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

Propagation of Errors

Category : 11th Class

(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)

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