Problem 56 - University Physics with Modern Physics

Question

A \(5.00-\mu F\) capacitor is initially charged to a potential of 16.0 \(\mathrm{V}\) . It is then connected in series with a \(3.75-\mathrm{mH}\) inductor. (a) What is the total energy stored in this circuit? (b) What is the maximum current in the inductor? What is the charge on the capacitor plates at the instant the current in the inductor is maximal?

Step-by-Step Solution

Answer

(a) 6.40 mJ, (b) 1.85 A maximum current, charge is 0.

Step 1: Understanding the Energy Stored in the Capacitor

The initial energy stored in a charged capacitor is given by the formula \( U = \frac{1}{2} C V^2 \). Here, \( C = 5.00 \times 10^{-6} \mu F \) and \( V = 16.0 \mathrm{V} \). Substitute these values into the formula to find the energy.

Step 2: Calculating the Total Initial Energy

Substitute \( C = 5.00 \times 10^{-6} \mathrm{F} \) and \( V = 16.0 \mathrm{V} \) into the energy formula: \[ U = \frac{1}{2} \, (5.00 \times 10^{-6}) \, (16.0)^2 \]Complete the calculation to find:\[ U = \frac{1}{2} \, (5.00 \times 10^{-6}) \, (256) \]\[ U = 6.40 \times 10^{-3} \mathrm{J} \]

Step 3: Understanding Maximum Current in the Inductor

In an LC circuit, the total initial energy stored in the capacitor is converted into the maximum kinetic energy of the oscillating charge when the current is at its maximum. The energy is given by \( U = \frac{1}{2} L I^2 \). Set this equal to the initial energy to find the maximum current, \( I_{max} \).

Step 4: Calculating Maximum Current

Using the equation \( U = \frac{1}{2} L I^2 \) and setting \( U = 6.40 \times 10^{-3} \mathrm{J} \), solve for \( I \):Given \( L = 3.75 \times 10^{-3} \mathrm{H} \),\[ 6.40 \times 10^{-3} = \frac{1}{2} (3.75 \times 10^{-3}) I^2 \]\[ I^2 = \frac{6.40 \times 10^{-3}}{1.875 \times 10^{-3}} \]\[ I = \sqrt{3.4133} \approx 1.85 \mathrm{A} \]

Step 5: Calculating Charge at Maximum Current

When the current is at its maximum, the charge on the capacitor is zero since all energy has been transferred to the inductor. Therefore, the charge \( q \) is 0 at maximum current.

Key Concepts

Energy Stored in Capacitors

In LC circuits, capacitors play a crucial role in storing electrical energy. When a capacitor is charged, it accumulates electrical energy due to the potential difference across its plates. The energy stored in a capacitor can be calculated using the formula:\[ U = \frac{1}{2} C V^2 \]where- \( U \) is the energy in joules,- \( C \) is the capacitance in farads,- \( V \) is the potential difference in volts.
This formula signifies that the stored energy depends on both the capacitance and the square of the voltage. In our example, a \(5.00-\mu F\) capacitor charged to \(16.0 \mathrm{V}\) results in an energy storage of \(6.40 \times 10^{-3} \mathrm{J}\).
The energy stored is crucial as the circuit utilizes this energy for oscillations when connected with an inductor.

Inductor Currents

Inductors in LC circuits convert energy between magnetic and electric forms. When the oscillation begins, the energy initially stored in the capacitor is transferred to the inductor, creating a current. An inductor opposes changes in the current flowing through it, which allows for the energy storage in the form of magnetic fields.
The maximum current, known as the peak current, occurs when all the energy from the capacitor is transferred. We use the formula:\[ U = \frac{1}{2} L I^2 \]where- \(I\) is the current in amperes,- \(L\) is the inductance in henrys.
For our example, with \( L = 3.75 \times 10^{-3} \mathrm{H} \), the peak current calculated is approximately \(1.85 \mathrm{A}\). This conversion highlights the profound relationship between capacitors' stored energy and inductors' current.

Oscillating Charge Energy

In an LC circuit, the charge oscillates between the capacitor and the inductor, converting between potential and kinetic forms of energy. Initially, the charge is fully stored as electrical potential within the capacitor. When released, it flows through the inductor, transforming into magnetic energy as current peaks.
As the process evolves, the energy continuously oscillates. At maximum current, all energy converts into the magnetic form, making the charge on the capacitor zero. This dynamic process manifests as electric oscillations, similar to a swinging pendulum, where energy shifts from potential to kinetic forms constantly.
Understanding this oscillation helps us grasp how energy transformations work in LC circuits, emphasizing the collaboration between capacitors and inductors in conserving total energy. This is essential for many electronics applications like radio frequency transmission.