Question

Even the best mirrors absorb or transmit some of the light incident on them. The highest-quality mirrors might reflect \(99.997 \%\) of incident light intensity. 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 is switched off.

Short answer

Based on the given information, the time required for the intensity of light in the cubical room to fall to 1.00% of its initial value is approximately 7.96 ms (milliseconds).

Step-by-step solution

Calculate the number of reflections

Initially, the light in the room is at \(100 \%\) intensity. We want to know the number of reflections required for the intensity to fall to \(1.00 \%\). Let \(n\) be the number of reflections. We have: Initial_intensity\(\times\)(Fraction_reflected)\(^{n} = \)Final_intensity \(1\times(0.99997)^{n}\) = \(0.01\) To solve for \(n\), we can take the natural logarithm of both sides: \(n\ln{(0.99997)} = \ln{(0.01)}\) Now we can solve for \(n\): \(n = \dfrac{\ln{(0.01)}}{\ln{(0.99997)}}\)

Calculate the distance traveled by light

In a cube, the longest straight path across the room is along the space diagonal. The length of the space diagonal can be calculated using the Pythagorean theorem in three dimensions: Space_diagonal = \(\sqrt{(\text{side_length})^2 + (\text{side_length})^2 + (\text{side_length})^2}\) Since the room is \(3.00\,\text{m}\) on each side, this gives: Space_diagonal = \(\sqrt{(3.00)^2 + (3.00)^2 + (3.00)^2}\) The total distance traveled by light is the product of the space diagonal and number of reflections: Total_distance = Space_diagonal \(\times n\)

Calculate the time required

The speed of light is approximately \(3.00\times10^8\,\text{m/s}\). We can now use this to find the time required for the intensity to decrease to the desired level: Time_required = \(\dfrac{\text{Total_distance}}{\text{Speed_of_light}}\) Now, let's perform the calculations to obtain the final answer. #Calculations# n = \(\dfrac{\ln{(0.01)}}{\ln{(0.99997)}} \approx 460506 \, \text{reflections}\) Space_diagonal = \(\sqrt{(3.00)^2 + (3.00)^2 + (3.00)^2} \approx 5.20\,\text{m}\) Total_distance = \(5.20 \times 460506 \approx 2.39\times10^6\,\text{m}\) Time_required = \(\dfrac{2.39\times10^6}{3.00\times10^8} \approx 7.96\times10^{-3}\,\text{s}\) So the time required for the intensity of light in the room to fall to \(1.00 \%\) of its initial value is approximately \(7.96\,\text{ms}\).

Key concepts

Light Reflection

Light reflection is a fundamental concept in optics, which refers to the phenomena where light waves bounce off surfaces. The law of reflection states that the angle of incidence (the angle at which incoming light hits a surface) is equal to the angle of reflection (the angle at which light bounces off). This principle explains why mirrors can redirect light in a predictable manner.

High-quality mirrors, such as those described in the exercise, have a very high reflection coefficient. This coefficient indicates the fraction of light intensity that is reflected by a mirror. In the stated problem, a mirror has a reflection coefficient of 99.997%, meaning it reflects nearly all of the incident light, thus requiring numerous reflections before the light's intensity is noticeably diminished inside the room.

To understand this better, imagine a game of billiards where, instead of balls, you're using beams of light, and instead of pockets on a table, you're dealing with absorption by the walls. With each 'bounce' or reflection off the mirror walls, a minuscule fraction of the light is lost, gradually dimming the room as if the light is being 'pocketed' at an exceedingly slow rate.

Intensity of Light

Intensity of light refers to the amount of light energy hitting a certain area over a specific time period, often quantified as watts per square meter. In our daily experience, we can think of intensity as the 'brightness' of light. As light travels and interacts with various materials, its intensity diminishes due to absorption, reflection, or scattering.

In the context of our cubical mirror room, every time light reflects off a surface, it retains almost all its intensity, falling by just a fraction of a percent. The exercise in question explores how many reflections it would take for the room's light intensity to fall to 1% of its initial value, demonstrating how the intensity of light changes as it interacts with reflective surfaces. This concept is pivotal in designing lighting for architectural spaces and understanding natural phenomena such as the fading of daylight during sunset or the dimming effects in water depths.

Natural Logarithms in Physics

Natural logarithms, symbolized by \(\ln\), are mathematical functions particularly useful in physics for dealing with exponential relationships. When analyzing phenomena like radioactive decay, sound intensity, or as in our exercise, light intensity diminishing over multiple reflections, natural logarithms help to linearize the exponential equations.

In the exercise, the use of natural logarithms transforms the exponential decay of light intensity into a linear equation that we can solve for the number of reflections needed to reach a specific intensity. The natural logarithm allows us to find that after about 460,506 reflections, the light intensity in the room would fall to just 1%. This mathematical tool is indispensable when working with exponential processes, providing clarity and simplicity in calculations and helping us understand how quickly or slowly changes occur in physical systems.