Sound Intensity & DecibelsJEE Main

Inverse-square spreading and the dB scale — interactive Physics simulation for IIT-JEE.

Concept

A point source spreads its power over ever-larger spheres, so intensity falls as $1/r^2$ . Ears sense ratios, not differences — hence the logarithmic decibel scale: every ×10 in intensity is +10 dB, every doubling of distance is −6 dB.

Key formula

I = \frac{P}{4\pi r^2}, \qquad \beta = 10\log_{10}\frac{I}{I_0}, \qquad I_0 = 10^{-12}\,\text{W/m}^2

Derivation

Energy conservation: power P crosses every sphere of area $4\pi r^2$ , so $I = P/4\pi r^2$ .

Doubling r: $I \to I/4$ , $\Delta\beta = 10\log(1/4) \approx -6\,$ dB. Two equal sources: $I \to 2I$ , $+3\,$ dB — 'twice the speakers' is barely louder.

Scenarios to explore

Decibel Scale — Inverse-square intensity and the logarithmic dB ladder.

Real-world applications

Concert & PA system design (line arrays fight the 1/r² loss).
Workplace noise limits (85 dB for 8 h).
Sonar & ultrasound ranging power budgets.

JEE exam tips

+10 dB = ×10 intensity, +3 dB = ×2, +6 dB = ×4 — arithmetic in your head.
n equal sources: β = β₁ + 10 log n.
I ∝ (amplitude)² ⇒ dB difference = 20 log(A₂/A₁).

Common mistakes

Adding decibels directly for two sources — add INTENSITIES, then convert.
−3 dB for doubling distance (that's for halving intensity; distance doubling gives −6 dB).
Using amplitude ratios with the 10log formula (amplitude uses 20log).

Exam traps to avoid

0 dB is not silence — it's the hearing threshold I₀.
dB is dimensionless; intensity carries the W/m² units.