Q5DQ - University Physics with Modern Physics

Question

In a diffraction experiment with waves of wavelength $\mathit{\lambda }$, there will be no intensity minima (that is, no dark fringes) if the slit width is small enough. What is the maximum slit width for which this occurs? Explain your answer.

Step-by-Step Solution

Answer

When maximum slit width is equal to wavelength of light, there will be no minima.

Step 1: Define single slit diffraction

When monochromatic light sent through a narrow slit, produces a diffraction pattern on a distant screen which is single slit diffraction.

Condition of dark fringes (destructive interference) in single slit diffraction is as follow,

 $\mathbf{asin\theta }\mathbf{=}\mathbf{n\lambda }$ 

Where   is slit width, $\mathbf{\theta }$ is the diffraction angle of minima;  is the order of minima and   is wavelength of the light.

Step 2: Determine no minima condition

If the diffraction angle for 1^st minima is $90°$ then there will be no minima in field of view as for 1^st minima diffracted light become parallel to slit and screen.

No minima occur when $\theta =90°$ .  From this one can say,

$$\begin{gathered} \mathrm{asin}90°=1\times \mathrm{\lambda } \\ \mathrm{a}=\mathrm{\lambda } \end{gathered}$$  

The above equation is the criterion for “no-minima”.

Therefore, Maximum slit width for which there will be no minima is equal to the wavelength of light.