Question

Electrons are accelerated through a potential difference of 100V and then shot through a slit of width 100 nm. They diffract and form a pattern on a screen 0.5 m distant from the slit. How far apart are the two dark spots on either side of the bright central spot in the pattern?

PLEASE SHOW ALL WORK AND EXPLANATIONS .(answer is 1.2 mm)

Answer 1

energy of the electrons=e*V

where V=potential through which it is energized

then 0.5*m*v^2=e*V

where m=mass of the electron

v=speed of electron

==>v=sqrt(2*e*V/m)

then momentum=m*v=sqrt(2*e*V*m)

wavelength of the electron wave=h/momentum

where h=planck's constant

==>wavelength=lambda=h/sqrt(2*m*e*V)

then distance between first two dark spots on either side of central maxima=2*lambda*D/d

where D=screen distance=0.5 m

d=slit width=100 nm=10^(-7) m

then distance between first two dark spot=2*h*D/(d*sqrt(2*m*e*V))

=6.626*10^(-34)*0.5/(10^(-7)*sqrt(2*9.1*10^(-31)*1.6*10^(-19)*100))=0.0012279 m

=1.2279 mm