Question

Using the expression $2dsin\theta =\lambda$ , one calculates the value of $d$ by measuring the corresponding angle $\theta$ in the range $0^{\circ }\text{to}90^{\circ }$ . The wavelength $\lambda$ is exactly known and the error in $\theta$ is constant for all values of $\theta$ , as $\theta$ increase from $0^{\circ }$ .

• A. The absolute error in $d$ remains constant.
• B. The absolute error in $d$ will increase.
• C. The fractional error in $d$ remains constant.
• D. The fractional error in $d$ decreases.

Answer 1

Answer: (D)

$2dsin\theta =\lambda$
$\Rightarrow d=\frac{\lambda }{2sin\theta }$
$logd=log\left(\frac{\lambda }{2}\right)-log(sin\theta )$
On differentiating,
$\left|\frac{\Delta d}{d}\right|=cot\theta d\theta$
$\theta \uparrow \Rightarrow cot\theta \downarrow$
$\left|\frac{\Delta d}{d}\right|$ will be decreased.