Coulomb repulsion in an electron beam Question (Verbatim)
Two narrow slits $S_{1}$ and $S_{2}$, each 1cm wide, positioned in a vacuum chamber are used to 'sharpen' an electron beam of energy 400eV (which is essentially the value of the kinetic energy of each electron in the beam). At which distance $z$ from the slit $S_{2}$ will the width of the electron beam double due to coulomb repulsion of the electrons? Assume the electron current per unit length along the slit $S_{2}$ is equal to $10^{-4}$ A/cm^2. You may assume the slits are infinitely long.
Solution Attempt
This problem has been a mess.
Let's say the slits extend infinity in the x-direction. The electrons are initially traveling in the z-direction. Let's define an "envelop" function for the electron beam, $f(z)$, such that the y-coordinate of the boundaries of the electron beam for a given $z$ would be from $y = -f(z)$ to $y = +f(z)$. We know $f(z=0)$ = 1cm. We're interested in finding $z'$ such that $f(z')$ = 2cm.
Coulomb force is: $F = \frac{q_1 q_2}{R^2}$
The force on some element $F(x,y,z) = \int_{0}^{\infty}\int_{-\infty}^{\infty}\int_{-f(z)}^{f(z)} \frac{q(x,y,z) \rho(x',y',z')}{(x-x') + (y-y') + (z-z')} dy' dx' dz'$
From here I could try integrating the force to get velocity (given a starting velocity of the electrons $v_{0}$, I might say since $z = v_0 t$ that $t = \frac{z}{v_0}$, so I could integrate force with respect to $\frac{z}{v_0}$ instead of $t$...), but I'm not sure how to handle using the velocities to figure out how the charge distribution $\rho(x,y,z)$ evolves or how to ultimately get $f(z)$ back out of that.
So far as i can tell, this form isn't really useful for anything, unless I want to solve for the force on each electron $q(x,y,z)$ individually then iteratively solve for velocity and position (in a computer simulation).
I've thought of using energy conservation in this, but that seems like it will run up to similar issues.
Might anyone point me in a more fruitful direction?
 A: Use symmetry. The slit is infinitely long in the x direction. Take a strip across the slit in the y direction. Everything is the same on both sides of the strip. so there is no net force in the x direction. 
Each electron has a fixed kinetic energy from a high velocity in the z direction. The velocity is fixed. Forces in the z direction are not part of the problem.
The only thing that changes is the y direction. The strip stretches. It is a 1D problem. Figure out the coulomb forces within a strip. 
It may help to give the strip a small rectangular cross section dx by dz.
A: Suppose the electron beam widens — e.g. due to the electrostatic repulsion of the electrons composing the beam.
Seen from above, the beam profile might thus resemble a funnel, the narrowest part of the funnel being, of course, the slit $S_2$.
Consider the plane containing the slit $S_2$, and another plane $P_2$, parallel to the aforementioned plane, and located, say, 2cm in front of it.
Consider the intersection of plane $P_2$ and the beam.
Consider the outermost electron in that intersection with $P_2$, on the left side (There's of course also an outermost electron on the right side of the beam, but, due to the problem's axial symmetry, a discussion of what happens on the left side of the beam is sufficient.)  We'll name that electron $e_{Leftmost2}$.
$e_{Leftmost2}$ will experience a repulsive force from all the other electrons located right of it; that repulsive force will tend to push $e_{Leftmost2}$ towards the left, and the beam might thus tend to widen.
One potential issue complicating the problem is that, if we now consider a plane $P_3$ located 3cm away from the slit plane, it's likely that the beam intersection with $P_3$ — or the beam width 3cm away from the slit plane, if you prefer — will be wider than the beam 2cm away from the slit.
That beam intersection with plane $P_3$ will also have an outermost electron on the left side.  Let's name that electron $e_{Leftmost3}$.
Consider the electrons that are in front of $e_{Leftmost2}$.  One of them will be $e_{Leftmost3}$.
As we assume the beam is widening, $e_{Leftmost3}$ must be in front, and on the left of $e_{Leftmost2}$.
This means that $e_{Leftmost3}$ will exert a repulsive electrostatic force pushing $e_{Leftmost2}$ towards the right.
So, it seems that, for a complete treatment of the balance of electrostatic forces for one particular electron, one should consider the electrons that are left and right of said electron, taking into account that these electrons might be distributed both behind (smaller $z$) and in front (larger $z$) of the particular electron we're considering.
The balance of electrostatic forces pushing a particular electron left or right will depend on the distance between the electrons, and therefore on the width of the beam.
The width of the beam at a certain distance $z$ in front of the slit plane will be a consequence of the balance of the left-pushing and right-pushing electrostatic forces exerted on the electrons while they are travelling the distance from slit $S_2$ to $z$.
Thus, the beam width (or its profile's curvature, if you will) depends on the electrostatic forces, and the electrostatic forces depend on the beam width.
This kind of interdependency is a bit reminiscent of the non-linear partial differential equations (the Einstein Field Equations) encountered in General Relativity: mass and energy tell spacetime how to curve, and the curvature of spacetime, in turn, recursively, influences the trajectories of mass.
This is getting complicated (^^;
