Problem 1 - Fundamentals of Physics

Question

An oscillating \(L C\) circuit consists of a \(75.0 \mathrm{mH}\) inductor and a \(3.60 \mu \mathrm{F}\) capacitor. If the maximum charge on the capacitor is \(2.90 \mu \mathrm{C},\) what are (a) the total energy in the circuit and (b) the maximum current?

Step-by-Step Solution

Answer

(a) Total energy is \( 1.17 \times 10^{-6} J \). (b) Maximum current is \( 0.109 A \).

Step 1: Understanding the Energy in an LC Circuit

The total energy stored in an LC circuit is conserved and can be given by the formula \( E = \frac{1}{2}C V^2 \), where \( V \) is the voltage across the capacitor. But we can also express energy in terms of the charge and capacitance: \( E = \frac{Q^2}{2C} \).

Step 2: Calculating Total Energy Using Charge and Capacitance

Given that the maximum charge \( Q \) on the capacitor is \( 2.90 \mu C \) and the capacitance \( C \) is \( 3.60 \mu F \), convert them to standard units: \( Q = 2.90 \times 10^{-6} C \), \( C = 3.60 \times 10^{-6} F \). The total energy \( E \) in the circuit is \( \frac{(2.90 \times 10^{-6})^2}{2 \times 3.60 \times 10^{-6}} \). Calculate to find \( E \).

Step 3: Introduction to Maximum Current Calculation

The maximum current in an LC circuit can be found using the relationship \( I_{max} = \omega Q \), where \( \omega = \frac{1}{\sqrt{LC}} \) is the angular frequency for the circuit.

Step 4: Calculate Angular Frequency

Find \( L = 75.0 \times 10^{-3} H \), \( C = 3.60 \times 10^{-6} F \). Calculate the angular frequency \( \omega = \frac{1}{\sqrt{LC}} = \frac{1}{\sqrt{75.0 \times 10^{-3} \times 3.60 \times 10^{-6}}} \).

Step 5: Calculate Maximum Current

With \( \omega \) calculated, find \( I_{max} = \omega \times Q = \omega \times 2.90 \times 10^{-6} \). Use the value of \( \omega \) from Step 4 to find \( I_{max} \).

Key Concepts

Energy Conservation in Circuits

In an LC circuit, energy conservation is a key principle. It implies that the total energy is always constant if there is no external influence like resistance or power supply. This energy toggles between the magnetic field of the inductor and the electric field of the capacitor.

Understanding this principle is crucial when dealing with circuits involving inductors and capacitors. When the capacitor discharges, it releases energy that builds up a magnetic field in the inductor. Conversely, as the inductor's magnetic field collapses, it recharges the capacitor.

The total energy in an LC circuit can be calculated using either the voltage across the capacitor or the charge on it. The formula in terms of charge is:

• \( E = \frac{Q^2}{2C} \)

where:

• \( E \) is the total energy.
• \( Q \) is the charge on the capacitor.
• \( C \) is the capacitance of the capacitor.

This equation highlights how energy is determined by the distribution of charge in the system.

Maximum Current in LC Circuit

The phenomenon of maximum current in an LC circuit is directly tied to the energy oscillating between the capacitor and inductor. As energy shifts, the current peaks when the energy stored in the capacitor is completely transferred to the inductor.

To find the maximum current, we use the relationship:

• \( I_{max} = \omega Q \)

Here,

• \( I_{max} \) is the maximum current.
• \( \omega \) is the angular frequency.
• \( Q \) is the maximum charge on the capacitor.

When the capacitor is fully discharged, all its energy has transitioned into kinetic energy represented by the current in the inductor. This is where the current reaches its maximum value.

Angular Frequency in LC Circuit

The angular frequency, denoted as \( \omega \), is crucial for analyzing LC circuits. It represents how fast the energy oscillates between the capacitor and inductor.

The expression for angular frequency in an LC circuit is:

• \( \omega = \frac{1}{\sqrt{LC}} \)

where:

• \( L \) is the inductance of the inductor.
• \( C \) is the capacitance of the capacitor.

Angular frequency gives insight into the rate of oscillation. Higher values suggest faster energy oscillation between the capacitor and inductor, while lower values indicate slower oscillation.

This concept ensures that you can understand how changes in either inductance or capacitance will affect the dynamic behavior of the circuit, hence allowing accurate predictions of its response.