Discuss in detail different types of errors arising in the measurement of the physical quantities. State the general method for estimating the error in a combined caldculation.

Explanation

In each experiment, number of measurements are carried out and in each measurement, some error is measured. Number of such errors are combined to get the final result. The maximum possible error in the final result in different cases occurs as :
(i) When final result involved sum of two observed physical quantities of the same type.
Suppose A and B are two quantities measured to get the final result Z, such that `Z=A+B`. Let `DeltaD` and `DeltaB` be absolute errors in the values of A and B. Then, the observed values of A and b are `A+-DeltaA` and `B+-DeltaB` respectively. If `DeltaZ` be absolute error in Z, then
`Z+-DeltaZ=(A+-DeltaA)+(B+-DeltaB)` or `Z+-DeltaZ)=(A+B)+-(DeltaA+DeltaB)`
As `Z=A+B`. Therefore, `DeltaZ=DeltaA+DeltaB`.
Hence, maximum possible absolute error in Z is given by `DeltaA+DeltaB`.
(ii) When the final result involves difference of two oberved physical quantities of the same type.
Suppose A and B are two measured physical quantities such that
`Z=A-B`
Let `DeltaA` and `DeltaB` be absolute errors in the values of A and B and `DeltaZ` be the absolute error in the measurement of Z , then
`Z+-(DeltaZ)=(A+-DeltaA)-(B+-DeltaB)`
`Z+-DeltaZ=(A-B)+-(DeltaA+DeltaB)`
`DeltaZ=DeltaA+DeltaB`
Which is maximum value of absolute error in value of final result.
(iii) When the final result is equal to product of two observed quantities.
Suppose final result `Z=AB`
Let measured value of A and B be `(A+-DeltaA)` and `(B+-DeltaB)`. If `DeltaZ` be the absolute error in Z, then `Z+-DeltaZ=(A+-DeltaA)(B+-DeltaB)=AB+-ADeltaB+-BDeltaA+-DeltaADeltaB`
Since `AB=Z`
Therefore, `+-DeltaZ=+-ADeltaB+-BDeltaA+-DeltaADeltaB`
Since, `DeltaA` and `DeltaB` are very small , so product of them, `DeltaA` `DeltaB` can be neglected
Hence, `+-DeltaZ=+-ADeltaB+-BDeltaA` ................(i)
Dividing equation (i) both sides by Z, we have
`(+-DeltaZ)/(Z)=(+-ADeltaB)/(Z)+-(BDeltaA)/(Z)=(ADeltaB)/(AB)+-(BDeltaA)/(AB)` (ignoring negative sign)
`(DeltaZ)/(Z)=(DeltaA)/(A)+(DeltaB)/(B)`.................(ii)
Therefore maximum possible relative error in `Z=` maximum possible relative error in `A+` maximum possible relative error in B.
(iv) When the final result is equal to the quotient of two observed quantities
Suppose final result is `Z=A//B`.
`Z=A//B`
Let `DeltaA`, `DeltaB` and `DeltaZ` be the abosolute errors in the measurement of `A,B ` and `Z` quantities respectively.
`Z+-DeltaZ=(A+-DeltaA)/(B+-DeltaB)=(A+-DeltaA)(B+-DeltaB)^(-1)`
`=A(1+-(DeltaA)/(A))B^(-1)(1+-(DeltaB)/(B))^(-1)=(A)/(B)(1+-(DeltaA)/(A))(1+-(DeltaB)/(B))^(-1) `
Expanding `(1+(DeltaB)/(B))^(-1)` by Binomial theorem and neglecting higher order terms, we have
`Z+-DeltaZ=(A)/(B)(1+-(DeltaA)/(A))(1+-(DeltaB)/(B))`, `Z+-DeltaZ=(A)/(B)(1+-(DeltaB)/(B)+-(DeltaA)/(A)+-(DeltaADeltaB)/(AB))`
Therefore , `Z+-DeltaZ=(A)/(B)(1+-(DeltaA)/(A)+-(DeltaB)/(B))`
or `Z+-DeltaZ=Z(1+-(DeltaA)/(A)+-(DeltaB)/(B))` ...............(iii)
Dividing eqn. (iii) by Z both sides, we have
`(Z+-DeltaZ)/(Z)=(Z)/(Z)(1+-(DeltaA)/(A)+-(DeltaB)/(B))implies(DeltaZ)/(Z)=(DeltaA)/(A)+(DeltaB)/(B)`
Hence maximum possible relative error in Z= maximum relative error in `A+` maximum relative error in B.
(v) Where final result is equal to some power of a physical quantity
Let `Z=A^(n)`
then `Z+-DeltaZ=(A+-DeltaA)^(n)=A^(n)[1+-(DeltaA)/(A)]^(n)`
`Z+-DeltaZ=Z[1+-(DeltaA)/(A)]^(n)`...........(iv)
Expanding `[1+(DeltaA)/(A)]^(n)` by Binomial theorem and neglecting higher order terms, we have
`[1+-(DeltaA)/(A)]^(n)=[1+-(nDeltaA)/(A)]`.
So equation (iv) becomes `Z+-DeltaZ=Z[(1+-(nDeltaA)/(A)]` .............(v)
Dividing equation (v) by Z both sides, we have
`(Z+-DeltaZ)/(Z)=(Z)/(Z)[1+-(nDeltaA)/(A)]`. Therefore `(DeltaZ)/(Z)=(nDeltaA)/(A)`.
Hence , maximum possible error in `Z=` power `xx` relative error in A.