Question

A \(2.00-\mu \mathrm{F}\) capacitor is fully charged by being connected to a 12.0-V battery. The fully charged capacitor is then connected to a \(0.250-\mathrm{H}\) inductor. Calculate (a) the maximum current in the inductor and (b) the frequency of oscillation of the LC circuit.

Short answer

Question: Calculate (a) the initial energy stored in the capacitor, (b) the maximum current in the inductor, and (c) the frequency of oscillation of the LC circuit for the given values: C = \(2.00 \times 10^{-6} \, \mathrm{F}\), V = 12.0 V, and L = 0.250 H. Answer: (a) The initial energy stored in the capacitor is \(1.44 \times 10^{-4} \ \mathrm{J}\). (b) The maximum current in the inductor is \(3.26 \times 10^{-2} \ \mathrm{A}\). (c) The frequency of oscillation of the LC circuit is \(159.2 \ \mathrm{Hz}\).

Step-by-step solution

(a) Initial energy stored in the capacitor

First, we need to calculate the initial energy stored in the capacitor when it's fully charged. The formula for the energy stored in a capacitor is: \(E_C = \frac{1}{2}CV^2\) Given, C = \(2.00 \times 10^{-6} \, \mathrm{F}\) and V = 12.0 V, Plug in the values to get: \(E_C = \frac{1}{2} (2.00 \times 10^{-6}\ \mathrm{F})(12.0\ \mathrm{V})^2 \) \(E_C = 1.44 \times 10^{-4} \ \mathrm{J}\)

(b) Maximum energy stored in the inductor

Now this energy will be transferred to the inductor in the form of magnetic energy. At the point when the maximum current flows through the inductor, the energy stored in the capacitor becomes zero and the inductor stores an equal amount of energy. We'll use the following formula for the energy stored in the inductor: \(E_L = \frac{1}{2}LI^2\) Given, L = 0.250 H and knowing that \(E_L = E_C\) Solve for the maximum current (I) flowing through the inductor: \(I = \sqrt{\frac{2E_L}{L}}\) \(I = \sqrt{\frac{2(1.44 \times 10^{-4} \ \mathrm{J})}{0.250 \ \mathrm{H}}}\) \(I = 3.26 \times 10^{-2} \ \mathrm{A}\) Answer (a): The maximum current in the inductor is \(3.26 \times 10^{-2} \ \mathrm{A}\).

(c) Frequency of oscillation

Now we need to calculate the frequency of oscillation of the LC circuit. We'll use the formula: \(f = \frac{1}{2\pi\sqrt{LC}}\) Plug in the values, L = 0.250 H and C = \(2.00 \times 10^{-6}\ \mathrm{F}\): \(f = \frac{1}{2\pi\sqrt{(0.250 \ \mathrm{H})(2.00 \times 10^{-6}\ \mathrm{F})}}\) \(f = 159.2 \ \mathrm{Hz}\) Answer (b): The frequency of oscillation of the LC circuit is \(159.2 \ \mathrm{Hz}\).

Key concepts

Capacitor Energy Formula

The energy stored in a capacitor is a fundamental concept in understanding LC circuit oscillation. It can be calculated using the capacitor energy formula, which is expressed as:

\( E_C = \frac{1}{2}CV^2 \)

where \( E_C \) is the energy in joules (J), \( C \) is the capacitance in farads (F), and \( V \) is the voltage in volts (V). When a capacitor is fully charged, it holds a maximum amount of energy, which can then be released to perform work or transferred to another component, like an inductor, in an LC circuit.

In the context of our exercise, the given values allow us to calculate the initial energy stored in the capacitor as: \( E_C = 1.44 \times 10^{-4} \, \mathrm{J} \) after plugging in the capacitance of \( 2.00-\mu \mathrm{F} \) and a voltage of 12.0 V. This energy will become important as we examine how it transfers to the inductor in the LC circuit.

Inductor Energy Formula

Similar to a capacitor, an inductor also stores energy, but it does so in its magnetic field. The formula for calculating the energy stored in an inductor is:

\( E_L = \frac{1}{2}LI^2 \)

where \( E_L \) represents the energy in joules (J), \( L \) is the inductance in henrys (H), and \( I \) is the current in amperes (A). This relationship shows that the energy is directly proportional to the square of the current flowing through the inductor.

Charming toward the practical side, the initial energy we previously calculated in the capacitor will completely transfer into the inductor during the oscillation when the current reaches its peak. Thus, the maximum energy in the inductor equals the initial energy stored in the capacitor, leading us to the maximum current. It is essential to know this to understand how energy oscillates between the two components in an LC circuit.

LC Circuit Frequency

The frequency of oscillation in an LC circuit is a crucial aspect of its operation. It indicates how fast the current oscillates back and forth between the capacitor and inductor. We calculate it using the following formula:

\( f = \frac{1}{2\pi\sqrt{LC}} \)

where \( f \) is the frequency in hertz (Hz), \( L \) is the inductance in henrys (H), and \( C \) is the capacitance in farads (F). The frequency depends only on the size of the inductor and the capacitor and is not influenced by the amount of energy initially stored.

For the problem at hand, when we insert the given inductance and capacitance into the formula, we discover that the LC circuit will oscillate at a frequency of \( 159.2 \, \mathrm{Hz} \). This number is pivotal because it defines the natural resonance frequency of the circuit, which has important implications in various applications such as radio transmitters and receivers.

Maximum Current in Inductor

The maximum current in an inductor within an LC circuit is reached when all the energy stored in the capacitor has fully transferred to the inductor. At this moment, the inductor's energy is at its peak. As per the energy conservation principle, this energy precisely matches the initial energy stored in the capacitor. By solving the inductor energy formula for the maximum current, we find:

\( I = \sqrt{\frac{2E_L}{L}} \)

From our exercise, the calculation yields a max current \( I \) of \( 3.26 \times 10^{-2} \, \mathrm{A} \). This value is paramount to comprehend because it helps predict how the systems such as electrical oscillators and circuits designed for filter applications will perform. It is also a key parameter in safeguarding components from potential damage due to excessive current.