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. (a) 6h
B. (b) 12h
C. (c) 24h
D. (d) 128 h

Explanation

To solve the problem, we need to determine the time required for the intensity of radiation from a radioactive source to decrease to a safe level. The intensity of radiation is proportional to the number of radioactive nuclei present.

### Step-by-Step Solution:

1. **Understand the Problem**:
- We are given that the half-life of the radioactive source is 2 hours.
- The initial intensity of radiation is 64 times the permissible safe level.

2. **Determine the Safe Level**:
- Let the permissible safe level of intensity be \( I_0 \).
- Therefore, the initial intensity \( I \) is \( 64 I_0 \).

3. **Use the Half-Life Formula**:
- The number of radioactive nuclei decreases by half every half-life period.
- The relationship can be expressed as:
\[ N(t) = N_0 \left( \frac{1}{2} \right)^{\frac{t}{t_{1/2}}} \]
- Here, \( N(t) \) is the number of radioactive nuclei at time \( t \), \( N_0 \) is the initial number of nuclei, and \( t_{1/2} \) is the half-life.

4. **Set Up the Equation**:
- We want to find the time \( t \) when the intensity becomes safe, i.e., when:
\[ I(t) = I_0 \]
- Since \( I \) is initially \( 64 I_0 \), we set up the equation:
\[ 64 I_0 \left( \frac{1}{2} \right)^{\frac{t}{2}} = I_0 \]

5. **Simplify the Equation**:
- Dividing both sides by \( I_0 \):
\[ 64 \left( \frac{1}{2} \right)^{\frac{t}{2}} = 1 \]
- This simplifies to:
\[ \left( \frac{1}{2} \right)^{\frac{t}{2}} = \frac{1}{64} \]

6. **Express 64 as a Power of 2**:
- We know that \( 64 = 2^6 \), so:
\[ \left( \frac{1}{2} \right)^{\frac{t}{2}} = \left( \frac{1}{2} \right)^{6} \]

7. **Equate the Exponents**:
- Since the bases are the same, we can equate the exponents:
\[ \frac{t}{2} = 6 \]

8. **Solve for \( t \)**:
- Multiplying both sides by 2 gives:
\[ t = 12 \text{ hours} \]

### Conclusion:
The minimum time after which it would be possible to work safely with this radioactive source is **12 hours**.