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In the above problem if each decay produ...

In the above problem if each decay produces `E_0` energy, then find
(a) power produced at time t
(b) total energy produced upto time t

Text Solution

Verified by Experts

The correct Answer is:
A, B, D
(a) :' `N=alpha/lambda(1-e^(-lambdat))`
At time t, number of decays per second `=lambdaN=alpha(1-e^(-lambdat))`
Each decay produces `E_0` energy. Therefore, energy produced per second or power.
`=(number of decays per second)(energy produced in each decay)
`=(lambdaN)E_0`
`=alphaE_0(1-e^(-lambdat))`
or
`P=alphaE_0(1-e^(-lambdat))`
(b) Power is a function of time. Therefore, total energy produced upto time t can be obtained by integrating this power or
`E_(Total) = int_0^tPdt`
Alternate Method
Energy is produced only in decay. Upto time t total `alphat` nuclei are produced and N nuclei are left. So, total number of nuclei decayed.
`N_d=alphat-N=alphat-alpha/lambda(1-e^(-lambdat))`
`=alpha[t-1/lambda(1-e^(-lambdat))]`
Each decay produces `E_0` energy. Therefore, total energy produced upto time t,
`E_(Total) = N_dE_0`
`=alphaE_0[t-1/lambda(1-e^(-lambdat))]`
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