Question

Safe Exposure The intensity of a radioactive source is 1.15 mrem/s $\cdot {\mathrm{m}}^2$ at a distance of 0.50 $\mathrm{m}$ . What is the minimum distance a person could be from the source to have a maximum exposure of 0.65 $\mathrm{mrem}/\mathrm{s}\cdot {\mathrm{m}}^2?$

Short answer

The minimum distance a person could be from the radioactive source to have a maximum exposure of 0.65 mrem/s⋅m² is approximately $0.665\mathrm{m}$.

Step-by-step solution

Write the initial intensity and desired intensity

The initial intensity (I₁) and distance (d₁) are 1.15 mrem/s·m² and 0.50 m, respectively. The desired maximum intensity (I₂) is 0.65 mrem/s·m². Our goal is to find the minimum distance (d₂) from the source that results in the desired intensity I₂.

Apply the inverse square law to find the ratio of intensities

The inverse square law states that the intensity (I) of radiation is inversely proportional to the square of the distance (d) from the source. Therefore, the ratio of the initial and desired intensities is equal to the square of the ratio of the distances: $$\frac{I_1}{I_2}=\frac{d_2^2}{d_1^2}$$

Substitute the given values into the equation

Next, substitute the given values for I₁, I₂, and d₁ into the equation and solve for d₂: $$\frac{1.15}{0.65}=\frac{d_2^2}{(0.50{)}^2}$$

Solve for d₂

Now we can solve for d₂, the minimum distance a person could be from the source to have the desired intensity: $d_2^2=\frac{1.15}{0.65}\times (0.50{)}^2$ $d_2^2\approx 0.4423$ $d_2\approx \sqrt{0.4423}$

Find the minimum distance

Taking the square root of $d_2^2$: $d_2\approx 0.665\mathrm{m}$ The minimum distance a person could be from the radioactive source to have a maximum exposure of 0.65 mrem/s⋅m² is approximately 0.665 m.

Key concepts

Radioactive Intensity

The concept of radioactive intensity refers to the strength of the radioactive source at a specific distance from it. Intensity tells us how much radiation is being emitted per unit area. In our problem, the initial intensity is given as 1.15 mrem/s·m². Simply put, this tells us how concentrated the radiation is when you are 0.50 meters away from the source.

Understanding intensity is important because it changes based on how far you are from the radioactive source. The farther you are, the more spread out the radiation becomes over a wider area, and hence, the intensity reduces. Always think of it as a flashlight beam getting dimmer as you move away. If you're too close, the radiation is stronger and more intense, but as you step back, it diminishes.

Remember, to change the intensity you experience, you can either move closer or farther from the source. Here, we are interested in reducing the intensity to make it safer for exposure.

Radiation Exposure

Radiation exposure measures how much radiation a person might receive. It's crucial to understand because exposure affects our health. The dose you receive depends on the intensity of the source and how close you are to it. In this task, the desired exposure is 0.65 mrem/s·m². To achieve this safer level of exposure, the intensity has to drop from its initial value.

It's worth noting that exposure is cumulative, meaning that even if the intensity is lower, spending a longer time near the source can increase your total absorbed dose. Safety regulations dictate limits on how much exposure is permissible to avoid potential health risks such as radiation sickness.

Managing radiation exposure effectively means either lowering the intensity or increasing the distance, or ideally, a combination of both.

Distance Calculation

Calculating the correct distance to reduce the radiation intensity to a safe level involves applying the inverse square law. This is a crucial principle in radiation physics. The law states that the intensity of radiation decreases with the square of the distance from a source. In simple terms, if you double the distance from the source, the intensity quarters.

To find our specific distance, we use the formula: $$\frac{I_1}{I_2}=\frac{d_2^2}{d_1^2}$$Here, $I_1$ is the initial intensity, $I_2$ is the desired lower intensity, $d_1$ is the initial distance, and $d_2$ is what we want to calculate.

By solving $d_2$ from $$d_2^2=\frac{I_1}{I_2}\times (d_1{)}^2$$we get $d_2\approx 0.665\text{ meters}$, meaning that moving to about 0.665 meters will safely lower the intensity to 0.65 mrem/s·m². This shows how even small adjustments in distance can significantly alter exposure levels and ensure safety.