Paragraph: If the measurement errors in all the independent quantities are known, then it is possible to…

Paragraph: If the measurement errors in all the independent quantities are known, then it is possible to determine the error in any dependent quantity. This is done by the use of series expansion and truncating the expansion at the first power of the error. For example, consider the relation z =xy. If the errors in x, y and z are ∆x, ∆y and ∆z , respectively, then
z±∆z=x±∆xy±∆y=xy1±∆xx1±∆yy-1
The series expansion for 1±∆yy-1, to first power in ∆y/y, is 1∓∆y/y. The relative errors in independent variables are always added. So the error in z will be
Δz=z( Δx x + Δy y ).
Question : In an experiment the initial number of radioactive nuclei is 3000 . It is found that \(1000 \pm 40\) nuclei decayed in the first \(1.0 \mathrm{~s}\). For \(|x| \ll 1, \ln (1+x)=x\) up to first power in \(x\). The error \(\Delta \lambda\), in the determination of the decay constant \(\lambda\), in \(s^{-1}\), is
  1. 0.04
  2. 0.03
  3. 0.02
  4. 0.01

Solution

N=N0e-λt
ln⁡N=ln⁡N0-λt
dNN=-dλt
Converting to error,
∆NN=∆λt
∴  ∆λ=402000×L=0.02 ( N is number of nuclei left undecayed) !

Asked in: JEE Advanced 2018 (Paper 1)

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