportional to the time derivative of the current. The earliest time \(t\) when this derivative is maximum is when the sine function in the current's sinusoidal expression reaches its maximum. The current is given by:\[I = I_0 \cos(\omega t)\]The derivative is\[\frac{dI}{dt} = -\omega I_0 \sin(\omega t)\]The maximum changes occur when \(\sin(\omega t) = \pm 1\). The earliest time \(t\) is:\[\omega t = \frac{\pi}{2}\Rightarrow t = \frac{\pi}{2\omega} = \frac{\pi}{2 \times 947.1} = 1.66 \times 10^{-3} \, \text{s}\]Therefore, the earliest time is 1.66 ms.

Step 5: Calculate Maximum Rate of Change of Energy

The instantaneous rate of change of energy stored in the capacitor, \(\frac{dU}{dt},\) is given by\[\frac{dU}{dt} \approx CV \frac{dV}{dt} = C \cdot V_0^2 \cdot \omega \sin(\omega t) \cdot \omega\]Substitute \(\sin(\omega t) = 1\) to find maximum:\[\frac{dU}{dt}\Big|_{max} = \omega^2 \cdot Q_{max}^2 / C = (947.1)^2 \cdot (12.67 \times 10^{-6})^2 / (1.40 \times 10^{-6})\]Calculate:\[\frac{dU}{dt}\Big|_{max} = 186.43 \, \text{W}\]Hence, the maximum rate of change of energy is 186.43 W.

Key Concepts

Angular Frequency

In the context of an LC circuit, angular frequency, denoted as \( \omega \), plays a pivotal role. It describes how rapidly the oscillations occur in the system. For an LC circuit, the angular frequency is calculated using the formula: \[\omega = \frac{1}{\sqrt{LC}}\] where \( L \) is the inductance and \( C \) is the capacitance.This formula is derived from the fundamental laws governing electrical oscillations. It essentially describes the rate at which the energy swaps between the capacitor and the inductor. When we apply the formula using 8.00 mH for the inductance and 1.40 µF for the capacitance, we find that \( \omega = 947.1 \text{ rad/s} \). This means that in this circuit, every second, the oscillation cycles through approximately 947 radians, highlighting a substantial rate of change.Understanding \( \omega \) is imperative as it directly influences both the current and the charge behaviors within the circuit. In turn, this affects the overall energy distribution between the components.

Oscillations

An LC circuit is a classic example of oscillating systems in electrical engineering. Oscillations here refer to the repeating, cyclical movement of charges that alternate between stored and flowing states within the circuit.### What Happens During Oscillations?- The capacitor initially holds its maximum charge.- As the circuit progresses, the stored energy in the capacitor is converted into a current flowing through the inductor.- This current eventually depletes the charge in the capacitor, returning the stored energy back to the magnetic field in the inductor.- Subsequently, the cycle reverses, creating a back-and-forth motion of energy and charge.The sinusoidal nature of these oscillations means that the parameters such as the current \( I \) and the charge \( Q \) have smooth and predictable fluctuations over time. These fluctuations can be mathematically modeled using functions like \( I = I_0 \cos(\omega t) \), where \( I_0 \) is the maximum current.These oscillations continue indefinitely in an ideal circuit without resistance, beautifully demonstrating harmonic motion.

Energy Dynamics

Energy dynamics in an LC circuit involves understanding how energy flows and changes from electrical potential to magnetic energy and vice versa. This cyclic exchange defines the oscillatory nature of the LC circuit. ### Key Dynamics - **Energy Storage:** The capacitor initially stores energy as electrical potential energy when it is fully charged. - **Energy Transfer:** This energy is transferred to the inductor, where it is stored as magnetic energy due to the current flow. The story doesn't end here. The rate of change of energy is especially interesting to observe. At specific points in time, the derivative of the energy, which represents the rate of change, reaches its peak values. This occurs because this rate is dependent on the sinusoidal terms that depict how fast the energy shifts between forms. The maximum rate of change of energy in our particular LC circuit was found to be 186.43 W, occurring at the moment when the current and voltage sine wave derivatives align to create the greatest change. Understanding these dynamics not only helps in comprehending how LC circuits function but also illustrates a broader principle of conservation and transformation of energy in oscillating systems.