A loudspeaker emits energy equally in all directions at the rate of 1.75J/s. What is the intensity level at a distance of 15 m?
surface area of sphere = 4 pi r^2
Watts/unit area at 15 m = 1.75/(4*pi*225)
To calculate the intensity level at a distance of 15 meters, we can use the formula:
\( IL = 10 \cdot \log_{10}(I) \),
where \( IL \) is the intensity level and \( I \) is the intensity of the sound wave.
Given that the loudspeaker emits energy at a rate of 1.75 J/s, we need to calculate the intensity at a distance of 15 meters.
The equation for sound intensity at a distance from a point source is:
\( I = \frac{P}{4 \pi r^{2}} \),
where \( P \) is the power emitted by the loudspeaker, and \( r \) is the distance from the loudspeaker.
First, let's calculate the intensity \( I \):
\( I = \frac{1.75 J/s}{4 \pi (15 m)^{2}} \).
\( I = \frac{1.75}{4 \pi (225)} \).
\( I = \frac{1.75}{900 \pi} \) J/m^2s.
Next, we can calculate the intensity level \( IL \):
\( IL = 10 \cdot \log_{10}\left(\frac{1.75}{900 \pi}\right) \).
\( IL = 10 \cdot \log_{10}\left(\frac{1.75}{900 \pi}\right) \).
Using a calculator, we find:
\( IL \approx 55.17 \) dB.
Therefore, the intensity level at a distance of 15 meters is approximately 55.17 dB.
To find the intensity level at a distance of 15m, we can use the formula:
I = P / (4πr^2)
Where:
- I is the intensity in watts per square meter (W/m^2)
- P is the power emitted by the loudspeaker, which is 1.75 J/s
- r is the distance from the source, which is 15m
- π is a constant approximately equal to 3.14
First, let's calculate the intensity:
I = 1.75 J/s / (4π * 15m^2)
To simplify the calculation, let's solve it step by step:
1. Calculate the area: 4π * 15m^2 = 4π * 225m^2 = 900π m^2
2. Divide the power (1.75 J/s) by the area we calculated: 1.75 J/s / 900π m^2 ≈ 0.00194 J/s/m^2
Now we have the Intensity (I) at a distance of 15m, which is approximately 0.00194 J/s/m^2.
Next, let's convert the intensity to the intensity level (β) using the formula:
β = 10 * log10(I / I0)
Where:
- β is the intensity level in decibels (dB)
- I is the calculated intensity (0.00194 J/s/m^2)
- I0 is the reference intensity, which is 10^-12 W/m^2 (a standard reference intensity for hearing threshold)
Plug in the values:
β = 10 * log10(0.00194 J/s/m^2 / 10^-12 W/m^2)
Simplifying the value:
β = 10 * log10(1940000) ≈ 78.4 dB
So, the intensity level at a distance of 15m from the loudspeaker is approximately 78.4 decibels.