Problem 56 - College Physics

Question

A 15.0\(\mu \mathrm{F}\) capacitor is charged to 175\(\mu \mathrm{C}\) and then connected across the ends of a 5.00 \(\mathrm{mH}\) inductor. (a) Find the maximum current in the inductor. At the instant the current in the inductor is maximal, how much charge is on the capacitor At this instant, what is the current in the inductor? (c) Find the maximum energy stored in the inductor. At this instant, what is the current in the circuit?

Step-by-Step Solution

Answer

(a) 2.41 A; Charge is 0 C when current is maximal. (c) Maximum energy is 1.02 mJ; Current at this energy is 2.41 A.

Step 1: Find Maximum Current in Inductor

The maximum current in the inductor can be found using the energy conservation principle for oscillating LC circuits. The initial energy in the capacitor is given by the formula \( U_C = \frac{1}{2} \frac{Q^2}{C} \). The maximum energy in the inductor when the current is at its peak is \( U_L = \frac{1}{2} L I^2 \). Setting these equal, i.e., \( U_C = U_L \), gives:\[ \frac{Q^2}{2C} = \frac{LI^2}{2} \]Solving for \( I \), we have:\[ I = \frac{Q}{\sqrt{LC}} \]Substitute given values \( Q = 175 \times 10^{-6} \, \mathrm{C} \), \( C = 15.0 \times 10^{-6} \, \mathrm{F} \), \( L = 5.00 \times 10^{-3} \, \mathrm{H} \):\[ I = \frac{175 \times 10^{-6}}{\sqrt{5.00 \times 10^{-3} \cdot 15.0 \times 10^{-6}}} = 2.41 \, \mathrm{A} \]Thus, the maximum current in the inductor is 2.41 A.

Step 2: Charge on Capacitor at Maximum Current

At the moment when the current is maximal, all energy is stored in the inductor and the charge on the capacitor is zero because this is an LC circuit at its resonant frequency. Therefore, the charge on the capacitor at this instant is 0 C.

Step 3: Maximum Energy Stored in Inductor

The maximum energy stored in the inductor is equal to the initial energy stored in the capacitor, which can be calculated using the formula: \[ U = \frac{1}{2} C V^2 \]The initial voltage across the capacitor can be found using \( V = \frac{Q}{C} \):\[ V = \frac{175 \times 10^{-6}}{15.0 \times 10^{-6}} = 11.67 \, \mathrm{V} \]Then the maximum energy is:\[ U = \frac{1}{2} \times 15.0 \times 10^{-6} \times (11.67)^2 = 1.02 \times 10^{-3} \, \mathrm{J} \]Thus, the maximum energy stored in the inductor is 1.02 mJ.

Key Concepts

Conservation of Energy

In an oscillating LC circuit, energy plays tag between the capacitor and the inductor.
The principle at work here is the Conservation of Energy, which states that energy cannot be created or destroyed but only transformed from one type to another. In this system, initially, the energy is fully stored in the charged capacitor as electrical potential energy. As the circuit begins to oscillate, this energy is transferred to the inductor, turning into magnetic energy as the current reaches its maximum.
The formula we use to express the conservation of energy in LC circuits is \[ \frac{1}{2} \frac{Q^2}{C} = \frac{1}{2} L I^2 \] where:

• \( Q \) is the initial charge on the capacitor
• \( C \) is the capacitance of the capacitor
• \( L \) is the inductance of the inductor
• \( I \) is the current at any given instant

This equation states that the electrical potential energy stored in a charged capacitor is equal to the magnetic energy stored in the inductor when current flows through it.

Capacitance and Inductance

Capacitance and inductance are fundamental properties within an LC circuit that affect how energy is transferred between the components.
Capacitance, denoted as \( C \), is a measure of how much charge a capacitor can store at a given voltage. The unit of capacitance is farads (\( \text{F} \)).
Capacitors with higher capacitance can store more electrical energy at a lower voltage. In our example, a 15.0 \( \mu\text{F} \) capacitor was used, storing the initial charge of 175 \( \mu\text{C} \). On the other hand, inductance, denoted by \( L \), measures the ability of an inductor to store energy in a magnetic field created by the flow of current. It's typically measured in henrys (\( \text{H} \)). Inductors with higher inductance slow down the rate at which the current changes, thus having a smoothing effect on the flow of current in the circuit. The example problem utilized a 5.00 \( \text{mH} \) inductor.
Both these properties decide how quickly the circuit oscillates and the energy transformation rate.

Maximum Current in Inductor

The maximum current in the inductor is a critical point of oscillation in an LC circuit. At this moment, all the energy initially stored in the capacitor has converted to magnetic energy in the inductor.
To find this maximum current \( I \), we use the derived energy equation \[ I = \frac{Q}{\sqrt{LC}} \] where \( Q \) is initial charge, \( L \) is inductance, and \( C \) is capacitance.
Insert the given values from the problem:

• \( Q = 175 \times 10^{-6} \text{ C} \)
• \( L = 5.00 \times 10^{-3} \text{ H} \)
• \( C = 15.0 \times 10^{-6} \text{ F} \)

After calculation, you get \( I = 2.41 \text{ A} \). This current is the peak that the inductor can reach under these conditions, showcasing the intricate balance of energy transformation within the circuit.