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Chapter 3: Problem 88

Calculate the decibels associated with a doorbell if the intensity is \(I=1 \times 10^{-45} \mathrm{W} / \mathrm{m}^{2}\).

Short Answer

Expert verified
The decibels associated with the given intensity are -330 dB.

Step by step solution

01

Understanding the Decibel Formula

The decibel level, denoted as \( L \), is calculated using the formula: \[ L = 10 imes ext{log}_{10}\left(\frac{I}{I_0}\right) \] where \( I \) is the intensity we are given, and \( I_0 \) is the reference intensity, typically \( I_0 = 1 imes 10^{-12} \, \text{W/m}^2 \).
02

Substitute the Given Intensity

Plug the given intensity \( I = 1 imes 10^{-45} \,\text{W/m}^2 \) and the reference intensity \( I_0 = 1 imes 10^{-12} \, \text{W/m}^2 \) into the decibel formula: \[ L = 10 imes ext{log}_{10}\left(\frac{1 imes 10^{-45}}{1 imes 10^{-12}}\right) \].
03

Calculate the Ratio Inside the Logarithm

Calculate the fraction inside the logarithm: \[ \frac{1 \times 10^{-45}}{1 \times 10^{-12}} = 1 \times 10^{-33} \].
04

Compute the Logarithm

Find the logarithm of the ratio: \[ \text{log}_{10} (1 \times 10^{-33}) = -33 \] because \( \text{log}_{10}(10^{-33}) = -33 \).
05

Calculate the Decibel Level

Finally, substitute the logarithm result into the decibel formula: \[ L = 10 imes (-33) = -330 \]. So, the decibel level associated with this intensity is -330 dB.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Intensity
Intensity in sound refers to the amount of energy that a sound wave carries per unit area. It is expressed in watts per square meter (W/m虏).
Think of intensity as how much power the sound is delivering to a surface. A higher intensity means more energy and a louder sound.
Some important points about intensity:
  • It always decreases with distance from the sound source.
  • Measured in a specific direction.
  • High intensity can result in discomfort or even pain.
In the exercise, we were given an intensity value of \(1 \times 10^{-45} \, \text{W/m}^2\). This is an extremely low intensity, even lower than typical sound levels found naturally on Earth, indicating an almost inaudible sound.
Logarithmic Scale for Sound
A logarithmic scale is used for measuring sound because of the vast range of sound intensities that can be heard by the human ear.
A logarithmic scale, as opposed to a linear one, allows us to compress these large differences into simpler numbers.
Key points about logarithmic scales:
  • They simplify calculations and comparisons.
  • The base 10 logarithm is commonly used in sound measurement.
  • Large changes in sound intensity correspond to smaller changes on a logarithmic scale.
This is why, in our exercise, we used a logarithmic scale to calculate the decibels from the given intensity. It provides a more practical way to compare sound levels.
Measuring Sound with Decibels
Sound is measured in decibels (dB), a unit that expresses the ratio of a given intensity to a reference intensity on a logarithmic scale.
The decibel formula is \( L = 10 \times \text{log}_{10}\left(\frac{I}{I_0}\right) \), where \( I \) is the intensity of the sound, and \( I_0 \) is the reference intensity.
Here's why decibels are handy:
  • They make it easier to handle very large or very small numbers.
  • No negative sections exist on a logarithmic scale for human audibility.
  • Provide a clearer understanding of the perceived loudness of sounds.
In our exercise, we calculated a decibel level of -330 dB, which indicates an extremely low intensity far below what humans can typically perceive.
Role of Reference Intensity in Sound Measurement
Reference intensity \( I_0 \) is crucial when converting a sound intensity to decibels.
It's the baseline or starting point for measuring sound intensity, usually set at \( 1 \times 10^{-12} \) W/m虏.
Aspects of reference intensity include:
  • It represents the threshold of hearing for the average human ear.
  • Allows comparison of different sound intensities on a consistent scale.
  • Serves as a fixed point to relate other sound measurements.
In our exercise, we used this standard reference intensity to find the decibel level of our given intensity, ensuring that the decibel value is meaningful in the context of human hearing.

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