Question

Two radioactive materials A and B have decay constants \(6\lambda\) and \(2\lambda\) respectively. If initially they have same number of nuclei, then the ratio of the number of nuclei of A to that of B will be \(\frac{1}{e}\) after time

• A. \(\frac{1}{3\lambda}\)
• B. \(\frac{1}{2\lambda}\)
• C. \(\frac{1}{\lambda}\)
• D. \(\frac{1}{4\lambda}\)

Hint:

Write the exponential decay law for both samples and take their ratio.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept
The decay law is \(N=N_0e^{-\lambda t}\).

Step 2: Ratio
\(\dfrac{N_A}{N_B}=\dfrac{e^{-6\lambda t}}{e^{-2\lambda t}}=e^{-4\lambda t}\), since \(N_0\) is the same.

Step 3: Set the ratio to 1/e
\(e^{-4\lambda t}=e^{-1}\), so \(4\lambda t=1\).

Step 4: Calculation
\(t=\dfrac{1}{4\lambda}\).

Final Answer:
The time required is \(\dfrac{1}{4\lambda}\). \[ \boxed{\text{(D) }\dfrac{1}{4\lambda}} \]