Question

In an \(L\)-\(C\) circuit, \(L = 85.0\) mH and \(C = 3.20 \, \mu \mathrm{F}\). During the oscillations the maximum current in the inductor is 0.850 mA. (a) What is the maximum charge on the capacitor? (b) What is the magnitude of the charge on the capacitor at an instant when the current in the inductor has magnitude 0.500 mA?

Short answer

(a) Use the maximum current to find the maximum charge through formula. (b) Calculate specific charge using relation between charge and given inductor current.

Step-by-step solution

Identify the formula for maximum charge

The maximum charge on the capacitor, denoted as \( Q_{max} \), is related to the maximum current \( I_{max} \) using the formula \( Q_{max} = I_{max} \times \sqrt{LC} \). This formula originates from the energy conservation principle in LC oscillations.

Calculate the resonant angular frequency

The resonant angular frequency \( \omega \) of the LC circuit is given by \( \omega = \frac{1}{\sqrt{LC}} \). Given \( L = 85.0 \) mH \( = 85.0 \times 10^{-3} \) H and \( C = 3.20 \) \( \mu \)F \( = 3.20 \times 10^{-6} \) F, we calculate \( \omega \).

Substitute to find Q_max

Now, calculate \( Q_{max} \) using the formula from Step 1: \( Q_{max} = (0.850 \times 10^{-3} \text{ A}) \times \sqrt{(85.0 \times 10^{-3} \text{ H}) (3.20 \times 10^{-6} \text{ F})} \). Solve for \( Q_{max} \).

Determine charge at a specific current

To find the charge on the capacitor when the inductor current is 0.500 mA, we use the formula \( Q = Q_{max} \cdot \sqrt{1-\left(\frac{I}{I_{max}}\right)^2} \). Substitute \( I = 0.500 \) mA and previously calculated \( I_{max} = 0.850 \) mA, and solve for \( Q \).

Key concepts

Maximum Charge on Capacitor

The maximum charge that a capacitor can hold in an LC circuit is crucial because it determines the circuit's performance during oscillations. In an LC circuit, when you hear about maximum charge, it refers to the scenario where the energy stored in the capacitor is at its peak. This maximum charge, denoted as \( Q_{max} \), can be directly related to the maximum current \( I_{max} \).

To find \( Q_{max} \), we use the formula:

• \( Q_{max} = I_{max} \times \sqrt{LC} \)

This formula comes from the principle of energy conservation, where the total energy in the system remains constant.

Additionally, this equation shows the interplay between the inductance \( L \), capacitance \( C \), and the maximum current. More inductance or capacitance means more energy stored, resulting in a larger maximum charge on the capacitor.

Resonant Angular Frequency

The resonant angular frequency is a fundamental concept in the analysis of LC circuits. When an LC circuit is oscillating naturally, it sits at a specific frequency where the circuit's impedance is minimized, and maximum energy transfer happens between the inductor and capacitor. This natural frequency ensures smooth oscillations, and it is defined by the circuit's inductance \( L \) and capacitance \( C \).

The formula to find the resonant angular frequency, \( \omega \), is:

• \( \omega = \frac{1}{\sqrt{LC}} \)

This equation highlights how both inductance and capacitance impact the frequency:

• A larger inductor or capacitor results in a lower resonant frequency.

Understanding this relationship helps engineers design circuits that perform optimally at the desired frequency.

Energy Conservation in Oscillations

In an LC circuit, energy conservation plays a vital role during the oscillation process. The stored energy alternates between the inductor and the capacitor, but the total energy remains constant. This principle allows the system to sustain oscillations without any external power input, given ideal conditions.

Here's what happens step-by-step in simple terms:

• Initially, the maximum energy is stored in the capacitor as electric potential energy.
• As time progresses, this energy shifts to the inductor, converting to magnetic energy while the capacitor discharges.
• The oscillation continues as the energy moves back and forth between the capacitor and inductor.

Because of this constant energy exchange, the maximum current and maximum charge are directly linked, reflecting the conservation of total energy. The formulas used to calculate the maximum charge and current derive from this fundamental conservation principle, illustrating how intertwined current, charge, capacitance, and inductance really are.