A freshly prepared radioactive source of half-life `2 h` emits radiation of intensity which is 64 times the permissible safe level. The minimum time after which it would be possible to work safely with this source is

Options

A. 128 h
B. 24 h
C. 6 h
D. 12 h

Answer

To solve the problem, we need to determine the minimum time after which the intensity of radiation from a radioactive source will be at or below the permissible safe level. The source has a half-life of 2 hours and initially emits radiation at an intensity that is 64 times the permissible safe level.

### Step-by-Step Solution:

1. **Understand the Initial Condition**:
- Let the initial intensity of radiation be \( I_0 = 64 \times I_{\text{safe}} \), where \( I_{\text{safe}} \) is the permissible safe level.

2. **Determine the Half-Life**:
- The half-life of the radioactive source is given as \( t_{1/2} = 2 \) hours. This means that every 2 hours, the intensity of radiation will reduce to half of its previous value.

3. **Calculate the Number of Half-Lives Required**:
- We need to find out how many half-lives it will take for the intensity to drop from \( 64 \times I_{\text{safe}} \) to \( I_{\text{safe}} \).
- After 1 half-life (2 hours):
\[ I = \frac{64 \times I_{\text{safe}}}{2} = 32 \times I_{\text{safe}} \]
- After 2 half-lives (4 hours):
\[ I = \frac{32 \times I_{\text{safe}}}{2} = 16 \times I_{\text{safe}} \]
- After 3 half-lives (6 hours):
\[ I = \frac{16 \times I_{\text{safe}}}{2} = 8 \times I_{\text{safe}} \]
- After 4 half-lives (8 hours):
\[ I = \frac{8 \times I_{\text{safe}}}{2} = 4 \times I_{\text{safe}} \]
- After 5 half-lives (10 hours):
\[ I = \frac{4 \times I_{\text{safe}}}{2} = 2 \times I_{\text{safe}} \]
- After 6 half-lives (12 hours):
\[ I = \frac{2 \times I_{\text{safe}}}{2} = 1 \times I_{\text{safe}} = I_{\text{safe}} \]

4. **Conclusion**:
- It takes 6 half-lives for the intensity to reduce from \( 64 \times I_{\text{safe}} \) to \( I_{\text{safe}} \).
- Since each half-life is 2 hours, the total time required is:
\[ \text{Total Time} = 6 \times 2 \text{ hours} = 12 \text{ hours} \]

Thus, the minimum time after which it would be possible to work safely with this source is **12 hours**.