The intensity of a wave, which represents the power it carries per unit area, is fundamentally proportional to the square of its amplitude. This relationship holds true for various types of waves, including sound waves, water waves, and importantly, electromagnetic waves like light.
According to the electromagnetic theory, the intensity of a wave is directly proportional to its amplitude, i.e., I ∝ A 2. Since light is also an electromagnetic wave, the intensity of light will depend upon the amplitude of the wave.
Understanding Wave Intensity
Intensity measures how much energy a wave transmits over a certain area per unit time. It's essentially the power density of the wave.
- Intensity (I): Power / Area (Watts / meter²)
- Power: Energy transmitted per unit time (Joules / second)
The Role of Amplitude
Amplitude is a measure of the maximum displacement or disturbance caused by a wave from its equilibrium position. For different types of waves, amplitude corresponds to:
- Sound Waves: The maximum change in pressure or displacement of air particles.
- Water Waves: The maximum vertical displacement of the water surface.
- Electromagnetic Waves (Light): The maximum strength of the electric or magnetic field vectors.
Why the Square Relationship?
The key reason intensity is proportional to the square of amplitude lies in the energy carried by the wave.
- Energy and Amplitude: The energy associated with an oscillator (like a particle or field oscillating in a wave) is proportional to the square of its amplitude of oscillation. Think of a spring: the potential energy stored is proportional to the square of the displacement (amplitude). Similarly, the kinetic energy involves velocity squared, which is also related to the amplitude and frequency squared.
- Intensity and Energy Flow: Intensity is a measure of the rate at which energy flows through a given area. Since the energy density (energy per unit volume) in a wave is proportional to the square of its amplitude, the rate of energy flow (power) across an area is also proportional to the square of the amplitude.
- Applying to Waves:
- For mechanical waves like sound, the kinetic and potential energy densities are proportional to the square of the velocity and displacement amplitudes, respectively. Both are proportional to the wave's amplitude squared.
- For electromagnetic waves, the energy density is proportional to the square of the electric field amplitude (E²) and the magnetic field amplitude (B²). Since E and B are proportional to the wave's overall amplitude (A), the energy density is proportional to A². The intensity (power per unit area, related to the Poynting vector) is therefore also proportional to A².
Electromagnetic Waves and Light
As the provided reference highlights, light is an electromagnetic wave. Its intensity is determined by the strength of its oscillating electric and magnetic fields.
- The energy density of an electromagnetic field is proportional to E² + B².
- In a traveling electromagnetic wave, E and B are proportional to each other and to the wave's amplitude (A).
- Thus, the energy density is proportional to A².
- The intensity of the light wave, being the average energy flow per unit area, is consequently proportional to the average energy density and the wave speed.
- This leads directly to the relationship: Intensity (I) ∝ Amplitude² (A²).
This fundamental proportionality explains why, for example, doubling the amplitude of a light wave quadruples its brightness, or doubling the amplitude of a sound wave makes it four times as loud (as perceived by power/intensity).
Key Takeaway
The intensity of any wave is directly proportional to the square of its amplitude because the energy transported by the wave, and thus the power flow, scales with the square of the disturbance magnitude. For light, an electromagnetic wave, this means the intensity (brightness) is proportional to the square of the amplitude of the electric and magnetic fields.
