Problem 61 - College Physics

Question

An electromagnetic car alarm. Your latest invention is a car alarm that produces sound at a particularly annoying frequency of 3500 \(\mathrm{Hz}\) . 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 \(\mathrm{V}\) (the same voltage as the car battery). To produce a sufficiently loud sound, the capacitor must store 0.0160 \(\mathrm{J}\) of energy. What values of capacitance and inductance should you choose for your car-alarm circuit?

Step-by-Step Solution

Answer

Capacitance \( C \approx 222 \, \mu\text{F} \), Inductance \( L \approx 5.79 \, \text{mH} \).

Step 1: Understanding Capacitor Energy Formula

The energy stored in a capacitor is given by the formula \( E = \frac{1}{2} C V^2 \), where \( E \) is the energy, \( C \) is the capacitance, and \( V \) is the voltage across the capacitor. We need to solve for \( C \) using \( E = 0.0160 \, \text{J} \) and \( V = 12.0 \, \text{V} \).

Step 2: Solving for Capacitance

Rearrange the formula to solve for \( C \):\[ C = \frac{2E}{V^2} \]Substituting the given values, we have:\[ C = \frac{2 imes 0.0160}{12.0^2} \approx 2.22 \times 10^{-4} \, \text{F} \] (or 222 \( \mu \text{F} \)).

Step 3: Understanding Resonant Frequency Formula

The resonant frequency \( f \) for an LC circuit is given by \( f = \frac{1}{2\pi\sqrt{LC}} \), where \( L \) is the inductance. We need to use this formula to find the inductance \( L \) given \( f = 3500 \, \text{Hz} \) and the \( C \) we calculated.

Step 4: Solving for Inductance

Rearrange the formula to solve for \( L \):\[ L = \frac{1}{(2\pi f)^2 C} \]Substitute \( f = 3500 \, \text{Hz} \) and \( C \approx 2.22 \times 10^{-4} \, \text{F} \):\[ L = \frac{1}{(2\pi \times 3500)^2 \times 2.22 \times 10^{-4}} \approx 5.79 \times 10^{-3} \, \text{H} \] (or 5.79 \( \text{mH} \)).

Key Concepts

Capacitance

Capacitance is a fundamental concept in electric circuits. It refers to the ability of a capacitor to store an electric charge. The measure of capacitance is given in farads (F). In our car alarm circuit, capacitance determines how much electric charge can be stored for a given voltage. This is crucial because the energy stored in the capacitor later helps in producing the sound.Let's break it down:

• Capacitance is determined by the size and properties of the capacitor plates and the dielectric material between them.
• The higher the capacitance, the more charge stored at a given voltage.The formula for energy stored is: \[ E = \frac{1}{2}CV^2 \] where \(C\) is the capacitance, \(V\) the voltage, and \(E\) the energy.
• In this exercise, rearranging to find \(C\) gives: \[ C = \frac{2E}{V^2} \]

This allows us to calculate the required capacitance based on the desired energy storage and voltage limits.

Inductance

Inductance is a key property of inductors in a circuit. It reflects the inductor's ability to resist changes in current and is measured in henrys (H). Inductance plays a vital role in determining how the LC circuit resonates at a specific frequency.Here's how it works:

• Inductance is affected by factors such as the number of turns in the coil and the core material used.
• The primary role of inductance in an LC circuit is to join with capacitance, enabling resonance at a specific frequency.
• To find the inductance needed, we use the formula for resonant frequency:\[ f = \frac{1}{2\pi\sqrt{LC}} \]

By rearranging this formula, we determine \(L\):\[ L = \frac{1}{(2\pi f)^2 C} \] helps us to compute the right inductance to safely and effectively create the annoying sound in the car alarm.

Resonant Frequency

The resonant frequency is an important characteristic of LC circuits. It's the frequency at which the circuit naturally oscillates. In this car alarm, resonant frequency allows the specific sound frequency of 3500 Hz to be generated.Some essential points:

• Resonant frequency is determined by both the capacitance \(C\) and inductance \(L\).
• The formula that relates them is:\[ f = \frac{1}{2\pi\sqrt{LC}} \]
• Achieving the desired resonant frequency involves precise calculations of \(C\) and \(L\).

By ensuring both components are correctly calculated, resonance allows the circuit to produce the exact sound frequency required for the car alarm.

Capacitor Energy

Capacitor energy is stored energy due to an electric field between its plates. This energy can be used to power circuits, like in our car alarm. Understanding how much energy a capacitor can store influences the loudness of the sound produced.Let's break it down further:

• Energy storage in a capacitor is defined by the formula:\[ E = \frac{1}{2}CV^2 \]
• This means the energy stored depends on both the capacitance and the square of the voltage.
• The stored energy in the car alarm's capacitor translates into the power emitted by the device, impacting the output sound volume.

This concept ensures that the capacitor in the circuit not only meets the voltage demands but also provides the necessary energy to achieve the desired sound level.