Problem 29 - University Physics with Modern Physics

Question

A \(7.50-\mathrm{nF}\) capacitor is charged up to \(12.0 \mathrm{V},\) then disconnected from the power supply and connected in series through a coil. The period of oscillation of the circuit is then measured to be \(8.60 \times 10^{-5} \mathrm{s}\) . Calculate: (a) the inductance of the coil; \((\mathrm{b})\) the maximum charge on the capacitor; (c) the total energy of the circuit; (d) the maximum current in the circuit.

Step-by-Step Solution

Answer

(a) 9.37 mH, (b) 90 nC, (c) 0.54 µJ, (d) 0.34 A.

Step 1: Identify known values

We know the following values from the problem: capacitance \( C = 7.50 \times 10^{-9} \) F, voltage \( V = 12.0 \) V, and the period \( T = 8.60 \times 10^{-5} \) s.

Step 2: Calculate inductance of the coil (a)

The period of oscillation \( T \) for an LC circuit is given by \( T = 2 \pi \sqrt{LC} \). We can rearrange this formula to solve for the inductance \( L \): \[L = \frac{T^2}{4 \pi^2 C}\]Substituting the values, \[ L = \frac{(8.60 \times 10^{-5})^2}{4 \pi^2 \cdot 7.50 \times 10^{-9}} \approx 9.37 \times 10^{-3} \text{ H} \].

Step 3: Determine the maximum charge on the capacitor (b)

The maximum charge \( Q_{max} \) on a capacitor is given by the formula \( Q = CV \). Substituting the known values:\[ Q_{max} = 7.50 \times 10^{-9} \times 12.0 = 9.0 \times 10^{-8} \text{ C} \].

Step 4: Calculate the total energy of the circuit (c)

The total energy \( E \) stored in the circuit is equal to the energy in the charged capacitor, given by:\[ E = \frac{1}{2} CV^2 \].Substitute the values into the formula:\[ E = \frac{1}{2} \times 7.50 \times 10^{-9} \times (12.0)^2 = 5.40 \times 10^{-7} \text{ J} \].

Step 5: Find the maximum current in the circuit (d)

The maximum current \( I_{max} \) can be found using the relationship between energy and current in an LC circuit: \[ E = \frac{1}{2} LI_{max}^2 \].Rearrange to solve for \( I_{max} \):\[ I_{max} = \sqrt{\frac{2E}{L}} \].Substituting the known values:\[ I_{max} = \sqrt{\frac{2 \times 5.40 \times 10^{-7}}{9.37 \times 10^{-3}}} \approx 0.34 \text{ A} \].

Key Concepts

Capacitor

A capacitor is a device that stores electrical energy. When we talk about capacitors in an LC circuit, we're looking at how these devices charge and discharge electricity to produce oscillations.
The capacitor can store a voltage across its plates, and in our example, it was charged to a voltage of 12.0 V. This stored voltage helps in initiating the energy exchange process that characterizes the LC circuit's behavior.
In simpler terms, once disconnected from the power supply, the charged capacitor in our example can release and absorb charge, creating the oscillations necessary for the LC circuit to function.

Inductance

Inductance is a property of coils or inductors, and it measures how much electrical energy can be stored in a magnetic field. In our LC circuit, the coil's inductance determines how the circuit oscillates when the charge moves back and forth between the capacitor and the coil.
The inductance is crucial in calculating the period of oscillation for an LC circuit. In the example exercise, the coil's inductance was calculated to be approximately 9.37 mH using the formula for the period of oscillation:

• The formula used is: \[L = \frac{T^2}{4 \pi^2 C}\]
• Where \( T \) is the period, \( \pi \) is a constant, and \( C \) is the capacitance.

This calculation shows that the coil’s properties significantly affect the circuit’s behavior.

Oscillation

The term oscillation in the context of LC circuits refers to the back-and-forth flow of electric charge. This movement happens as the capacitor discharges its stored energy into the inductor and then recharges, maintaining continual cyclic motion.
The oscillation period is the time it takes for the circuit to go through one complete cycle of charge movement. In our example, this was given as 8.60 x 10^{-5} seconds. This time dictates how quickly the circuit can transfer energy back and forth. Understanding oscillations helps to predict the behavior of LC circuits in various applications, including radio tuning and signal processing.

Charge

Charge in an LC circuit usually refers to the quantity of electricity held by the capacitor. The maximum charge can be calculated using the formula:- \( Q_{max} = CV \)In the exercise, the maximum charge on the capacitor after charging to 12.0 V was calculated as 9.0 x 10^{-8} C.
Charge is integral to understanding how and when capacitors release energy, giving us insight into the oscillatory behavior of the LC circuit.
When the circuit begins to oscillate, the charge continuously fluctuates between the capacitor and the inductor, facilitating the energy oscillation.

Energy

Energy in an LC circuit is the total capability to do work, stored in both the electric field of the capacitor and the magnetic field of the inductor. It can be calculated using:

• Total energy formula: \( E = \frac{1}{2} CV^2 \)

In step-by-step solutions, the total energy stored at any given time was found to be 5.40 x 10^{-7} J.
This energy is then partially converted into magnetic energy in the inductor and back to electrical energy in the capacitor as the circuit oscillates. The conservation of energy principle tells us that while energy might fluctuate between these two forms, the total energy in the ideal LC circuit remains constant, ignoring resistive losses.