A radioactive sample has an average life of 3 | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(C) The charge on the capacitor at time $t$ is given by $q = q_0 e^{-\frac{t}{RC}}$,where $RC$ is the time constant of the $RC$ circuit.The activity $

Vedclass · Accepted Answer

$150$

Vedclass · Answer

(C) The charge on the capacitor at time $t$ is given by $q = q_0 e^{-\frac{t}{RC}}$,where $RC$ is the time constant of the $RC$ circuit.The activity $A$ of the radioactive sample at time $t$ is given by $A = A_0 e^{-\lambda t}$,where $A_0$ is the initial activity and $\lambda$ is the decay constant.Given that the ratio $\frac{q}{A}$ is constant with respect to time,we have:$\frac{q}{A} = \frac{q_0 e^{-\frac{t}{RC}}}{A_0 e^{-\lambda t}} = 	ext{constant}$For this ratio to be independent of time,the exponential terms must cancel out,which implies:$-\frac{t}{RC} = -\lambda t \implies \lambda = \frac{1}{RC}$We know that the average life $	au = \frac{1}{\lambda} = 30 \, ms = 30 	imes 10^{-3} \, s$.Therefore,$\lambda = \frac{1}{30 	imes 10^{-3}} \, s^{-1}$.Substituting $\lambda = \frac{1}{RC}$ into the equation for $R$:$R = \frac{1}{\lambda C} = 	au 	imes \frac{1}{C}$Given $C = 200 \, \mu F = 200 	imes 10^{-6} \, F$:$R = \frac{30 	imes 10^{-3}}{200 	imes 10^{-6}} = \frac{30000}{200} = 150 \, \Omega$.

Similar Questions

At any instant,two elements $X_1$ and $X_2$ have the same number of radioactive atoms. If the decay constants of $X_1$ and $X_2$ are $10\lambda$ and $\lambda$ respectively,then the time when the ratio of their atoms becomes $\frac{1}{e}$ will be:

$A$ heavy nucleus $Q$ of half-life $20 \text{ minutes}$ undergoes alpha-decay with a probability of $60 \%$ and beta-decay with a probability of $40 \%$. Initially,the number of $Q$ nuclei is $1000$. The number of alpha-decays of $Q$ in the first one hour is:

The activity of a radioactive sample is $64 \times 10^{-5} \text{ Ci}$. Its half-life is $3 \text{ days}$. The activity will become $5 \times 10^{-6} \text{ Ci}$ after how many days?

Radon-$222$ has a half-life of $3.8$ days. If one starts with $0.064 \ kg$ of radon-$222$, the quantity of radon-$222$ left after $19$ days will be (in $kg$)

The half-life of a radioactive substance is $5$ years. After $x$ years,a given sample of the radioactive substance gets reduced to $6.25 \%$ of its initial value. The value of $x$ is ...............

Vedclass Test Series

Exam Paper Generator

Online Exam Module