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 \(2.75 \mathrm{m}\) from a surface produces an illumination of \(528 \mathrm{lux}\) on that surface. Find the illumination if the distance is changed to \(1.55 \mathrm{m}\).

Short answer

The new illumination is approximately 1661.43 lux at a distance of 1.55 m.

Step-by-step solution

Understand the Inverse Square Law

The inverse square law for illumination can be expressed as the formula, I = k/d^2, where I is the intensity of illumination, k is a proportionality constant, and d is the distance from the light source to the surface.

Calculate the Proportionality Constant (k)

Given the initial distance d1 = 2.75 m and initial illumination I1 = 528 lux, use the formula to solve for k: k = I1 * d1^2.

Plug in Given Values and Solve for k

Substitute the given values into the equation to find k: k = 528 * (2.75)^2 = 528 * 7.5625 = 3990.9 lux*m^2.

Write the Formula with the Proportionality Constant

Now that k is known, the formula for illumination becomes I = 3990.9 / d^2.

Calculate New Illumination (I2)

Given the new distance d2 = 1.55 m, use the formula to find the new illumination I2: I2 = 3990.9 / (1.55)^2.

Solve for New Illumination (I2)

Substitute the new distance into the equation to get I2: I2 = 3990.9 / (1.55)^2 = 3990.9 / 2.4025 ≈ 1661.43 lux.

State the Final Answer

The new illumination on the surface when the distance is 1.55 m is approximately 1661.43 lux.

Key concepts

Intensity of Illumination

Understanding the intensity of illumination is essential when studying how light behaves as it spreads out from a source. The intensity, measured in lux, refers to the amount of light that strikes a surface per unit area. Imagine a light bulb hanging in a room; the surfaces closer to the bulb receive more light directly and thus have a higher intensity of illumination.

Whether it is a bulb illuminating a room or a sunbeam striking the ground, the principle holds true: the closer you are to the light source, the brighter the light appears. The inverse square law quantitatively explains this phenomenon: as the distance from the light source increases, the intensity of light decreases rapidly, specifically, by the square of the distance. Understanding this relationship helps us in practical situations, such as determining the lighting required to adequately illuminate spaces of various shapes and sizes.

Proportionality Constant

The proportionality constant, typically denoted as 'k', plays a crucial role in the inverse square law equation \( I = \frac{k}{d^2} \). It represents the brightness of a light source before considering the effect of distance on illumination. When you understand this constant, you can predict how a change in distance will affect brightness.

The proportionality constant is determined when the conditions are known at a certain distance. By solving for 'k' with known values of illumination and distance, as done in the given exercise, this constant becomes a key value to use whenever the light source remains unchanged but the distance varies. This allows us to determine the light intensity at any other distance without needing to directly measure it each time, a useful trait for planning lighting in architectural design or conducting physics experiments.

Distance and Illumination Relationship

The distance and illumination relationship highlights how modifications in the distance from a light source can substantially alter the level of illumination on a given surface. When the distance is halved, the illumination increases by a factor of four; conversely, doubling the distance results in illumination decreasing by a quarter. This inversely proportional relationship shows the incredible impact distance has over how we perceive brightness.

In practical applications, such as in photography or stage lighting, professionals adjust the position of the lights relative to subjects or objects to achieve the desired illumination effect. This concept is also vital in energy conservation, as it helps to optimize the placement of light sources to ensure efficient usage while achieving the right ambient light for safety, comfort, or aesthetic purposes.