Difference between revisions of "Diffraction"
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| − | '''Diffraction''' occurs ''“when [[waves]] pass through small openings, around obstacles, or past sharp edges. When an opaque object is placed between a point source of [[light]] and a screen, no sharp boundary exists on the screen between a [[shadow | + | '''Diffraction''' occurs ''“when [[waves]] pass through small openings, around obstacles, or past sharp edges. When an opaque object is placed between a point source of [[light]] and a screen, no sharp boundary exists on the screen between a [[shadow]]ed region and an illuminated region. The illuminated region above the shadow of the object contains alternating light and dark fringes. Such a display is called a '''diffraction pattern'''.” ''<ref>Halliday, Resnik, Walker: ''Fundamentals of Physics, 8th edition. p. 1200</ref> |
| + | |||
| + | ==Young's Double Slit Experiment== | ||
| + | |||
| + | Consider two slits a distance <math>d</math> apart and a screen a distance <math>R</math> from the slits. To find the positions of the maxima (brightest points) on the screen, the path difference from each of the slits, <math>\Delta</math> must be an [[integer]] number of wavelengths <math>n \lambda</math>. The the angle between a ray of light from one slit and the normal of that slit is <math>\theta</math>, then for constructive interference: | ||
| + | |||
| + | <math>d \sin{\theta} = n \lambda</math> | ||
| + | |||
| + | If <math>y</math> is the distance from the centre of the screen to a maxima, then <math>y= R \tan{\theta} </math>. | ||
| + | |||
| + | If we assume the screen is much further away than the slits, such that <math>R>>d</math> then using the small angle approximation, <math>\tan{\theta} \approx \sin{\theta}</math>, rearranging gives: | ||
| + | |||
| + | <math>y_n = \frac{nR \lambda}{d}</math> | ||
| + | |||
| + | where | ||
| + | :<math>y_n</math> is the distance on the screen to the n<sup>th</sup> maxima | ||
| + | :<math>\lambda</math> is the [[wavelength]] | ||
| + | |||
| + | For destructive interference, the path difference must be <math>(n+\frac{1}{2}) \lambda</math>. This gives: | ||
| + | |||
| + | <math>y_n = \frac{(n+ \frac{1}{2})R \lambda}{d}</math> | ||
| + | |||
| + | for the n<sup>th</sup> minima. | ||
==References== | ==References== | ||
<references /> | <references /> | ||
| − | ==See | + | ==See also== |
[[Double-slit experiment]] | [[Double-slit experiment]] | ||
| − | [[Category: | + | [[Category:Optics]] |
Latest revision as of 17:08, April 7, 2017
Diffraction occurs “when waves pass through small openings, around obstacles, or past sharp edges. When an opaque object is placed between a point source of light and a screen, no sharp boundary exists on the screen between a shadowed region and an illuminated region. The illuminated region above the shadow of the object contains alternating light and dark fringes. Such a display is called a diffraction pattern.” [1]
Young's Double Slit Experiment
Consider two slits a distance Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d} apart and a screen a distance Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} from the slits. To find the positions of the maxima (brightest points) on the screen, the path difference from each of the slits, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta} must be an integer number of wavelengths Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n \lambda} . The the angle between a ray of light from one slit and the normal of that slit is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \theta} , then for constructive interference:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d \sin{\theta} = n \lambda}
If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y} is the distance from the centre of the screen to a maxima, then .
If we assume the screen is much further away than the slits, such that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R>>d} then using the small angle approximation, , rearranging gives:
where
- is the distance on the screen to the nth maxima
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda} is the wavelength
For destructive interference, the path difference must be Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (n+\frac{1}{2}) \lambda} . This gives:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y_n = \frac{(n+ \frac{1}{2})R \lambda}{d}}
for the nth minima.
References
- ↑ Halliday, Resnik, Walker: Fundamentals of Physics, 8th edition. p. 1200