Question

Your latest invention is a car alarm that produces sound at a particularly annoying frequency of 3500 Hz. To do this, the car-alarm circuitry must produce an alternating electric current of the same frequency. That's why your design includes an inductor and a capacitor in series. The maximum voltage across the capacitor is to be 12.0 V. To produce a sufficiently loud sound, the capacitor must store 0.0160 J of energy. What values of capacitance and inductance should you choose for your car-alarm circuit?

Short answer

Capacitance \( \approx 2.22 \times 10^{-4} \, \text{F} \) and Inductance \( \approx 2.08 \times 10^{-3} \, \text{H} \).

Step-by-step solution

Find Capacitance from Energy Stored in Capacitor

We know the formula for the energy stored in a capacitor is \( U = \frac{1}{2} C V^2 \), where \( U \) is the energy, \( C \) is the capacitance, and \( V \) is the voltage. Here, \( U = 0.0160 \, \text{J} \) and \( V = 12.0 \, \text{V} \). Rearrange the formula to solve for \( C \):\[ C = \frac{2U}{V^2} = \frac{2 \times 0.0160}{12.0^2} \]Calculate \( C \):\[ C = \frac{0.0320}{144.0} \approx 2.22 \times 10^{-4} \, \text{F} \]

Find Inductance Using Resonant Frequency Formula

The resonant frequency formula for a series LC circuit is given by \( f = \frac{1}{2\pi\sqrt{LC}} \), where \( f \) is the frequency, \( L \) is the inductance, and \( C \) is the capacitance. We know \( f = 3500 \, \text{Hz} \) and \( C \approx 2.22 \times 10^{-4} \, \text{F} \). Rearrange the formula to solve for \( L \):\[ L = \frac{1}{(2\pi f)^2 C} \]Substitute the known values:\[ L = \frac{1}{(2\pi \times 3500)^2 \times 2.22 \times 10^{-4}} \]Calculate \( L \):\[ L \approx 2.08 \times 10^{-3} \, \text{H} \]

Final Step: Required Values for Circuit

Based on the calculations, the necessary values for the car-alarm circuit are a capacitance of approximately \( 2.22 \times 10^{-4} \, \text{F} \) and an inductance of approximately \( 2.08 \times 10^{-3} \, \text{H} \).

Key concepts

Capacitance

Capacitance is a fundamental property of capacitors, which are components in electrical circuits that store electrical energy. Imagine a capacitor as a tiny container that holds electric charge. The ability of a capacitor to store charge is called its capacitance, and it is measured in farads (F).

When a voltage is applied across a capacitor, it accumulates electrical energy. The energy stored is given by the formula \( U = \frac{1}{2} C V^2 \), where \( U \) is energy in joules, \( C \) is capacitance in farads, and \( V \) is the voltage in volts. In the case of our car alarm, the capacitor needs to store 0.0160 J of energy at 12.0 V.

By rearranging the formula to \( C = \frac{2U}{V^2} \), you can calculate the necessary capacitance. For the car alarm, this results in a capacitance of approximately \( 2.22 \times 10^{-4} \text{ F} \). This value ensures that the capacitor holds enough energy to produce the loud sound needed for the alarm.

Inductance

Inductance is a property of inductors in a circuit. It describes the inductor's ability to oppose changes in electric current passing through it.

Think of inductance as the momentum of an electric circuit; it resists changes just like a car resists changes in speed due to its mass. Inductors are typically coil-shaped and their inductance is measured in henries (H).

In an LC circuit, the inductor works together with the capacitor to determine the resonant frequency of the circuit. When you're designing an LC circuit, you choose the inductance so the circuit resonates at your desired frequency. In the context of the car alarm, the inductance was chosen so that the circuit produces a tone at 3500 Hz, which is known as the resonant frequency.

Using the formula \( f = \frac{1}{2\pi\sqrt{LC}} \), you can rearrange to solve for the inductance, giving \( L = \frac{1}{(2\pi f)^2 C} \). This calculation ensures the circuit has the desired frequency, resulting in an inductance of approximately \( 2.08 \times 10^{-3} \text{ H} \).

Resonant Frequency

The resonant frequency is a crucial concept for understanding how LC circuits work. It is the frequency at which the circuit naturally oscillates when not driven by an external source.

In a series LC circuit, the inductor and capacitor interact in such a way that at resonance, they store and release energy back and forth. This creates a continuous oscillation of electric current, and the circuit "resonates" at a particular frequency.

The resonant frequency \( f \) in a series LC circuit is given by the formula \( f = \frac{1}{2\pi\sqrt{LC}} \). This relationship shows that the frequency depends on both the capacitance \( C \) and the inductance \( L \).

For a car alarm, tuning the LC circuit to 3500 Hz ensures it produces the desired sound frequency. This results from carefully choosing the correct values for the capacitor and inductor, identified earlier, ensuring the alarm is both effective and loud enough to capture attention.