The number of undecayed nuclei N in a sample | vedclass

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(A) The law of radioactive decay is given by $N = N_0 e^{-\lambda t}$.Activity $A$ is defined as the rate of decay of nuclei, given by $A = -\frac{dN}

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(A) The law of radioactive decay is given by $N = N_0 e^{-\lambda t}$.Activity $A$ is defined as the rate of decay of nuclei, given by $A = -\frac{dN}{dt}$.Differentiating $N$ with respect to $t$, we get:$\frac{dN}{dt} = -\lambda N_0 e^{-\lambda t} = -\lambda N$.Therefore, the magnitude of activity is $A = |-\frac{dN}{dt}| = \lambda N$.This equation $A = \lambda N$ represents a linear relationship between $N$ and $A$, where $\lambda$ is the decay constant.Since $A$ is directly proportional to $N$ $(A \propto N)$, the graph of $N$ versus $A$ will be a straight line passing through the origin. Thus, option $A$ is correct.

The number of undecayed nuclei $N$ in a sample of radioactive material as a function of time $t$ is shown in the figure. Which of the following graphs correctly shows the relationship between $N$ and the activity $A$?

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