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 $
$\Rightarrow d=\frac{\lambda }{2 sin \theta }$
$logd=log\left(\frac{\lambda }{2}\right)-log\left(\right.sin\theta \left.\right)$
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.