Question

A 7.50-nF capacitor is charged up to 12.0 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; (d) the maximum current in the circuit.

Short answer

(a) The inductance is 3.12 mH. (b) Max charge is 90.0 nC. (c) Total energy is 540 nJ. (d) Max current is 11.7 mA.

Step-by-step solution

Given Values

First, note the values provided in the problem: - Capacitance of the capacitor, \( C = 7.50 \times 10^{-9} \) F.- Voltage across the capacitor, \( V = 12.0 \) V.- Period of oscillation, \( T = 8.60 \times 10^{-5} \) s.

Find the Inductance (L)

The period of oscillation \( T \) in a series LC circuit is given by the formula: \[ T = 2\pi \sqrt{LC} \]Rearrange this to solve for the inductance \( L \): \[ L = \frac{T^2}{4\pi^2C} \]Substitute the known values:\[ L = \frac{(8.60 \times 10^{-5})^2}{4 \pi^2 (7.50 \times 10^{-9})} \]\[ \Rightarrow L \approx 3.12 \times 10^{-3} \text{ H} \]

Maximum Charge on the Capacitor (Q_max)

The maximum charge \( Q_{\text{max}} \) on the capacitor is calculated using:\[ Q_{\text{max}} = CV \]Substitute the given values:\[ Q_{\text{max}} = (7.50 \times 10^{-9} \text{ F})(12.0 \text{ V}) \]\[ \Rightarrow Q_{\text{max}} = 9.00 \times 10^{-8} \text{ C} \]

Total Energy of the Circuit (E)

The total energy \( E \) stored in the LC circuit is given by:\[ E = \frac{1}{2}CV^2 \]Using the known values:\[ E = \frac{1}{2}(7.50 \times 10^{-9})(12.0)^2 \]\[ \Rightarrow E = 5.40 \times 10^{-7} \text{ J} \]

Maximum Current in the Circuit (I_max)

The maximum current \( I_{\text{max}} \) in the circuit is related to the maximum energy in the inductor:\[ E = \frac{1}{2}LI_{\text{max}}^2 \]Solve for \( I_{\text{max}} \):\[ I_{\text{max}} = \sqrt{\frac{2E}{L}} \]Substitute the values:\[ I_{\text{max}} = \sqrt{\frac{2(5.40 \times 10^{-7})}{3.12 \times 10^{-3}}} \]\[ \Rightarrow I_{\text{max}} \approx 1.17 \times 10^{-2} \text{ A} \]

Key concepts

Capacitance

Capacitance is a fundamental concept in electromagnetism. It refers to a capacitor's ability to store an electrical charge. A capacitor, in simple terms, is a device that can hold electric charge, consisting typically of two metal plates separated by an insulating material known as a dielectric. The unit of capacitance is the farad (F).
The capacitance of a capacitor can be calculated using the formula:

• \( C = \frac{Q}{V} \)

where

• \( C \) is the capacitance in farads,
• \( Q \) is the charge in coulombs, and
• \( V \) is the voltage in volts.

In our example, the capacitance was given as 7.50 nF (nanofarads). Understanding capacitance is crucial in various circuits, such as RC circuits and especially in LC circuits where capacitors play a significant role in the oscillation process.

Inductance

Inductance is a property of an electrical conductor whereby a change in current flowing through it induces an electromotive force (EMF) in it and possibly also in nearby conductors. It is a key principle in circuits that contain coils of wire, called inductors.
The unit of inductance is the henry (H), and it can be calculated using the formula derived from the LC circuit:

• \[ L = \frac{T^2}{4\pi^2C} \]

where

• \( L \) is the inductance,
• \( T \) is the period of oscillation, and
• \( C \) is the capacitance.

In the LC circuit described in the exercise, the inductance was found to be approximately 3.12 mH (millihenries). Inductance is crucial because it determines how the circuit will react to changes in current, which affects the circuit's resonance and oscillation characteristics.

Resonance Frequency

Resonance frequency is a natural frequency at which a system oscillates with the largest amplitude. In an LC circuit, it is defined as the frequency at which the reactance of the inductor equals the reactance of the capacitor, causing the impedance to be minimal.
The formula for the resonance frequency \( f_0 \) in an LC circuit is:

• \[ f_0 = \frac{1}{2\pi \sqrt{LC}} \]

where

• \( L \) is the inductance, and
• \( C \) is the capacitance.

Resonance is significant because at this frequency, the circuit can exchange energy efficiently between the capacitor and the inductor, leading to sustained oscillations without loss. This principle is employed in various applications like radio transmitters and receivers.

Energy Storage in Capacitors

Energy storage in capacitors is a concept where capacitors store energy in the form of an electric field between their plates. When connected to a circuit, this stored energy can be released to do work.
The energy \( E \) stored in a capacitor is given by the formula:

• \[ E = \frac{1}{2} C V^2 \]

where

• \( C \) is the capacitance, and
• \( V \) is the voltage across the capacitor.

In the example from the exercise, the energy stored was calculated to be 5.40 µJ (microjoules). Capacitors are widely used in electronic circuits to maintain power supply while switching devices, provide timing functions, and filter signals.

Current in Inductors

Current in inductors is an essential aspect of circuit analysis, particularly in circuits containing both inductors and capacitors. When an inductor is part of an LC circuit, it stores energy in the form of a magnetic field as the current passes through.
The relationship between the maximum current \( I_{\text{max}} \) and the stored energy \( E \) in an inductor is:

• \[ I_{\text{max}} = \sqrt{\frac{2E}{L}} \]

where

• \( E \) is the total energy in the circuit, and
• \( L \) is the inductance.

In the example, the maximum current was found to be approximately 11.7 mA. Understanding how current behaves in inductors is critical for designing and analyzing AC circuits, transformers, and other electromagnetic devices.