Infinite Line of Charge and Electric Potential Difference

1. The problem statement, all variables and given/known data
1. Consider a line of charge (with λ charge per unit length which extends along the x axis from x=-∞ to x=0
(a) Find all components of the electric field vector at any point along the positive x-axis

(b) Find the electric potential difference between any point on the positive x-axis and x=1 m

(c) If λ=0.1 μC per meter and a proton is placed at x=1 m with zero initial speed, does the proton’s speed ever reach 106 m/s. If so, where does it reach this speed?

2. Relevant equations

Charge of a proton=1.602[itex]\cdot[/itex]10-19 C
Mass of a proton=1.673[itex]\cdot[/itex]10-27 kg
Gauss’ Law Flux=∫E[itex]\cdot[/itex]dA=Q/ε0
Q=λ[itex]\cdot[/itex]L
ΔV=VB-VA

3. The attempt at a solution
For part a, I got
E=λ/(2∏ε0r) when I took the integral of ∫2∏rE[itex]\cdot[/itex]dl from 0 to length L and set it equal to Q/ε0

For part b, not sure if this is correct, but I used the equation
ΔV=VB-VA=-∫E[itex]\cdot[/itex]dR from A to B
I substituted the answer I found in part A for E and got -λ[itex]\cdot[/itex]lnX/2∏ε0
I am not sure if this is the correct answer, it sort of makes sense, the further the particle is on the x-axis, the greater the potential

(c) For part c, I presume that since the line of charge is infinitely long, the proton will eventually reach that speed. I tried to find the charge Q using the equation Q=λL, but since this is an infinitely long line, Q increases as L increases. The equation for electrical potential energy is U=q0/(4∏ε0[itex]\cdot∑[/itex]qn/rn)
Would the sum of all the point charges just be λ since q/l is just charge over lenght?

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