Which equation correctly represents the inverse square law for radiation intensity with distance?

Which equation correctly represents the inverse square law for radiation intensity with distance?

• A. I1/I2 = (D2)^2/(D1)^2
• B. I1/I2 = (D1/D2)^2
• C. I1/I2 = D2-D1
• D. I1/I2 = (D1)^2/(D2)^2

Answer: (A)

Radiation intensity falls off with the square of distance. If the distance from the source is r, the intensity I is proportional to 1/r^2. For two positions with distances D1 and D2 and corresponding intensities I1 and I2, you can write I1 = k/(D1^2) and I2 = k/(D2^2). Taking the ratio gives I1/I2 = (k/D1^2)/(k/D2^2) = D2^2/D1^2. This is exactly the form I1/I2 = (D2)^2/(D1)^2, which matches the inverse-square relationship. If you use (D1/D2)^2, you’d get I1/I2 = D1^2/D2^2, the reciprocal of the correct ratio (it corresponds to I2/I1). A linear difference like D2 − D1 or other non-squared forms don’t reflect the inverse-square behavior.

Radiation intensity falls off with the square of distance. If the distance from the source is r, the intensity I is proportional to 1/r^2. For two positions with distances D1 and D2 and corresponding intensities I1 and I2, you can write I1 = k/(D1^2) and I2 = k/(D2^2). Taking the ratio gives I1/I2 = (k/D1^2)/(k/D2^2) = D2^2/D1^2. This is exactly the form I1/I2 = (D2)^2/(D1)^2, which matches the inverse-square relationship. If you use (D1/D2)^2, you’d get I1/I2 = D1^2/D2^2, the reciprocal of the correct ratio (it corresponds to I2/I1). A linear difference like D2 − D1 or other non-squared forms don’t reflect the inverse-square behavior.