Question

In a single slit diffraction experiment, a slit of width $(0.016 \pm 0.002)$ mm is used to measure the wavelength of a monochromatic light source. In the diffraction pattern, the angular distance between the central maximum and first minimum is measured to be $(2^\circ \pm 40')$. The value of the fractional error in the measurement of wavelength is:
(Given: $\sin(2^\circ) = 0.035$)

Hint:

When dealing with trigonometric functions in error analysis, absolute errors in angles ($\Delta \theta$) must always be converted to radians. For small angles, $\sin \theta \approx \theta$ and $\cos \theta \approx 1$, but using $\cot \theta \Delta \theta$ is more precise.

Answer 1

Correct Answer: 0.46

Step 1: Understanding the Question:
The question asks for the fractional error in wavelength ($\Delta\lambda / \lambda$) using the formula for single-slit diffraction. We must combine the relative error in slit width and the error propagated through the angular measurement.

Step 2: Key Formula or Approach:

• Condition for first minimum: $a \sin \theta = \lambda$.

• Wavelength: $\lambda = a \sin \theta$.

• Error propagation: $\frac{\Delta\lambda}{\lambda} = \frac{\Delta a}{a} + \frac{\cos \theta \Delta \theta}{\sin \theta} = \frac{\Delta a}{a} + \cot \theta \Delta \theta$.

Step 3: Detailed Explanation:

• Slit width error:
$a = 0.016$ mm, $\Delta a = 0.002$ mm.
$\frac{\Delta a}{a} = \frac{0.002}{0.016} = \frac{1}{8} = 0.125$.

• Angular error:
$\theta = 2^\circ$. $\Delta \theta = 40' = \frac{40}{60}$ degrees $= \frac{2}{3}^\circ$.
To use in the formula, convert $\Delta \theta$ to radians:
$\Delta \theta = \frac{2}{3} \times \frac{\pi}{180} = \frac{\pi}{270}$ rad $\approx 0.01163$ rad.

• Calculate $\cot \theta$:
Given $\sin 2^\circ = 0.035$, then $\cos 2^\circ = \sqrt{1 - (0.035)^2} \approx 0.9994$.
$\cot 2^\circ = \frac{0.9994}{0.035} \approx 28.55$.

• Total Fractional Error:
$\frac{\Delta\lambda}{\lambda} = 0.125 + (28.55 \times 0.01163)$.
$\frac{\Delta\lambda}{\lambda} = 0.125 + 0.332 = 0.457$.
Rounding off to two decimal places, we get $0.46$.

Step 4: Final Answer:
The fractional error in the measurement of wavelength is 0.46.