Problem 15 - Fundamentals of Physics

Question

An oscillating \(L C\) circuit consisting of a \(1.0 \mathrm{nF}\) capacitor and a \(3.0 \mathrm{mH}\) coil has a maximum voltage of \(3.0 \mathrm{~V}\). What are (a) the maximum charge on the capacitor, (b) the maximum current through the circuit, and (c) the maximum energy stored in the magnetic field of the coil?

Step-by-Step Solution

Answer

(a) 3.0 nC, (b) 3.0 mA, (c) 1.35 x 10^{-8} J.

Step 1: Calculating the Maximum Charge on the Capacitor

The maximum charge on a capacitor is given by the equation: \[ Q_{max} = C \cdot V_{max} \]where \( C = 1.0 \times 10^{-9} \text{ F} \) (1.0 nF) and \( V_{max} = 3.0 \text{ V} \). Thus, \[ Q_{max} = (1.0 \times 10^{-9} \,\text{F}) \times (3.0 \, \text{V}) = 3.0 \times 10^{-9} \,\text{C} \].

Step 2: Calculating the Resonant Frequency of the LC Circuit

The resonant frequency \( \omega \) of an LC circuit is given by the formula: \[ \omega = \frac{1}{\sqrt{LC}} \]We have \( L = 3.0 \times 10^{-3} \text{ H} \) and \( C = 1.0 \times 10^{-9} \text{ F} \). Calculating it gives:\[ \omega = \frac{1}{\sqrt{(3.0 \times 10^{-3} \, \text{H} \cdot 1.0 \times 10^{-9} \, \text{F})}} = \frac{1}{\sqrt{3.0 \times 10^{-12}}} \,\text{rad/s} \].

Step 3: Calculating the Maximum Current in the Circuit

The maximum current \( I_{max} \) in an LC circuit can be found with:\[ I_{max} = \omega \cdot Q_{max} \]Since we calculated \( Q_{max} = 3.0 \times 10^{-9} \text{ C} \), and substituting \( \omega \) from step 2:\[ I_{max} = \frac{1}{\sqrt{3.0 \times 10^{-12}}} \cdot 3.0 \times 10^{-9} \text{ C} \].This simplifies to approximately \( 3.0 \times 10^{-3} \text{ A} \).

Step 4: Calculating the Maximum Energy Stored in the Magnetic Field

The maximum energy \( U_{max} \) stored in the magnetic field of the coil is given by:\[ U_{max} = \frac{1}{2} L I_{max}^2 \]Substituting \( L = 3.0 \times 10^{-3} \text{ H} \) and \( I_{max} = 3.0 \times 10^{-3} \text{ A} \):\[ U_{max} = \frac{1}{2} \times 3.0 \times 10^{-3} \text{ H} \times (3.0 \times 10^{-3} \text{ A})^2 = 1.35 \times 10^{-8} \text{ J} \].

Key Concepts

Oscillating Circuit

An oscillating circuit, often called an LC circuit, is a basic type of electrical circuit consisting of an inductor (L) and a capacitor (C). These components work together to produce electromagnetic oscillations. They store and transfer energy back and forth between the electric field of the capacitor and the magnetic field of the inductor.
In an LC circuit, the capacitor initially holds the electrical charge. As it discharges, the current flows through the inductor, creating a magnetic field. Once the capacitor is completely discharged, the magnetic field collapses, inducing a current back into the capacitor in the opposite direction. This process repeats, creating voltage oscillations in the circuit.
Key points to remember about LC circuits include:

• Energy conservation: Energy oscillates between the capacitor and the inductor without any loss, assuming no resistance.
• Natural frequency: The circuit oscillates at its resonant frequency, which depends on the values of L and C.

Maximum Charge

The maximum charge on a capacitor in an LC circuit can be determined using the formula: \[ Q_{max} = C \cdot V_{max} \]where \( C \) is the capacitance and \( V_{max} \) is the maximum voltage across the capacitor. This equation comes from the basic definition of capacitance, which is the charge per unit voltage.
In this exercise, with a capacitance of \( 1.0 \text{ nF} = 1.0 \times 10^{-9} \text{ F} \) and a maximum voltage of \( 3.0 \text{ V} \), the maximum charge \( Q_{max} \) can be calculated as:\[ Q_{max} = (1.0 \times 10^{-9} \text{ F}) \times (3.0 \text{ V}) = 3.0 \times 10^{-9} \text{ C} \]
This indicates that at its peak, the capacitor will hold a charge of \( 3.0 \times 10^{-9} \text{ C} \).

Resonant Frequency

The resonant frequency of an LC circuit is a critical concept in understanding how these circuits function. It is the frequency at which the circuit naturally oscillates when not subject to any external energy input. This frequency is given by the formula:\[\omega = \frac{1}{\sqrt{LC}}\]where \( L \) is the inductance and \( C \) is the capacitance.
The resonant frequency plays a key role because:

• At this frequency, the impedance of the circuit is minimized, thus maximizing current flow.
• It dictates the oscillation speed of the charge and energy between the capacitor and inductor.

In the current scenario with \( L = 3.0 \times 10^{-3} \text{ H} \) and \( C = 1.0 \times 10^{-9} \text{ F} \), this frequency can be calculated. The result is a rapid oscillation, indicative of high-frequency electrical behavior.

Magnetic Field Energy

The energy stored in the magnetic field of the inductor at the peak of its charge cycle is an important aspect of the LC circuit. This magnetic field energy can be calculated using the formula:\[U_{max} = \frac{1}{2} L I_{max}^2\]where \( L \) is the inductance and \( I_{max} \) is the maximum current through the circuit.
In this context, once you have calculated the maximum current, you can determine the energy stored at its peak. Energy is a crucial concept because it represents the system's ability to do work or generate a response.
This equation shows that energy depends mainly on both the inductance and the square of the current. In our exercise, using \( L = 3.0 \times 10^{-3} \text{ H} \) and \( I_{max} = 3.0 \times 10^{-3} \text{ A} \), the energy stored comes out to \( 1.35 \times 10^{-8} \text{ J} \). This represents the peak energy the system can store magnetically.