# 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 10^{6} 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]d**A**=Q/ε_{0}

Q=λ[itex]\cdot[/itex]L

ΔV=V_{B}-V_{A}

**3. The attempt at a solution**

For part a, I got

E=λ/(2∏ε_{0}r) when I took the integral of ∫2∏rE[itex]\cdot[/itex]dl from 0 to length L and set it equal to Q/ε_{0}

For part b, not sure if this is correct, but I used the equation

ΔV=V_{B}-V_{A}=-∫**E**[itex]\cdot[/itex]d**R** from A to B

I substituted the answer I found in part A for **E** and got -λ[itex]\cdot[/itex]lnX/2∏ε_{0}

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=q_{0}/(4∏ε_{0}[itex]\cdot∑[/itex]q_{n}/r_{n})

Would the sum of all the point charges just be λ since q/l is just charge over lenght?

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