A β-emitting radiation source gives 250 units of radiation at a distance of 4.0 m. At what distance does the radiation drop to one-tenth its original value?

Verified step by step guidance

1. Understand that the intensity of radiation follows the inverse square law, which states that intensity is inversely proportional to the square of the distance from the source.
2. Set up the equation using the inverse square law: \( I_1 \times d_1^2 = I_2 \times d_2^2 \), where \( I_1 \) is the initial intensity, \( d_1 \) is the initial distance, \( I_2 \) is the final intensity, and \( d_2 \) is the final distance.
3. Substitute the known values into the equation: \( 250 \times (4.0)^2 = (\frac{250}{10}) \times d_2^2 \).
4. Simplify the equation to solve for \( d_2^2 \).
5. Take the square root of both sides to find \( d_2 \), the distance at which the radiation intensity is one-tenth of its original value.

Key Concepts

Inverse Square Law

The Inverse Square Law states that the intensity of radiation from a point source decreases with the square of the distance from the source. This means that if you double the distance from the source, the intensity of radiation is reduced to one-fourth. Understanding this principle is crucial for calculating how radiation levels change with distance.

Radioactive Decay

Radioactive decay refers to the process by which unstable atomic nuclei lose energy by emitting radiation. The rate of decay is characterized by the half-life, which is the time required for half of the radioactive atoms in a sample to decay. This concept is important for understanding how radiation levels decrease over distance and time.

Exponential Decay

Exponential decay describes the process where a quantity decreases at a rate proportional to its current value. In the context of radiation, this means that as distance increases, the radiation intensity diminishes exponentially. This concept helps in determining the distance at which radiation levels fall to a specific fraction of their original value.