$("$Planck$\text{'}$s constant from Feynman's double$-$slit experiment"$)$ In Feynman's double$-$slit experiment we discussed in class, a weak radioactive source emits electrons, one at a time. The electrons are passing through a very narrow slit, on which they diffract. The diffracted electrons next hit two other narrow slits, at equal distance from the first slit. The separation between the slits is $\backslash ($ d $\backslash ).$ In a modern version of this experiment, we detect an interference pattern with a flat sheet of silicon, sliced into very small pixels and instrumented so that it's acting like a CCD camera. The screeen with the two slits is parallel with the plane of the CCD camera, and the distance between them is $\backslash ($ L $\backslash ),$ where $\backslash ($ L $\backslash$gg d $\backslash ).$

$($a$)$ We first need to measure the momentum of the electrons emitted by the radioactive source. $($The electrons are a product of radioactive decays, and have a fixed kinetic energy.$)$

To do that, we remove the screens with slits, and turn on a magnetic field $\backslash (\backslash$vec$\{$B$\}\backslash )$ perpendicular to the direction of the electron beam. From the deflection of the electron beam we measure the radius of the curvature of its trajectory, $\backslash ($ R $\backslash ).$ What is the momentum of the electrons?

$($b$)$ What is deBroglie wavelength of the electrons?

$($c$)$ Next, we reinstall the screens and perform Feynman's experiment. We observe the "central maximum" $($corresponding to no phase difference$)$ at the mid$-$point between the slits, projected onto the plane of the CCD camera. The first interference maximum is observed at distance $\backslash (\backslash$Delta y$(\backslash$Delta y $\backslash$ll L$)\backslash ).$ What is the phase condition for a constructive interference corresponding to the first maximum?

$($d$)$ Determine the Planck's constant from a combination of these two experiments.