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. 187
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? 402