Sound Intensity — Home Speaker (Worked Example)

Given

• P = 2.0 W

• r = 3.0 m

Find

The intensity I.

Principle

I = P/(4πr²): the same power spread over a bigger sphere means weaker sound.

Formula

Solution

1. Write the formula: $I = P/(4\pi r^2)$

2. Substitute: $I = 2.0/(4\pi \times 3.0^2)$

3. Denominator: $4\pi \times 9 = 113.1$

4. Divide: $I = 2.0/113.1 = 0.0177\ \mathrm{W/m^2}$

Unit check

W ÷ m² = W/m². Power spread over the spherical area gives intensity — dimensionally consistent.

What this means physically

0.0177 W/m² ≈ 102 dB — loud for a living room. Moving to 6 m would drop it to a quarter, about 96 dB.