Question

30.24 A 2.00 - \(\mu\) F capacitor is fully charged by being connected to a 12.0 - \(\mathrm{V}\) 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: Determine the maximum current in the inductor and the frequency of oscillation of the given LC circuit with a capacitance of 2.00 μF, voltage of 12.0 V, and inductance of 0.250 H. Answer: The maximum current in the inductor is 1.20 A, and the frequency of oscillation of the LC circuit is 50.3 Hz.

Step-by-step solution

Identify the given values

We are given the following values: - Capacitance, C = 2.00 μF - Voltage of the fully charged capacitor, V = 12.0 V - Inductance, L = 0.250 H

Find the initial energy stored in the capacitor

We can calculate the initial energy stored in the capacitor when it is fully charged using the formula: \(E_C = \dfrac{1}{2} CV^2\) Where, \(E_C\) is the initial energy stored in the capacitor, C is the capacitance and V is the voltage across the capacitor. Plugging the given values, we get: \(E_C = \dfrac{1}{2}(2.00 \times 10^{-6}\,\mathrm{F})(12.0\,\mathrm{V})^2\) \(E_C = 0.144\,\mathrm{J}\)

Calculate the maximum energy stored in the inductor

Since the energy in an LC circuit oscillates between the capacitor and the inductor, the maximum energy stored in the inductor will be the initial energy stored in the capacitor. Therefore, the maximum energy stored in the inductor is: \(E_L = E_C = 0.144\,\mathrm{J}\)

Calculate the maximum current in the inductor

The energy stored in the inductor can be represented as: \(E_L = \dfrac{1}{2}LI^2\) Where \(I\) is the maximum current in the inductor and L is the inductance. Now, rearrange the formula to find the maximum current: \(I = \sqrt{\dfrac{2E_L}{L}}\) Substitute the given values, \(I = \sqrt{\dfrac{2(0.144\,\mathrm{J})}{0.250\,\mathrm{H}}}\) \(I = 1.20\,\mathrm{A}\) So, the maximum current in the inductor is 1.20 A.

Calculate the frequency of oscillation of the LC circuit

The frequency of oscillation of an LC circuit can be determined using the formula: \(f = \dfrac{1}{2\pi\sqrt{LC}}\) Substitute the given values, \(f = \dfrac{1}{2\pi\sqrt{(0.250\,\mathrm{H})(2.00 \times 10^{-6}\,\mathrm{F})}}\) \(f = 50.3\,\mathrm{Hz}\) Hence, the frequency of oscillation of the LC circuit is 50.3 Hz. In summary, the maximum current in the inductor is 1.20 A and the frequency of oscillation of the LC circuit is 50.3 Hz.

Key concepts

Capacitor Charge and Energy

When dealing with electrical circuits, capacitors play a vital role in storing and releasing energy. In an LC circuit, a capacitor's ability to hold a charge is central to the energy transformations that take place. The charge on a capacitor, denoted by the symbol 'Q', is a product of its capacitance, 'C', and the voltage, 'V', across it (\(Q = CV\)). Capacitance is a measure of how much charge the capacitor can store per unit of voltage.

Once the capacitor is fully charged by a power source, like a battery, it holds a certain amount of energy that can be described by the formula \(E_C = \frac{1}{2}CV^2\), where \(E_C\) is the energy in joules. In our case, the 2.00-\(\mu\)F capacitor charged to 12.0 V holds an initial energy of 0.144 joules. Understanding this energy storage is crucial as it sets the stage for the energy oscillations that occur once the capacitor is connected to the inductor in an LC circuit.

Inductor Maximum Current

After delving into capacitor charge and energy, it's equally important to understand the role of inductors in an LC circuit. An inductor typically opposes changes in current flow through it, leading to energy being stored in its magnetic field when current flows.

The maximum current in the inductor is a critical point in the oscillation of the LC circuit because it represents the moment when all the energy stored in the capacitor's electric field has been transferred to the inductor's magnetic field. The energy in the inductor at maximum current is given by \(E_L = \frac{1}{2}LI^2\), where 'L' is the inductance and 'I' is the current. Equal to the initial energy stored in the capacitor, this energy can be used to solve for the maximum current, \(I = \sqrt{\frac{2E_L}{L}}\). In the exercise, we found this current to be 1.20 A. This concept is pivotal because it illustrates energy conservation within the circuit, as well as the dynamic relationship between electric and magnetic fields in oscillatory systems.

Oscillation Frequency Calculation

The natural rhythm of an LC circuit comes to life in the oscillation frequency. Oscillation occurs because the energy in the LC circuit alternates between the electric field of the capacitor and the magnetic field of the inductor. The frequency of these oscillations can be found using the formula \(f = \frac{1}{2\pi\sqrt{LC}}\), where 'f' represents the frequency in hertz (Hz), 'L' is the inductance in henries (H), and 'C' is the capacitance in farads (F).

For students who may struggle with this concept, visualizing the oscillation can be helpful. Imagine the flow of energy as a pendulum swinging back and forth without friction. The frequency calculation determines how fast this energy 'pendulum' swings, or in other words, how quickly the energy moves between the capacitor and inductor. In our example, we calculated the frequency to be 50.3 Hz, indicating that the energy oscillates between the capacitor and inductor over 50 times per second.