A concert loudspeaker suspended high above the ground emits 35 W of sound power. A small microphone with a 1.0 cm² area is 50 m from the speaker. a. What is the sound intensity at the position of the microphone?

Verified step by step guidance

1. Identify the formula for sound intensity, which is given by $I = \frac{P}{A}$, where $I$ is the intensity, $P$ is the power of the sound source, and $A$ is the area over which the power is spread.
2. Calculate the area over which the sound power spreads. Since the sound spreads spherically, use the formula for the surface area of a sphere, $A = 4\pi r^2$, where $r$ is the distance from the source to the microphone.
3. Substitute the distance from the speaker to the microphone into the area formula to find the total area at that distance. Use $r = 50$ m.
4. Substitute the values of $P = 35$ W and the area calculated in the previous step into the intensity formula to find the sound intensity at the microphone.
5. Express the final intensity in appropriate units, such as watts per square meter (W/m²).

Key Concepts

Sound Intensity

Sound intensity is defined as the power per unit area carried by a sound wave. It is measured in watts per square meter (W/m²) and represents how much sound energy passes through a given area in a specific time. The intensity of sound decreases with distance from the source due to the spreading of sound waves.

Inverse Square Law

The inverse square law states that the intensity of a physical quantity (like sound) is inversely proportional to the square of the distance from the source. This means that as you move away from the sound source, the intensity decreases rapidly, specifically by a factor of the square of the distance. This principle is crucial for calculating sound intensity at different distances.

Area of the Microphone

The area of the microphone is important for determining how much sound power it can capture. In this case, the microphone has a specified area of 1.0 cm², which can be converted to square meters for calculations. The sound intensity at the microphone's location can be used to find the total sound power received by the microphone based on its area.