In a single slit diffraction experiment, a slit of width (0.016 ± 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° ± 40′). The value of the fractional error in the measurement of wavelength is:
(Given: sin(2°) = 0.035)

The correct answer is 0.46.

In a single slit diffraction experiment, the condition for the first minimum is:

d·\sin \left(\theta \right)=\lambda

where d is the slit width, θ is the angular position of the first minimum (measured from the central maximum), and λ is the wavelength. Rearranging for wavelength:

\lambda =d·\sin \left(\theta \right)

Step 1: Identify given values and their uncertainties

Slit width: d=0.016\text{mm}, with \mathrm{\Delta d}=0.002\text{mm}
Angular position: \theta =\mathrm{2°}, with \mathrm{\Delta \theta }=40\prime (40 arcminutes)
\sin \left(\mathrm{2°}\right)=0.035 (given)

Step 2: Write the fractional error formula

Taking the logarithm of \lambda =d·\sin \left(\theta \right) and differentiating, the fractional error in \lambda is:

\frac{\mathrm{\Delta \lambda }}{\lambda }=\frac{\mathrm{\Delta d}}{d}+\frac{\Delta }{\left(}

Since \Delta \left(\sin \theta \right)=\cos \left(\theta \right)·\mathrm{\Delta \theta }, the second term becomes:

\frac{\Delta }{\left(}=\frac{\cos }{\left(}

Step 3: Calculate the fractional error from slit width

\frac{\mathrm{\Delta d}}{d}=\frac{0.002}{0.016}=0.125

Step 4: Convert Δθ to radians

40 arcminutes must be converted to radians (since calculus-based error propagation requires radians):

\mathrm{\Delta \theta }=40\prime =\frac{40}{60}°=\frac{40}{60}\times \frac{\pi }{180}\text{rad}=\frac{40}{10800}\pi \text{rad}=\frac{\pi }{270}\text{rad}\approx 0.01164\text{rad}

Step 5: Calculate the fractional error from angular measurement

Since \theta =\mathrm{2°} is very small, \cos \left(\mathrm{2°}\right)\approx 1. Therefore:

\frac{\cos }{\left(}\approx \frac{\mathrm{\Delta \theta }}{\sin }=\frac{0.01164}{0.035}\approx 0.3326

Step 6: Calculate the total fractional error

\frac{\mathrm{\Delta \lambda }}{\lambda }=0.125+0.3326\approx 0.4576

Rounding to two decimal places:

\frac{\mathrm{\Delta \lambda }}{\lambda }\approx 0.46

Therefore, the fractional error in the measurement of wavelength is 0.46. The dominant contribution (about 72%) comes from the uncertainty in the angular measurement, while the uncertainty in slit width contributes the remaining 27%, highlighting how critical precise angular measurement is in such experiments.