A loud speaker emits energy in all directions at a rate of 2.75 J/s . What is the intensity level at a distance of 25 m?
Divide the total power emitted (2.75 W) by the area of the sphere at that distance, 4*pi*R^2
With R in meters, that will give you the sound intensity in W/m^2 Usually designated by I) . I get I = 3.5*10^-4 W/m^2
There is a formula for converting that to decibels. I get 85 dB
dB = 10 log(10) I/10^-12
Why sphere?
To calculate the intensity level at a given distance from the loudspeaker, we need to know the initial intensity at the source and take into account the inverse square law, which states that the intensity decreases inversely proportionally to the square of the distance.
The formula for intensity level (L) is given by:
L = 10 * log10(I / I0)
Where:
L is the intensity level in decibels (dB)
I is the intensity at a given distance
I0 is the reference intensity (threshold of hearing), which is usually set at 10^(-12) watts per square meter (W/m^2).
In this case, we are given the energy emission rate of the speaker, which is 2.75 J/s. To convert this to intensity, we need to divide it by the area over which the energy is distributed.
Assuming that the loudspeaker emits energy uniformly in all directions, the energy is spread over the surface of a sphere. The surface area of a sphere is given by the formula:
A = 4πr^2
Where:
A is the surface area of the sphere
r is the distance from the loudspeaker
In this case, the distance (r) is given as 25 m. So, we can calculate the surface area as:
A = 4π * (25)^2 = 4 * π * 625 = 2500π m^2
Now, we can calculate the intensity (I) by dividing the energy emission rate by the surface area:
I = 2.75 J/s / 2500π m^2
Finally, we can calculate the intensity level (L) using the formula mentioned earlier:
L = 10 * log10(I / I0)
Substituting the values, we can calculate the intensity level at a distance of 25 m.