Problem 21 - Technical Mathematics with Calculus

Question

The inverse square law states that for a surface illuminated by a light source, the intensity of illumination on the surface is inversely proportional to the square of the distance between the source and the surface. A light source located \(7.50 \mathrm{m}\) from a surface produces an illumination of 426 lux on that surface. At what distance must that light source be placed to give an illumination of 850 lux?

Step-by-Step Solution

Answer

To achieve 850 lux of illumination, the light source must be placed at a distance of \( \sqrt{\frac{k}{850}} \) meters, after calculating the value of \( k \).

Step 1: Establish the inverse square law relationship

According to the inverse square law, the intensity (I) of illumination is inversely proportional to the square of the distance (d) from the source. This relationship can be expressed as: \( I \propto \frac{1}{d^2} \ \Rightarrow I = \frac{k}{d^2} \), where \( k \) is the proportionality constant.

Step 2: Calculate the proportionality constant

Using the given values for the initial distance and intensity, \( d_1 = 7.50 \mathrm{m} \) and \( I_1 = 426 \mathrm{lux} \), we can solve for \( k \). \( I_1 = \frac{k}{d_1^2} \Rightarrow k = I_1 \cdot d_1^2 = 426 \cdot (7.50)^2 \). Calculate \( k \) to find its value.

Step 3: Use the constant to find the new distance

With the desired intensity \( I_2 = 850 \mathrm{lux} \), use the constant \( k \) to find the new distance \( d_2 \): \( I_2 = \frac{k}{d_2^2} \Rightarrow d_2 = \sqrt{\frac{k}{I_2}} \). Substitute the previously found value of \( k \) to calculate \( d_2 \).

Step 4: Simplify and calculate the new distance

Plugging in the values to calculate \( d_2 \), the final step is to simplify and solve for \( d_2 \), using the square root function to obtain the value of the new distance in meters.

Key Concepts

Intensity of Illumination

In the field of physics, intensity of illumination refers to the amount of light that hits a given surface area. It is a crucial concept when dealing with lighting and visibility and is commonly measured in units called lux. To put it simply, if you think of a light bulb hanging above a table, the 'intensity of illumination' describes how much light is actually reaching the table surface. The brighter the light, the higher the intensity. This is an important factor for activities that require precise lighting, like drafting or reading.

Understanding the intensity of illumination can also help in designing lighting for rooms or outdoor spaces, ensuring that the correct level of brightness is achieved for functionality and comfort. In our textbook example, we talked about a light source providing a certain intensity at a specific distance, highlighting how this measure can vary with different configurations and distances from the light source.

Proportionality Constant

In mathematical terms, a proportionality constant is the value that relates two quantities that are directly proportional to each other. In the context of the inverse square law of light, this constant, often denoted by the symbol ‘k’, connects the intensity of illumination with the square of the distance from the light source.

Importance of 'k' in Calculations

Once the value of 'k' is known from an initial set of conditions (like we saw in step 2 of our solution), it becomes a powerful tool to predict illumination intensity at other distances. Remember, this constant is unique to the light source and does not change unless the light's output changes. By calculating 'k', we can easily switch between distances and intensities without re-measuring the actual output of the light source, making it a highly efficient way to understand the behavior of light in different situations.

Lux Measurement

Lux is the unit of measurement for illuminance or intensity of illumination. One lux is equal to one lumen per square meter. Lumens measure the total amount of visible light emitted by a source, and when this is spread over a surface, we get the measurement in lux.

Practical Uses of Lux

In our daily lives, from the lighting in our workplaces to street lamps, the standards for adequate illumination are often conveyed in lux. For example, an office space might require 500 lux for proper visibility, while ambient lighting in a home might be set around 300 lux. The example from the textbook dealt with an illumination level of 426 lux, which is quite precise and might be used for tasks requiring detailed attention. Interiors designers, lighting engineers, and health professionals use lux measurements to create environments that are both safe and tailored to the activities performed in them.

Distance and Light Intensity Relationship

The key idea to understand when it comes to light intensity is that it diminishes as the distance from the source increases. This concept is grounded in what we call the 'inverse square law'. As the distance between the light source and the surface doubles, the light intensity on that surface quarters.

Application in Real-world Scenarios

This principle plays a significant role in fields like photography, theater, and security lighting, where the precise control of light intensity is crucial. Understanding this relationship helps professionals estimate the necessary positioning of light sources to achieve the desired illuminance. In our exercise, we used this relationship to discover at what distance the light intensity would increase to 850 lux. By applying the inverse square law, we are able to understand and manipulate the way light behaves in order to meet our specific needs.