The electric potential inside a 10-m-long linear particle accelerator is given b

Question

The electric potential inside a 10-m-long linear particle accelerator is given by
V = 3150 V - (3 V/m2) x2, where x is the distance from the left plate along the accelerator tube.
(a) Determine an expression for the electric field along the accelerator tube. (Use the following as necessary: x.
(b) A proton is released (from rest) at x = 4 m. Calculate the acceleration of the proton just after it is released. (Enter the magnitude of the acceleration.)
(c) What is the impact speed of the proton when (and if) it collides with the plate? (If the proton does not collide with the plate, enter 0.)

Explanation / Answer

a) Electric field is the negative gradient of potential. Take the derivative dV/dx. E = -dV/dx. b) F = ma and also F = qE. Use F = qE to calculate the force knowing q and E, and F = ma to calculate the acceleration. c) Energy E from being accelerated through a potential difference V is E = qV. Calculate the potential at x = 0 and at the other end (x = 10). The difference between those numbers is the potential difference that accelerates the proton. Once you know its energy, you can use (1/2)mv^2 to get the speed.

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