A single slit diffraction experiment is performed to determine the slit width using the equation, bd /D =mλ , where b is the D slit width, D the shortest distance between the slit and the screen, d the distance between the m^th diffraction maximum and the central maximum, and λ is the wavelength. D and d are measured with scales of least count of 1 cm and 1 mm, respectively. The values of λ and m are known precisely to be 600 nm and 3, respectively. The absolute error (in μm) in the value of b estimated using the diffraction maximum that occurs for m = 3 with d = 5 mm and D = 1 m is

The correct answer is 94.50.

To find the absolute error in the estimated slit width $b$, we can use the given equation for single-slit diffraction:
$\frac{bd}{D}=m\lambda$
Rearranging the formula to solve for the slit width $b$:
$b=\frac{m\lambda D}{d}$

The parameters and their values given in the problem are:
- Wavelength of light ($\lambda$) = 600 nm = $600\times {10}^{-9}\text{ m}$ (known precisely)
- Diffraction order ($m$) = 3 (known precisely)
- Distance to screen ($D$) = 1 m, measured with a scale of least count 1 cm ($\Delta D=0.01\text{ m}$)
- Distance of maximum from central maximum ($d$) = 5 mm = $0.005\text{ m}$, measured with a scale of least count 1 mm ($\Delta d=0.001\text{ m}$)

First, we calculate the nominal value of $b$:
$b=\frac{3\times (600\times {10}^{-9}\text{ m})\times 1\text{ m}}{0.005\text{ m}}=3.6\times {10}^{-4}\text{ m}=360\text{ μm}$

Now, let's analyze the absolute error in $b$ using the two standard approaches:

Approach 1: Maximum Absolute Deviation Method
To find the maximum possible calculated value of $b$, we take the maximum value of the numerator and the minimum value of the denominator:
$D_{\text{max}}=D+\Delta D=1+0.01=1.01\text{ m}$
$d_{\text{min}}=d-\Delta d=5-1=4\text{ mm}=0.004\text{ m}$
Calculating the maximum possible value of $b$:
$b_{\text{max}}=\frac{3\times (600\times {10}^{-9}\text{ m})\times 1.01\text{ m}}{0.004\text{ m}}=4.545\times {10}^{-4}\text{ m}=454.5\text{ μm}$
The maximum positive deviation (absolute error) in the measurement is:
$\Delta b_1=b_{\text{max}}-b_{\text{nominal}}=454.5\text{ μm}-360\text{ μm}=94.50\text{ μm}$

Approach 2: Fractional/Relative Error Approximation Method
Using the logarithmic differentiation method for error propagation:
$\frac{\Delta b}{b}=\frac{\Delta D}{D}+\frac{\Delta d}{d}$
Substituting the given experimental uncertainties:
$\frac{\Delta b}{b}=\frac{1\text{ cm}}{100\text{ cm}}+\frac{1\text{ mm}}{5\text{ mm}}=0.01+0.20=0.21$
This gives the absolute error:
$\Delta b=0.21\times 360\text{ μm}=75.60\text{ μm}$

Thus, depending on the error model used, the absolute error is estimated to be either 75.60 μm or 94.50 μm (representing the maximum possible deviation from the nominal value).