Power-Intensity Relationship

The power-intensity relationship is a fundamental concept in physics that relates the power carried by a wave to its intensity. This relationship is particularly important in the fields of optics and acoustics, where it is used to describe the energy transfer of light and sound waves, respectively.

Definitions

• Power (P): The rate at which energy is transferred by the wave. It is measured in watts (W).
• Intensity (I): The power per unit area perpendicular to the direction of wave propagation. It is measured in watts per square meter (W/m²).

Relationship Formula

The intensity ( I ) of a wave is directly proportional to the power ( P ) of the wave and inversely proportional to the area ( A ) over which the power is distributed:

[ I = \frac{P}{A} ]

If the wave is spreading out uniformly in all directions (spherically), the area ( A ) can be represented by the surface area of a sphere:

[ A = 4\pi r^2 ]

where ( r ) is the radius of the sphere. Therefore, the intensity at a distance ( r ) from a point source is:

[ I = \frac{P}{4\pi r^2} ]

Table of Differences and Important Points

Property	Power (P)	Intensity (I)
Definition	Rate of energy transfer	Power per unit area
SI Unit	Watts (W)	Watts per square meter (W/m²)
Dependence	Independent of distance	Inversely proportional to area (A)
Formula	P = I * A	I = P / A
Example	Total energy emitted by a lightbulb	Energy received per m² from the bulb

Examples

Example 1: Lightbulb

A 100 W lightbulb emits light uniformly in all directions. What is the intensity of light 2 meters away from the bulb?

Using the formula for intensity:

[ I = \frac{P}{4\pi r^2} = \frac{100\, \text{W}}{4\pi (2\, \text{m})^2} \approx \frac{100}{50.27} \approx 1.99\, \text{W/m}^2 ]

Example 2: Sound Waves

A loudspeaker emits a sound power of 10 W. If the sound spreads uniformly in all directions, what is the intensity 5 meters away?

[ I = \frac{P}{4\pi r^2} = \frac{10\, \text{W}}{4\pi (5\, \text{m})^2} \approx \frac{10}{314.16} \approx 0.032\, \text{W/m}^2 ]

Conclusion

Understanding the power-intensity relationship is crucial for analyzing the behavior of waves and their energy distribution. It is a key concept in designing and interpreting experiments and applications involving light, sound, and other types of waves. The inverse square law, which arises from this relationship, explains why the intensity of a wave decreases with the square of the distance from the source, a principle that has profound implications in various scientific and engineering disciplines.