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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.

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  • $\begingroup$ By repel, do you mean bring it to a stop? $\endgroup$ – Skawang Jul 18 at 5:37
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I will assume that by repel, you mean "reduce the velocity component in the original direction to zero." You can modify the following arguments accordingly if this is not the case.

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 $$ vt = \frac{1}{2}\pi r = \frac{\pi m v}{2 q B} $$ $$ B = \frac{\pi m}{2 q t}. $$

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