Question

When do we use I=P/a or I=P/4pi(r)^2?

for example question 1:

30 seconds of exposure to 115 dB sound can damage your hearing, but a much quieter 94 dB may begin to cause damage after 1 hour of continuous exposure.You are going to an outdoor concert, and you'll be standing near a speaker that emits 50W of acoustic power as a spherical wave. What minimum distance should you be from the speaker to keep the sound intensity level below 94 dB?

- we used I=P/4pir^2

Question 2:

A woman wearing an in-ear hearing aid listens to a television set at a normal volume of approximately 60 dB. To hear it, she requires an amplification of 30 dB, so the hearing aid supplies sound at 90 dB to the ear canal, which we assume to be circular with a diameter of 7 mm. What is the output power of the hearing aid?

why did we use I=P/a not I=P/4pir^2?

Answer 1

In the first question we wanted that the intensity falling on the year to be less than 94 dB. The given power of the speakers is 50 W. Considering the speakers to be a point isotropic source of a sound wave. We say that the intensity at a distance r away from the speakers will be given by this formula

I = P/4?r²

This formula will give us the intensity of the sound at a distance r.

Whereas in the second question they asked about the output power of the hearing aid. In this case the intensity was known to us i.e. 90 dB. We wanted to calulate the output power. In this case we don't want to be concerned about the source distance from the hearing aid. We just want the cross sectional area of the ear canal which can be given easily since the radius is given 7mm. So the output power will be just given by the product of intensity and the area of cross section.

Hence here I = P/A or P(output) = I x A

Hope it helps.