Question

An infinitely long line of charge has a linear charge density of 8.00×10−12 C/m . A proton is at distance 13.5 cm from the line and is moving directly toward the line with speed 1500 m/s . How close does the proton get to the line of charge?

Answer 1

The following solution assumes that the proton approaches the line of charge perpendicularly so that the charge is 8.00x10^-12 C regardless of the distance.
To solve this problem you need to know what the potential will be as the proton approaches the line of charge. Using Coulomb's law you can calculate the potential energy, U, created by the interaction between the two like-charges:

U = k*(Q*q) / x

where Q is the charge of the line of charge, q is the charge of the proton (1.6x10^-19 C), k = 8.99x10^9 J*m/(C^2).

Now, all you need to do is determine the total kinetic energy of the proton:
U = 8.99x10^9 x 8.00x10^-12 C*1.6x10^-19 C/x
KE = 1/2mv^2

and finally, the moment when the proton stops moving (before being repelled back by the line charge) will correspond to the moment when the potential energy equals the kinetic energy of the proton:

U = KE

With this equation you can solve for x and determine how far the proton traveled.