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Chapter 16: Problem 2303

The width of a single slit, if the first minimum is observed at an angle of \(2^{\circ}\) with a wavelength of light \(6980 \AA\) is \(\mathrm{mm}\) (A) \(0.2\) (B) \(2 \times 10^{-5}\) (C) \(2 \times 10^{5}\) (D) \(0.02\)

Short Answer

Expert verified
The width of the single slit is calculated using the formula \(a = \dfrac{mλ}{sin(θ)}\), where \(a\) is the width of the slit, \(m = 1\) for the first minimum, \(λ = 6.98 × 10^{-7} m\) is the wavelength of light, and \(θ = 0.0349\) radians is the angle of the first minimum. Plugging in the values, we find the width, \(a = 0.02 mm\). Hence, the answer is (D) \(0.02 mm\).

Step by step solution

01

Understand the formula

The angular position of the first minimum in a single-slit diffraction pattern is given by the formula: \(asin(θ)=mλ/L\) where - \(a\) is the width of the slit - \(θ\) is the angle of the first minimum - \(m\) is the order of the minimum (in this case, \(m=1\) for the first minimum) - \(λ\) is the wavelength of light - \(L\) is the distance between the slit and the screen (Not needed in this problem, since we only care about the angular position) We need to find the value of \(a\) given the angle and wavelength of light.
02

Convert the angle to radians

The given angle is in degrees, which we need to convert into radians. To do this, use the following formula: \(θ_{rad} = (π/180) × θ_{degrees}\) In our case, \(θ_{degrees} = 2°\) \(θ_{rad} = (π/180) × 2 = (π/90) = 0.0349\, radians\)
03

Convert the wavelength to meters

The wavelength is given in Angstroms (\(Å\)), which needs to be converted to meters. There are \(10^{-10} m\) in \(1 Å\). So, \(λ = 6980 Å × (10^{-10} m/Å) = 6.98 × 10^{-7} m\)
04

Calculate the width of the slit

Now we have all the necessary information to calculate the width of the slit. From the minimum position formula, we can isolate \(a\): \(a = \dfrac{mλ}{sin(θ)}\) Then, by plugging in the values \(m = 1\), \(λ = 6.98 × 10^{-7} m\), and \(θ = 0.0349\, radians\), we have: \(a = \dfrac{(1)(6.98 × 10^{-7} m)}{sin(0.0349)} = 2 × 10^{-5} m\) To convert to millimeters, we have \(2 × 10^{-5} m × (10^{3} mm/m) = 0.02 mm\).
05

Choose the correct answer

Comparing our result with the given options, the correct answer is (D) 0.02 mm.

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Most popular questions from this chapter

$$ \begin{array}{|l|l|} \hline \text { Column - I } & \text { Column - II } \\ \hline \text { (i) Minimum deviation } & \text { (a) }(\mathrm{n}-1) \mathrm{A}+\left(\mathrm{n}^{\prime}-1\right) \mathrm{A}^{\prime}=0 \\ \text { (ii) Angular dispersion } & \text { (b) }\left[\left(\mathrm{n}_{\mathrm{v}}-\mathrm{n}_{\mathrm{r}}\right) /(\mathrm{n}-1)\right] \\ \text { (iii) Dispersive power } & \text { (c) }\left(\mathrm{n}_{\mathrm{v}}-\mathrm{n}_{\mathrm{r}}\right) \mathrm{A} \\ \text { (iv) Condition for no deviation } & \text { (d) }(\mathrm{n}-1) \mathrm{A} \\ \hline \end{array} $$ (A) \(i-c\), ii \(-d\), ii \(-b\), iv-a (B) \(i-a, 1 i-b\), iii \(-c\), iv-d (C) \(i-c\), ii \(-b\), ii \(-a\), iv-d (D) $\mathrm{i}-\mathrm{d}, \mathrm{i} i-\mathrm{c}, \mathrm{ii}-\mathrm{b}, \mathrm{iv}-\mathrm{a}$

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