# Simon Says with protons and electrons

An electron is accelerated by a potential difference of $$U_e= 1~\mbox{mV}$$. It then enters a region with an inhomogeneous magnetic field $$\vec{B}(x,y,z)$$ generated by a system of coils carrying currents $$I_{1}, I_{2}\ldots$$.

We then perform a similar experiment with a proton. We first reverse the current in all the coils generating the magnetic field. We then accelerate the proton with a potential difference of $$U_p$$ and it enters the region with the magnetic field. What must $$U_p$$ be in Volts so that the trajectory of the proton is the same as that of the electron? Don't forget any sign changes!

Hint: Compute $\frac{d}{ds} \frac{\vec{v}}{v},$ where $$s$$ is the arc length along the trajectory:

$$s=\int_0^t v(t') dt'$$.

Details and assumptions

The proton to electron mass ratio is approximately $$6 \pi^{5}$$.

×