Respuesta :
Explanation:
Formula for the electric field due to the infinite sheet of charge is as follows.
E = [tex]\frac{\sigma}{2 \epsilon_{o}}[/tex]
where, [tex]\sigma[/tex] = surface charge density
Now, formula for electric force acting on the proton is as follows.
F = eE
where, e = charge of the proton
According to the Newton's second law of motion, the net force acting on the proton is as follows.
F = ma
a = [tex]\frac{eE}{m}[/tex]
= [tex]\frac{e(\frac{\sigma}{2 \epsilon_{o}})}{m}[/tex]
= [tex]\frac{e \sigma}{2m \epsilon_{o}}[/tex]
According to the kinematic equation, speed of the proton in perpendicular direction is as follows.
[tex]v_{f} = v_{i} + at[/tex]
= [tex](0 m/s) + \frac{e \sigma}{2 m \epsilon_{o}}t[/tex]
= [tex]\frac{1.6 \times 10^{-19}C \times 2.34 \times 10^{-9} C/m^{2} \times 5.40 \times 10^{-8}s}{2 \times (1.67 \times 10^{-27} kg)(8.85 \times 10^{-12} C^{2}/Nm^{2}}[/tex]
= 683.974 m/s
Hence, total speed of the proton is as follows.
v' = [tex]\sqrt{(960 m/s)^{2} + (683.974 m/s)^{2}}[/tex]
= [tex]\sqrt{921600 + 467820.43}[/tex]
= [tex]\sqrt{1389420.43}[/tex]
= 1178.73 m/s
Therefore, we can conclude that speed of the proton is 1178.73 m/s.
The speed of the proton after the given time is 1178.75 m/s.
Speed of the proton:
The electric field due to a large sheet with surface charge density (σ) is given by:
E = σ/2ε₀
Therefore the force exerted on the proton will be:
F = qE
where q is the charge of the proton
ma = qE
so acceleration is:
a = qE/m
m is the mass of the proton
Let the sheet is in the horizontal direction, so the electric field and force will be in a vertical direction.
The initial speed of proton in vertical direction is zero.
from the third equation of motion, the vertical speed of proton:
v' = 0 + at
given that time t = 5.4 × 10⁻⁸s
v' = qEt/m
v' = [tex]\frac{1.6\times10^{-19}\times(2.34\times10^{-9}/2\times8.85\times10^{-12})\times5.4\times10^{-8}}{1.67\times10^{-27}}[/tex]
v' = 684 m/s
Given that the horizontal speed is 960 m/s.
The resultant speed of the proton is :
v = [tex]\sqrt{(960)^2+(684)^2}\;m/s[/tex]
v = 1178.75 m/s
Learn more about equations of motion:
https://brainly.com/question/5955789?referrer=searchResults
