hi can i really get some help on this one. Thankssssss.
When electrons pass through a single slit, they form a diffraction pattern. The central bright fringe extends to either side of the midpoint, according to an angle è given by sin =ë/W, where ë is the de Broglie wavelength. of the electron and W is the width of the slit. When ë is the same size as W, ë=90o, and the central fringe fills the entire observation screen. In this case, an electron passing through the slit has roughly the same probability of hitting the screen either straight ahead or anywhere off to one side or the other. Now, imagine yourself in a world where Planck's constant is large enough so you exhibit similar effects when you walk through a 0.90m doorway. Your mass is 82kg and you walk at a speed of 050m/s. How large would Planck's constant have to be in this hypothetical world?
Divide your momentum by the momentum of the electron. Plank's constant would have to be larger by this factor.
However, this problem is flawed for the following reason: Dimensionful constants like Planck's constant, the speed of light, the gravitational constant are not fundamental constants at all. They appear in equations of physics simply because our choice of units.
Basically, Planck's constant, the speed of light and the Gravitational constant are just conversion factors to convert kilograms, meters and seconds into each other. You can define a new set of units in which Planck's constant= speed of light = Gravitational constant = 1. Such units are often used in theoretical physics.
Changing the value that Planck's constant takes in SI units therefore simply amounts to changing the definition of SI units. So, in this problem, when you consider the world with the larger value for Plank's constant, that world is actually exactly the same as our world. All that has changed is that the definition of the kilogram, meter and second have changed.
The momentum of an object of 82 kilogram and 0.5 m/s now corresponds to the momentum of an electron.
To calculate the value of Planck's constant in this hypothetical world, we need to divide the momentum of the object by the momentum of an electron.
The momentum of an object is given by the formula: momentum = mass x velocity.
In this case, the mass is given as 82 kg and the velocity is given as 0.5 m/s.
So, the momentum of the object is: momentum_object = 82 kg x 0.5 m/s.
Next, we need to find the momentum of an electron. The momentum of an electron is given by the formula: momentum_electron = mass_electron x velocity_electron.
The mass of an electron is approximately 9.11 x 10^-31 kg, and the velocity of an electron is approximately the speed of light, which is 2.998 x 10^8 m/s.
So, the momentum of an electron is: momentum_electron = 9.11 x 10^-31 kg x 2.998 x 10^8 m/s.
Now, to find the factor by which Planck's constant would have to be larger, we divide the momentum of the object by the momentum of an electron:
factor = momentum_object / momentum_electron.
This gives us the factor by which Planck's constant would have to be larger in this hypothetical world.
However, as mentioned earlier, changing the value of Planck's constant in SI units is essentially changing the definition of the kilogram, meter, and second. So, in this problem, the world with the larger value of Planck's constant is actually the same as our world, but with a different definition of the units.