Question

The practical limit to an electric field in air is about 3.00 × 10^6 N/C . Above this strength, sparking takes place because air begins to ionize.
(a) At this electric field strength, how far would a proton travel before hitting the speed of light (100% speed of light) (ignore relativistic effects)?

(b) Is it practical to leave air in particle accelerators?

Answer 1

Answer:

(a) x=157 m

(b) No

Explanation:

Given Data

Mass of proton m=1.67×10⁻²⁷kg

Charge of proton e=1.6×10⁻¹⁹C

Electric field E=3.00×10⁶ N/C

Speed of light c=3×10⁸ m/s

For part (a) distance would proton travel

Apply the third equation of motion

$(v_{f})^{2} =(v_{i})^{2}+2ax$

In this case vi=0 m/s and vf=c

so

$c^{2}=(0)^{2}+2ax\\ c^{2}=2ax\\x=\frac{c^{2} }{2a}$

$x=\frac{c^{2}}{2a}--------Equation (i)$

From the electric force on proton

$F=qE\\where\\ F=ma\\so\\ma=qE\\a=\frac{qE}{m}$

put this a(acceleration) in Equation (i)

So

$x=\frac{c^{2} }{2(qE/m)}\\ x=\frac{mc^{2}}{2qE} \\x=\frac{(1.67*10^{-27})*(3*10^{8})^{2} }{2*(1.6*10^{-19})*(3*10^{6})}\\ x=157m$

For part (b)

No the proton would collide with air molecule