Question

An infinitely long line of charge has linear charge density 5.50 10-12 C/m. A proton (mass 1.67 10-27 kg, charge +1.60 10-19 C) is 18.0 cm from the line and moving directly toward the line at 1.50 103 m/s. How close does the proton get to the line of charge? Incorrect: 15.17 cm

Answer 1

Suppose proton stops at x m from line of charge

Work done by electric field = kinetic energy=qU
$dE=\frac{kdQ}{r^{2}}\Rightarrow E=\frac{2k\sigma }{r}$

$dU=-E\cdot dr=-\frac{2k\sigma }{r} dr\rightarrow U=-2k\sigma* ln(r)$

$K=\frac{mv^{2}}{2}=\frac{1.67*10^{-27}kg(1500^{2}\frac{m}{s})}{2}=1.879*10^{-21}J$

$1.60*10^{-19}*-2k\sigma* ln(r)=1.879*10^{-21}\rightarrow ln(r1)-ln(r2)=-\frac{1.879*10^{-21}}{1.60*10^{-19}*2k\sigma}=-0.1186$

$ln(\frac{r_{1}}{r_{2}})=-0.1186\Rightarrow(\frac{r_{1}}{r_{2}})=0.8882\Rightarrow r_{1}=0.8882*0.18m=0.1599m$