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If I have a proton in a uniform field between two parallel oppositely charged plates and the proton accelerates, the electric force acting on it stays constant seeing it is a uniform field and as a result the acceleration of the particle is also constant.

But does the electric force on the proton increase as it nears the negative plate? I would think yes seeing the distance is getting smaller (cf. Coulomb's law) so the electric force (which originates from Coulomb interactions between the charged particle and the plates) must be getting bigger. But this conflicts with the fact that the force is constant. Any help?

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Your brain/mind might be processing the information in the wrong way. As field is uniform, force remains constant and acceleration remains constant. You see, the acceleration remains constant, but velocity doesn't remain constant, as the time to which your proton is accelerated increases, your proton's velocity also increases. So, your proton will be drifting at higher and higher velocity as it goes nearer and nearer to your plate, even though force remains constant. Blame your brain for misdirection!

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NO the force does not change with distance

                             as  E=F/Q
                                 F=QE

electric field is constant and charge is also constant then force is also constant.

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