11th Class Physics Physical World / भौतिक जगत Propagation of Errors

(1) Error in sum of the quantities : Suppose \[x=a+b\]

Let \[\Delta a=\] absolute error in measurement of a

\[\Delta b=\] absolute error in measurement of b

\[\Delta x=\] absolute error in calculation of x i.e. sum of a and b.

The maximum absolute error in x is \[\Delta x=\pm (\Delta a+\Delta b)\]

Percentage error in the value of \[x=\frac{(\Delta a+\Delta b)}{a+b}\times 100%\]

(2) Error in difference of the quantities : Suppose \[x=a-b\]

Let \[\Delta a=\] absolute error in measurement of a,

\[\Delta b=\] absolute error in measurement of b

\[\Delta x=\] absolute error in calculation of x i.e. difference of a and b.

The maximum absolute error in x is \[\Delta x=\pm (\Delta a+\Delta b)\]

Percentage error in the value of \[x=\frac{(\Delta a+\Delta b)}{a-b}\times 100%\]

(3) Error in product of quantities : Suppose \[x=a\times b\]

Let \[\Delta a=\] absolute error in measurement of a,

\[\Delta b=\] absolute error in measurement of b

\[\Delta x=\] absolute error in calculation of x i.e. product of a and b.

The maximum fractional error in x is \[\frac{\Delta x}{x}=\pm \,\left( \frac{\Delta a}{a}+\frac{\Delta b}{b} \right)\]

Percentage error in the value of x         

= (% error in value of a) + (% error in value of b)

(4) Error in division of quantities : Suppose \[x=\frac{a}{b}\]

Let \[\Delta a=\] absolute error in measurement of a,

\[\Delta b=\] absolute error in measurement of b

\[\Delta x=\] absolute error in calculation of x i.e. division of a and b.

The maximum fractional error in x is \[\frac{\Delta x}{x}=\pm \,\left( \frac{\Delta a}{a}+\frac{\Delta b}{b} \right)\]

Percentage error in the value of x

= (% error in value of a) + (% error in value of b)

(5) Error in quantity raised to some power : Suppose \[x=\frac{{{a}^{n}}}{{{b}^{m}}}\]

Let \[\Delta a=\] absolute error in measurement of a,

\[\Delta b=\] absolute error in measurement of b

\[\Delta x=\] absolute error in calculation of x

The maximum fractional error in x is \[\frac{\Delta x}{x}=\pm \,\left( n\frac{\Delta a}{a}+m\frac{\Delta b}{b} \right)\]

Percentage error in the value of x

= n (% error in value of a) + m (%error in value of b)