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 \(6 \pi^{5}\).

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