How do I calculate how strong of a field I need to repel a proton in a span of time

How do I calculate how strong of a field I need to repel a single proton in a given span of time, given that I know it's velocity.

• By repel, do you mean bring it to a stop? – Skawang Jul 18 at 5:37

The proton can be repelled using a static, uniform E-field of magnitude $$E$$. Assuming the E-field is in the opposite direction to that of the proton velocity, by using Newton's 2nd law and the Lorentz force law, $$qE = F = ma = m \frac{v}{t}$$ $$E = \frac{mv}{qt}$$ where $$m$$ is the proton mass, $$q$$ is the elementary charge, $$v$$ is the initial velocity and $$t$$ is the time within which the proton must be stopped after "entering" the E-field.
The proton can be "repelled" by not stopping it, but redirecting it using a magnetic field. Let a uniform magnetic field of magnitude $$B$$ be orthogonal to the initial velocity of the proton. After entering the magnetic field, the proton moves in a circle. Equating the centripetal force for this motion to the Lorentz force due to the magnetic field, $$qvB = \frac{mv^2}{r}$$ $$r = \frac{mv}{qB}$$ where $$r$$ is the radius of the circle. The proton will move a quarter of a full circle (a distance of $$\frac{1}{2}\pi r$$) before the velocity component in the original direction is zero. The magnitude of the velocity during this motion will not change, so we have $$vt = \frac{1}{2}\pi r = \frac{\pi m v}{2 q B}$$ $$B = \frac{\pi m}{2 q t}.$$