The intensity of illumination at a given point is directly proportional to the intensity of the light source and inversely proportional to the square of the distance from the light source. If a desk is properly illuminated by a 74.0 w lamp 8.00 ft from the desk, what size lamp will be needed to provide the same lighting at a distance of 12.0 ft?
To find the size of the lamp needed to provide the same lighting at a different distance from the desk, we need to use the inverse square law of illumination.
The inverse square law states that the intensity of illumination is inversely proportional to the square of the distance from the light source. Mathematically, it can be written as:
I₁ / I₂ = (D₂)² / (D₁)²
Where:
I₁ and I₂ are the intensities of illumination at distances D₁ and D₂ from the light source, respectively.
In this case, we know that the intensity of illumination at the desk is directly proportional to the intensity of the light source. Let's call the size of the lamp needed to provide the same lighting at a distance of 12.0 ft as L.
So, we have:
I₁ / I₂ = L₁ / L₂
Since the intensity of illumination is directly proportional to the size of the lamp, we can write it as:
I₁ / I₂ = (L₁) / (L₂)
Now, let's plug in the given values:
I₁ / I₂ = (74.0 W) / L₂ [Distance D₁ = 8.00 ft; Lamp size L₁ = 74.0 W]
(D₂)² / (D₁)² = (12.0 ft)² / (8.00 ft)²
Now we can solve for L₂ by rearranging the equation:
I₁ / I₂ = (L₁) / (L₂)
(74.0 W) / L₂ = (12.0 ft)² / (8.00 ft)²
Let's calculate the values on both sides of the equation:
(74.0 W) / L₂ = (12.0 ft)² / (8.00 ft)²
(74.0 W) / L₂ = 1.5
To isolate L₂, we can multiply both sides of the equation by L₂:
(74.0 W) = 1.5 * L₂
Now, divide both sides by 1.5 to solve for L₂:
L₂ = (74.0 W) / 1.5
L₂ ≈ 49.3 W
Therefore, a lamp size of approximately 49.3 W will be needed to provide the same lighting at a distance of 12.0 ft.