Question

Consider the ratio $r=\frac{(1-a)}{(1+a)}$ to be determined by measuring a dimensionless quantity $a$. If the error in the measurement of a is $\Delta a(\frac{\Delta a}{a}<<1)$, then what is the error $\Delta r$ in determining $r$?

• A. $\frac{\Delta a}{(1+a{)}^2}$
• B. $\frac{2\Delta a}{(1+a{)}^2}$
• C. $\frac{2\Delta a}{(1-a^2)}$
• D. $\frac{2a\Delta a}{(1-a^2)}$

Answer 1

Answer: (B), (D)

$\frac{2\Delta a}{(1+a{)}^2}$
$\frac{2a\Delta a}{(1-a^2)}$

$r=\left(\frac{1-a}{1+a}\right)$
$\frac{\Delta r}{r}=\frac{\Delta (1-a)}{(1-a)}+\frac{\Delta (1+a)}{(1+a)}$
$=\frac{\Delta a}{(1-a)}=\frac{\Delta a}{(1+a)}$
$=\frac{\Delta a(1+a+a-a)}{(1-a)(1+a)}$
$\therefore \Delta r=\frac{2\Delta a}{(1-a)(1+a)}\frac{(1-a)}{(1+a)}=\frac{2\Delta a}{(1+a{)}^2}$