How to calculate velocity from the attraction of two charges?

I need to calculate the acceleration of an proton to a metal plate with a charge of 16 Coulombs (negative charge), 1 meter away. Then I need to calculate the velocity at the moment it passes the plate. Starting Velocity is 0 m/s. What is the velocity and what is the acceleration? This is not homework. When I did this, I got an acceleration of $1.376 \times 1018\:\mathrm{ms^{-2}}$. The velocity is then $1.65891531\times109\:\mathrm{ms^{-1}}$. Also, how long does it take for the proton to get to the plate.

It exceeded the speed of light. the size of the plate is two inches in diameter and there is a hole in the middle that is 1/2 inch in diameter and the proton will go through it.

What about 1 microcoulomb

• Your question may not literally be homework but by this site's policy it's a homework&exercises style question. – Gert Oct 26 '16 at 15:24
• How and what are the steps that you used to find your acceleration? You should provide the equations and values that you used to come about this answer. Then people can help you figure out where in your process you went wrong. – scrappedcola Oct 26 '16 at 15:25
• This is for a particle accelerator design – Arvin Singh Oct 26 '16 at 18:25

1 Answer

Use conservation of energy, and relativistic equations.

The total energy of the proton increases by the electrical potential energy from the attraction of the plate (for an infinite plate, E field is constant with distance).

For energy, use:

$$E^2 = p^2 c^2 + m_0 c^4$$

from this you find the momentum of the proton as it passes the plate; and the momentum is related to the velocity by

$$p = \gamma m v$$

and finally,

$$\gamma = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}$$

• Could you clarify? I do not understand – Arvin Singh Oct 30 '16 at 16:23