Even the best mirrors absorb or transmit some of the light...

Even the best mirrors absorb or transmit some of the light incident on them. The highest-quality mirrors might reflect $99.997 \%$ of the incident light. Suppose a cubical "room" $3.00 \mathrm{~m}$ on an edge, were constructed with such mirrors for the walls, floor, and ceiling. How slowly would such a room get dark? Estimate the time required for the intensity of light in such a room to fall to $1.00 \%$ of its initial value after the only light source in the room was switched off.

Key Concepts

Exponential Decay

When a quantity decreases by a fixed percentage over equal intervals (in this case, each reflection of light), the process can be modeled as exponential decay. In a reflective cavity, the fractional loss of light per encounter with the surface leads to an exponential reduction in the overall light intensity over time.

Reflectivity and Absorption Loss

Reflectivity is the fraction of the incident light that is reflected by a surface, while the remainder is absorbed or transmitted. Even very high-quality mirrors, which have reflectivity close to 100%, still absorb a small percentage of light at each reflection. Over many cycles, these small losses accumulate to produce a significant decrease in light intensity.

Photon Lifetime in an Enclosed Cavity

The photon lifetime in a cavity refers to the average time a photon remains in the cavity before its energy is significantly diminished by absorption or transmission losses. In a closed, mirrored room, the time between reflections is determined by the geometry and the speed of light, and the cumulative effect of losses at each reflection governs how quickly the intensity decays.

Geometric Considerations and Mean Free Path

The physical dimensions and shape of an enclosure determine the average distance a photon travels between successive reflections—its mean free path. In a perfectly enclosed, cubical space, this distance along with the speed of light gives an estimate of the time between reflections, which is essential for calculating the overall rate of decay in light intensity.

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