Problem 33 - University Physics with Modern Physics

Question

A 7.50 -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 \(8.60 \times 10^{-5}\) s. Calculate: (a) the inductance of the coil; (b) the maximum charge on the capacitor; (c) the total energy of the circuit; \((\mathrm{d})\) the maximum current in the circuit.

Step-by-Step Solution

Answer

(a) L = 2.0 mH, (b) Q_max = 90.0 nC, (c) U = 5.4 µJ, (d) I_max = 1.04 A.

Step 1: Understand the LC Circuit

The problem describes an LC circuit, which is a circuit containing an inductor (coil) and a capacitor. Once charged, the capacitor will discharge through the inductor, resulting in an oscillating circuit. The electrical energy is exchanged between the electric field of the capacitor and the magnetic field of the inductor.

Step 2: Use the Formula for the Period of LC Circuit Oscillation

The period of oscillation \( T \) for an LC circuit is given by the formula: \( T = 2\pi \sqrt{LC} \), where \( L \) is the inductance, and \( C \) is the capacitance. We can rearrange this equation to solve for \( L \):\[ L = \frac{T^2}{4\pi^2 C} \]. Substitute \( T = 8.60 \times 10^{-5} \) s and \( C = 7.50 \times 10^{-9} \) F into the formula.

Step 3: Calculate the Inductance

Substitute the values into the equation: \[ L = \frac{(8.60 \times 10^{-5})^2}{4\pi^2 \times 7.50 \times 10^{-9}} \]. Calculating this gives the inductance of the coil.

Step 4: Determine the Maximum Charge on the Capacitor

The maximum charge \( Q_{max} \) on the capacitor can be calculated using the formula \( Q = CV \), where \( C \) is the capacitance and \( V \) is the voltage. Substitute \( C = 7.50 \times 10^{-9} \) F and \( V = 12.0 \) V to find \( Q_{max} \).

Step 5: Calculate the Total Energy in the Circuit

The total energy stored in the capacitor when fully charged is given by \( U = \frac{1}{2}CV^2 \). Use \( C = 7.50 \times 10^{-9} \) F and \( V = 12.0 \) V to calculate the energy \( U \).

Step 6: Find the Maximum Current in the Circuit

When the energy is transferred to the inductor, the maximum current \( I_{max} \) is given by \( U = \frac{1}{2}LI_{max}^2 \). Solve for \( I_{max} \): \[ I_{max} = \sqrt{\frac{2U}{L}} \]. Insert the values of \( U \) and \( L \) calculated in previous steps to find \( I_{max} \).

Key Concepts

capacitor

The capacitor is a fundamental component in electronics that stores electrical energy. In the simplest terms, it consists of two conductive plates separated by an insulating material known as the dielectric. When you apply voltage, the capacitor stores energy in the form of an electric field. When connected to a circuit, like in our LC (inductor-capacitor) circuit example, it can release that stored energy, creating a flow of charge.

In an LC circuit, the capacitor plays a crucial role in creating oscillations. Initially charged up to a specific voltage, the capacitor discharges through the inductor, starting the oscillating process. During this cycle, the energy alternates between the electric field of the capacitor and the magnetic field of the inductor.

• Capacitance is measured in farads (F), and in our exercise, it was given as 7.50 nF (nanofarads).
• The maximum charge stored, calculated using the formula \( Q = CV \), translates our understanding of how much energy the capacitor can produce at its peak.

Understanding the capacitor's role helps in grasping how LC circuits work and the nature of this energy interchange.

inductor

An inductor is another essential electronic component that stores energy, but instead of in an electric field, it stores it in a magnetic field. This magnetic energy storage becomes critical in an LC circuit.

Inductors are often coils of wire, and when the current flows through them, a magnetic field is created. In the LC circuit, when the capacitor discharges, its energy flows into the inductor, generating a magnetic field and causing a current to oscillate back and forth.

• The inductance, measured in henrys (H), determines how effectively the inductor manages the energy transfer. In our example, the inductance \( L \) can be calculated using rearranged oscillation formulas.
• This calculation tells us how the coil responds to changes in current, affecting the overall oscillation period of the circuit.

Understanding the role of inductors essentializes the concept of energy transformation within the circuit.

oscillation period

In an LC circuit, the oscillation period is the time it takes for the energy to complete one full cycle of transferring from the capacitor to the inductor and back again. This period is critical in circuits that require timing, such as radio transmitters or harmonic oscillators.

The formula for the period of oscillation, \( T = 2\pi \sqrt{LC} \), relates the capacitance and inductance to the time it takes for a full cycle. Rearranging this formula allows us to determine the inductance when the period is known:

\[ L = \frac{T^2}{4\pi^2 C} \]

By understanding how changing either the capacitance or the inductance affects the oscillation period, engineers can design circuits with precise timing features.

• Knowing the oscillation period is vital in applications where control over timing is essential, influencing how signals are processed and managed.
• In our scenario, the time was given as \( 8.60 \times 10^{-5} \) seconds, providing a basis to compute the inductance and further explore the circuit's dynamics.

Recognizing the significance of the oscillation period offers insight into the precise workings of an LC circuit.

energy in circuit

Energy plays a fundamental role in the functionality of an LC circuit. It moves between potential energy in the capacitor and kinetic energy in the inductor, driving the oscillations in the circuit.

This energy transfer is quantified by specific formulas. The initial energy stored in the capacitors is given by \( U = \frac{1}{2}CV^2 \). As the energy shifts to the inductor, we explore how that energy allows current to flow and oscillate.

• The total energy in the circuit remains constant, assuming there are no energy losses, such as resistance. This principle of conservation is fundamental in understanding how circuits maintain operation over time.
• In the exercise, knowing the energy allows us to solve for maximum current or understand changes if the capacitor or inductor values change.

Grasping how energy moves and transforms within the circuit is key to mastering its operation, showing the beauty of conservation principles in electronics.