What is the effect on a diffraction pattern when reducing the width of the slit?

[hidemi]

Homework Statement

The width of single slit slowly reduced. As a result?

Relevant Equations

a*sinθ = mλ

Monochromatic light is normally incident on a diffraction grating. The mth order line is at an angle of diffraction angle θ and has width w. A wide single slit is now placed in front of the grating and its width is then slowly reduced. As a result:
A. both θ and w increase
B. both θ and w decrease
C. θ remains the same and w increases
D. θ remains the same and w decreases
E. θ decreases and w increases

Ans: C
I think I can use the equation above but I don't know where to start.
Thanks

 

[BvU]

What is it that changes if a wide slit is placed in front of the grating ? Compare with here

 

[hidemi]

What is it that changes if a wide slit is placed in front of the grating ? Compare with here

I saw your link, but I still cannot relate it to the question. Could you explain more clearly, please?

 

[BvU]

That would be telling.
What does you relevant equation tell you about $\theta$ ?
In the (four!) pictures, what has a relation with w ? (hint: does not feature in your equation)

 

[hidemi]

That would be telling.
What does you relevant equation tell you about $\theta$ ?
In the (four!) pictures, what has a relation with w ? (hint: does not feature in your equation)

I still don't understand :(

 

[Steve4Physics]

I saw your link, but I still cannot relate it to the question. Could you explain more clearly, please?

@BvU's link is extremely useful if you read through all of it.

A simple optical diffraction grating has hundreds of 'lines' each mm. Each grating-line is basically a very narrow slit.

If you cover the grating with a 'wide single slit' you leave only a small number of grating-lines exposed, e.g. 50 lines. When you shine light at the grating, the diffraction pattern is produced by only 50 grating-lines.

If you reduce the width of the 'wide single slit', this reduces the number of grating-lines exposed, e.g. to 20 lines. When you shine light at the grating, the diffraction pattern is now produced by only 20 grating-lines.

Read through @BvU's link and see how the changing the number of lines affects the diffraction pattern.

 

[Gordianus]

Again. Can you write the relationship between slit width w, wavelength $\lambda$ and diffraction order m?

 

[BvU]

I still don't understand :(

Does the wide slit appear in your relevant equation ?

 

[hidemi]

@BvU's link is extremely useful if you read through all of it.

A simple optical diffraction grating has hundreds of 'lines' each mm. Each grating-line is basically a very narrow slit.

If you cover the grating with a 'wide single slit' you leave only a small number of grating-lines exposed, e.g. 50 lines. When you shine light at the grating, the diffraction pattern is produced by only 50 grating-lines.

If you reduce the width of the 'wide single slit', this reduces the number of grating-lines exposed, e.g. to 20 lines. When you shine light at the grating, the diffraction pattern is now produced by only 20 grating-lines.

Read through @BvU's link and see how the changing the number of lines affects the diffraction pattern.

Thanks for explaining it!
However, your explanation sounds like D is the correct answer, instead of C. Could you explain further? Thank you

 

[Steve4Physics]

Thanks for explaining it!
However, your explanation sounds like D is the correct answer, instead of C. Could you explain further? Thank you

Look at @BvU's link: http://hyperphysics.phy-astr.gsu.edu/hbase/phyopt/mulslid.html

Suppose a 'single wide slit' of width L is placed against a diffraction grating so 5 grating-lines of the diffraction grating are left exposed.. The diffraction image produced is shown by the bottom image in the link - the diagram labelled 'Five Slit Diffraction'.

Note there are five sharp 'maxima’ in the central region (the 0-th order line, two 1-st order lines and two 2nd order lines). Each maximum has width w, and w is quite small because the maximum is narrow.

Now reduce L so only 2 grating-lines on the diffraction grating are exposed. The diffraction image produced is shown on the diagram labelled 'Double Slit Diffraction'. You can easily see the five maxima are now wider but at the same angular positions. The value of $\theta$ for each maximum is unchanged but its width (w) has increased.

Reducing L leaves θ unchanged and increases w - exactly matching choice C.

The real challenge is to understand why changing the number of slits changes w. To do this you have to have a good understanding of how multiple slits produce a diffraction image.

 

[hidemi]

Look at @BvU's link: http://hyperphysics.phy-astr.gsu.edu/hbase/phyopt/mulslid.html

Suppose a 'single wide slit' of width L is placed against a diffraction grating so 5 grating-lines of the diffraction grating are left exposed.. The diffraction image produced is shown by the bottom image in the link - the diagram labelled 'Five Slit Diffraction'.

Note there are five sharp 'maxima’ in the central region (the 0-th order line, two 1-st order lines and two 2nd order lines). Each maximum has width w, and w is quite small because the maximum is narrow.

Now reduce L so only 2 grating-lines on the diffraction grating are exposed. The diffraction image produced is shown on the diagram labelled 'Double Slit Diffraction'. You can easily see the five maxima are now wider but at the same angular positions. The value of $\theta$ for each maximum is unchanged but its width (w) has increased.

Reducing L leaves θ unchanged and increases w - exactly matching choice C.

The real challenge is to understand why changing the number of slits changes w. To do this you have to have a good understanding of how multiple slits produce a diffraction image.

Thank you so much for the explanation! I got it :)

 

[BvU]

The take away from a*sinθ = mλ or sinθ = mλ/a is:

small a $\Rightarrow$ big $\theta$​

or: small features $\Rightarrow$ big pattern​

and conversely.

That's why the single grating slit envelope is a wide feature. and -- in this exercise -- the wide slit has an effect on the smaller w.

It will come back later in Fourier transforms, Heisenberg uncertainty, signal processing and what have you.

$\$