A speaker produces sound waves. You double the amplitude of oscillation of the air particles near the speaker. By what factor does the sound intensity change?
AIt doubles — intensity is proportional to amplitude
BIt quadruples — intensity is proportional to amplitude squared
CIt increases by a factor of 8 — intensity depends on both amplitude and the cube of frequency
DIt remains unchanged — intensity depends only on frequency, not amplitude
Intensity scales with the square of amplitude: I ∝ A². Doubling A gives (2A)² = 4A², so intensity quadruples. This follows directly from the energy stored in simple harmonic motion, which goes as kA². Intensity measures the power delivered per unit area, and that power scales with the square of the displacement amplitude. Option D is the common misconception — amplitude and intensity are tightly linked.
Question 2 Multiple Choice
A point source of sound radiates uniformly in all directions. A microphone at 3 m from the source measures an intensity of 0.04 W/m². A second microphone is placed 9 m from the source. What intensity does it measure?
A0.013 W/m² — intensity falls as 1/r, so tripling distance cuts it to a third
B0.020 W/m² — intensity falls as 1/√r
C0.0044 W/m² — intensity falls as 1/r², so tripling distance cuts it to one-ninth
D0.04 W/m² — intensity is conserved and independent of distance
For a point source, the same total power spreads over a sphere whose area grows as 4πr². Since intensity = power/area, and area ∝ r², intensity ∝ 1/r² — the inverse square law. Tripling distance (from 3 to 9 m) means r² increases by 9, so intensity becomes 0.04/9 ≈ 0.0044 W/m². Option A is the most common distractor: students confuse 1/r (which describes wave amplitude) with 1/r² (which describes intensity).
Question 3 True / False
Wave intensity is proportional to both the square of amplitude and the square of frequency.
TTrue
FFalse
Answer: True
The full expression is I ∝ A²f², where A is amplitude and f is frequency. Each oscillation cycle carries energy, and higher frequency means more cycles per second. Combined with the amplitude-squared dependence from the energy in each oscillation, both factors contribute to intensity. This means a wave with twice the frequency and the same amplitude carries four times the intensity.
Question 4 True / False
The Poynting vector of an electromagnetic wave points in a direction perpendicular to the wave's direction of propagation.
TTrue
FFalse
Answer: False
The Poynting vector S = E × B / μ₀ points in the *same* direction as wave propagation, not perpendicular to it. It represents the direction and magnitude of energy flow. The electric (E) and magnetic (B) fields are each perpendicular to the propagation direction and to each other; their cross product therefore points along the propagation direction. Confusing the orientation of the fields with the direction of energy transport is the common error here.
Question 5 Short Answer
Why does wave intensity obey an inverse square law for a point source, even if the medium absorbs no energy at all?
Think about your answer, then reveal below.
Model answer: Because the total power emitted by the source is constant and spreads over spherical wavefronts whose surface area grows as 4πr². Intensity is power per unit area, so as the same power is divided among an ever-larger area, intensity must decrease as 1/r². No energy is lost — it is simply spread more thinly.
This is a geometry argument, not an energy-loss argument. Energy is conserved: the total power crossing any sphere centered on the source is the same. But the area of that sphere is 4πr², so intensity — power per unit area — must fall as 1/r². This principle applies to light, sound, gravity, and any other quantity that radiates isotropically from a point source.