Lecture 38

Readings

http://umdberg.pbworks.com/w/page/73088900/Two-slit%20interference%20%282013%29

http://umdberg.pbworks.com/w/page/125895722/Workout%3A%20Two-slit%20interference

https://phet.colorado.edu/sims/html/wave-interference/latest/wave-interference_en.html

http://umdberg.pbworks.com/w/page/53179895/Diffraction

http://umdberg.pbworks.com/w/page/53181746/Interference%20from%20two%20wide%20slits

2 - Slit Interference

• Superposition principle - 2 waves can propagate through each other , and at any point , they add
• Destructive interference is a consequence of negative part of wave adding to a positive part
• Geometry :

• $Y=L\ast tan(\theta )$$Y = L * tan(\theta)$ , but if $\theta$$\theta$ is small ( means << L ) , then $tan(\theta )\approx \theta$$tan(\theta) \approx \theta$ and $y=L\ast \theta$$y = L * \theta$

  • $\theta$$\theta$ is in radians!
  • Also , $\mathrm{\Delta }r=r_2-r_1$$\Delta r = r_2 - r_1$

• Remember :

  • If $\mathrm{\Delta }r=0,\lambda ,2\lambda ,3\lambda ,etc=n\ast \lambda$$\Delta r = 0 , \lambda , 2\lambda , 3\lambda , etc = n * \lambda$

    • Then constructive interference
    • Crests add to crest
    • Bright

  • If $\mathrm{\Delta }r=\frac{\lambda }{2},\frac{3\ast \lambda }{2},\frac{5\ast \lambda }{2},etc=(n+\frac{1}{2})\ast \lambda$$\Delta r = \frac{\lambda}{2} , \frac{3*\lambda}{2} , \frac{5 *\lambda}{2} , etc = \left(n + \frac{1}{2}\right) * \lambda$

    • Then Destructive interference
    • Crests adds to trough
    • Dark

• So if you know $\mathrm{\Delta }r$$\Delta r$ to a point on the screen , you can say if bright or dark

• Doing trigonometry , $\mathrm{\Delta }r\approx \alpha \ast \theta$$\Delta r \approx \alpha *\theta$ ( for small $\theta$$\theta$ )

  • Eliminating $\theta$$\theta$ gives : $y=\frac{\mathrm{\Delta }r\ast L}{\alpha }$$y = \frac{\Delta r * L}{\alpha}$

   

• Constructive Rule = $y=\frac{n\ast \lambda \ast L}{\alpha }$$y = \frac{n * \lambda * L}{\alpha}$

• Destructive Rule = $y=\frac{(n+\frac{1}{2})\ast \lambda \ast L}{\alpha }$$y = \frac{\left(n + \frac{1}{2}\right) * \lambda * L}{\alpha}$

• Above equations are used to find location of bright and dark parts on the screen

Diffraction

• Interference of 1 part of the slit with another part of the same slit

• When top of slit and middle of slit have correct geometry for destructive interference , entire slit will destructively interfere with itself $\to$$\rightarrow$ produce dark region

• From previous equation for destructive interference, but use $\frac{d}{2}$$\frac{d}{2}$ in place of $\alpha$$\alpha$ and 1st minimum at n = 0 :

  $$y=\frac{\lambda \ast L}{2\ast \frac{d}{2}}=\frac{\lambda \ast L}{2}$$

• This is location on screen ( measured from mid point ) of the 1st "squash point" ( dark regions ) due to diffraction

• Note : the "central bright region" in diffraction gets wider as $d$$d$ gets smaller

• Note : if opening is very large ( think doorway or other macroscopic object )

  • Then the "bright region" will match up with ray prediction ( normal shadows )

    • You would only see diffraction effects at the edge of the doorway

      

  • And only if the edge of doorway sharp and light source is small and monochromatic ( single wavelength )

Clicker Questions