Two infinitely long slits, S1 and S2, each of width d=0.1 cm are used to prepare a box-shaped electron beam with energy E=400 eV. At what distance x from the slit S2 will the thickness of this electron beam double due to Coulomb's repulsion? The linear density of electron current (as measured right after the slit S2) is i=.0001 A/cm. (Hint: linear density of current i is the total electron current I divided by the length of the slit L. Consider the electron beam as an infinitely large plane of charge, in such conditions surface charge density can be found as i/v, where v is the electron's velocity after the slit S2).
To find the distance from the slit S2 at which the thickness of the electron beam doubles due to Coulomb's repulsion, we need to calculate the surface charge density and use it to determine the location where the thickness doubles.
1. Calculate the surface charge density:
The linear density of the electron current (i) is given as 0.0001 A/cm. This represents the total electron current (I) divided by the length of the slit (L).
i = I/L
2. Determine the electron's velocity after the slit S2:
The linear density of current (i) is also equal to the surface charge density (σ) multiplied by the electron's velocity after the slit (v).
i = σ * v
3. Rearrange the equation to solve for v:
v = i / σ
4. Calculate the velocity of the electrons (v) using the given values:
v = 0.0001 A/cm / σ
5. Once you have the velocity (v), you can use it to find the distance x from the slit S2 at which the thickness of the electron beam doubles.
To do this, consider the Coulomb's repulsion acting on the electron beam. As the electrons repel each other, the thickness of the beam tends to expand. At a distance x from the slit S2, if the thickness doubles, the repulsion between electrons must have influenced the beam at that point.
The repulsion between electrons can be calculated using Coulomb's Law:
F = k * ((q1 * q2) / r^2)
Where:
- F is the electrostatic force between the two electrons
- k is Coulomb's constant
- q1 and q2 are the charges of the electrons
- r is the distance between the electrons
6. Set up an equation for the repulsive force. Considering two adjacent electrons in the electron beam, they will experience Coulomb's repulsion force:
F = ((e^2) / (4πε0 * r^2))
Where:
- e is the charge of one electron
- ε0 is the permittivity of free space
7. Find an expression for the electric field E at a distance r from the slit S2. This electric field results from the repulsive force between the electrons:
E = F / e = ((e^2) / (4πε0 * r^2)) / e
8. The electric field (E) created by a charged particle is given by:
E = Electric potential (V) / Distance (r)
9. Rearrange the equation to solve for the distance (r) in terms of the electric potential (V):
r = sqrt((e^2) / (4πε0 * V))
10. Using the electron's velocity (v) from step 4 and considering the electric potential (V) as the electron's kinetic energy (E), substitute the values into the equation from step 9 to find the distance (r):
r = sqrt((e^2) / (4πε0 * E))
11. Convert the distance (r) to a displacement (x) by subtracting the distance between the two slits (d):
x = r - d
Now, plug in the appropriate values for the known variables (e, ε0, E, and d) to find the value of x, which represents the distance from the slit S2 at which the thickness of the electron beam doubles due to Coulomb's repulsion.