Question

Many physical quantities are connected by inverse square laws, that is, by power functions of the form f(x)=kx^(-2). In particular, the illumination of an object by a light source is inversely proportional to the square of the distance from the source. Suppose that after dark you are in a room with just one lamp and you are trying to read a book. The light is too dim and so you move halfway to the lamp. How much brighter is the light?

Answer 1

The light becomes four times brighter when you move halfway closer to a light source, due to the inverse square law.

Step-by-step explanation:

When you move halfway closer to a light source, the illumination on the book you are reading increases dramatically because of the inverse square law for light propagation. According to this law, the brightness or intensity of light is inversely proportional to the square of the distance from the light source. If you reduce the distance to half, the new brightness becomes (1/0.5)^2 = 4 times the initial brightness.

For example, if the initial intensity of the light is I at a distance d, when you move to a distance of d/2, the intensity becomes 4I. This is because the new distance squared is (d/2)^2 which is 1/4th of the original distance squared (d^2), hence reversing the effect we get four times the initial intensity due to the inverse law.

Answer 2

• 4 times

Step-by-step explanation:

Since the equation for the illumination of an object, i.e. the brightness of the light, is inversely proportional to the square of the distance from the light source, the form of the function is:

• f(x) = k.x⁻²

Where x is the distance between the object and the light force, k is the constant of proportionality, and f(x) is the brightness.

Then, if you move halfway to the lamp the new distance is x/2 and the new brightness (call if F) is :

`F=k(x/2)^(-2)=(k)/((x/2)^2)= (k)/(x^2). 4=f(x).4`

Then, you have found that the light is 4 times as bright as it originally was.