Question

Red light of wavelength of 700 nm falls on a double slit separated by 400 nm. (a) At what angle is the first-order maximum in the diffraction pattern? (b) What is unreasonable about this result? (c) Which assumptions are unreasonable or inconsistent?

Short answer

(a) There will be no first-order maximum in the spectrum.

(b) It produces no viable solution for $\mathrm{\theta }$between zero and$90.0°$, implying that there will be no spectrum.

(c) The $\mathrm{d}$ must be bigger than or equal to$\mathrm{\lambda }$in order to produce fringes.

Step-by-step solution

Concept Introduction

The wavelength of a waveform signal conveyed in space or down a wire is the distance between identical points (adjacent crests) in adjacent cycles.

Given data

The distance between the two slits is $\mathrm{d}=400\hspace{0.33em}\mathrm{nm}\left(\frac{{10}^{‐9}\mathrm{m}}{1\mathrm{nm}}\right)=4.00\times {10}^{‐7}\hspace{0.33em}\mathrm{m}$

The wavelength of the light is $\mathrm{\lambda }=700\hspace{0.33em}\mathrm{nm}\left(\frac{{10}^{‐9}\mathrm{m}}{1\mathrm{nm}}\right)=7.00\times {10}^{‐7}\hspace{0.33em}\mathrm{m}$

Find angle is the first-order maximum in the diffraction pattern (a)

The equation for the constructive interference can be given as

$\mathbf{dsin\theta }\mathbf{=}\mathbf{m\lambda }$…………………..(1)

Solve equation (1) for $\mathrm{\theta }$ and substitute the value of m=1 and $\mathrm{\lambda }$, we get

$$\begin{gathered} \mathrm{\theta }={\sin }^{‐1}\left(\frac{\mathrm{m\lambda }}{\mathrm{d}}\right) \\ ={\sin }^{‐1}\left(\frac{1\times 7.00\times {10}^{‐7}\mathrm{m}}{4.00\times {10}^{‐7}\mathrm{m}}\right) \\ ={\sin }^{‐1}\left(1.75\right) \end{gathered}$$

There is no way to solve it since the sine function can have values in the range of

$\left[0_1\pm 1\right]$

Therefore, the value of $\mathrm{\theta sin}$ cannot exceed $1$.

Explain unreasonable about this result (b)

It gives no reasonable solution for$\mathrm{\theta }$between zero and $90.0°$, implying that no formed fringes can exist.

Therefore, it produces no viable solution for $\mathrm{\theta }$between zero and $90.0°$, implying there will be no spectrum.

Write assumptions that are unreasonable or inconsistent (c)

The $\mathrm{d}$must be greater than or equal to$\mathrm{\lambda }$in order to produce fringes.

Therefore,$\mathrm{d}$must be bigger than or equal to $\mathrm{\lambda }$in order to produce fringes.