Understanding the relationship between electric energy and force

Question

I'm trying to do the following problem:

One of the three types of radioactive decay is "β decay", during which protons decay into neutrons or vice­versa, emitting either electrons (β­) or positrons (β+) at high velocity as a result, and neutrinos. In one experiment, a β­ source and β+ source are placed 10 cm apart from each other. At a certain time, both sources decay simultaneously, with the electron being emitted along the $x$ ­axis and the positron being emitted along the $y$ ­axis (i.e. the paths of the two particles are at a right angle). Both particles are emitted with 5 keV(kilo­electron volts) of kinetic energy and start on the $x$ ­axis. What is magnitude and direction of the total force on each particle? (Do not ignore electric force, assume electrons and positrons have the same mass.)

I attempted to solve this using conservation of energy. Initially, we have both kinetic and potential energy. Kinetic energy is given to us and we can find electric potential energy. I'm assuming the total final energy (kinetic + potential) is zero since Vf = 0 and Uf = 0 (since distance between the two particles gets "infinitely" long). I then tried to use the equation

$\Delta E=Ui+Uf=-\int F.dr=> F=-\nabla E$

But I'm not sure how this would help. I feel like I'm missing something. The fact that the question states "don't ignore electric force" makes me think I have to use it, but then if I'm using the electric force formula what's the point of having the initial energy? Any help would be appreciated.

Answer 1

When they say "Do not ignore electric force", they mean that there is both a magnetic and an electric force on the electron/positron, and you should not forget the electric force.

In other words, you are asked to compute, for the $\vec v_+$, $q_+$ of the positron, the effect on the electron of its $\vec E$ and $\vec B$ field. Fortunately, 5 keV (kinetic energy of the positron) is much much smaller than 511 keV (mass of the positron), so we can use non-relativistic physics, which means that the positron's magnetic field is given by a quasi-Biot-Savart expression:

$\vec B_+ ~=~ \mu_0 ~ q_+ ~ (\vec v_+ \times \hat r) ~ /~ (4 ~ \pi ~ r^2) $

with $\hat r$ being the unit vector pointing from the positron to the field position, where the electron is. (If the particle speed gets higher, you need to start from the reference frame of the positron, where you've got a well-behaved electric field, and Lorentz-transform the field to the new reference frame.)

Similarly the electric field is just:

$\vec E_+ ~=~ q_+ ~/~ (4 ~ \pi ~ \epsilon_0 ~ r^2) $

Computing the Lorentz force on the electron $\vec F_- ~=~ q_- ~ (\vec E_+ ~+~ \vec v_- \times \vec B_+)$ then gives you the force you're looking for.

You can attempt to use the Coulomb potential field to calculate the energy, but the derivative will simply give you Coulomb's law, so why bother?