What sound level in dB is produced by earphones that create an intensity of 4.50 ✕ 10−2 W/m2?

To determine the sound level in decibels (dB) produced by earphones with a given intensity, we can use the formula:

\[
L = 10 \, \log \left(\frac{I}{I_0}\right)
\]

where:
- \( L \) is the sound level in dB,
- \( I \) is the intensity of the sound in watts per square meter (W/m²),
- \( I_0 \) is the reference intensity, which is the threshold of hearing, typically \( 1 \times 10^{-12} \, \text{W/m}^2 \).

Given:
- \( I = 4.50 \times 10^{-2} \, \text{W/m}^2 \)
- \( I_0 = 1 \times 10^{-12} \, \text{W/m}^2 \)

Let's substitute these values into the formula:

\[
L = 10 \, \log \left(\frac{4.50 \times 10^{-2}}{1 \times 10^{-12}}\right)
\]

Calculating inside the logarithm:

\[
L = 10 \, \log \left(4.50 \times 10^{10}\right)
\]

Breaking it down further:

\[
L = 10 \times \left(\log(4.50) + \log(10^{10})\right)
\]
\[
L = 10 \times \left(0.6532 + 10\right)
\]
\[
L = 10 \times 10.6532
\]
\[
L = 106.532 \, \text{dB}
\]

**Answer: The sound level produced by the earphones is approximately \( 106.5 \, \text{dB} \).**

Would you like a further breakdown of these calculations, or do you have any questions?

---

1. How is the reference intensity \( I_0 \) of \( 1 \times 10^{-12} \, \text{W/m}^2 \) determined?
2. What are common dB levels for different everyday sounds?
3. How does distance from the source affect perceived sound intensity?
4. Why is a logarithmic scale used for measuring sound levels?
5. How can prolonged exposure to high dB levels affect hearing?

**Tip:** When dealing with logarithmic calculations, remember that each increase of 10 dB represents a tenfold increase in intensity.

What sound level in dB is produced by earphones that create an intensity of 4.50 ✕ 10−2 W/m²?

Discover how to calculate the sound level in decibels (dB) produced by earphones with an intensity of 4.50 ✕ 10⁻² W/m². This problem covers logarithmic functions and the decibel scale, providing a detailed step-by-step solution suitable for students in Grades 9-12.

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