Question

An electron in a vacuum chamber is fired with a speed of 8300 km/s toward a large, uniformly charged plate 75 cm away. The electron reaches a closest distance of 15 cm before being repelled. What is the plate’s surface charge density?

Answer 1

To find the plate's surface charge density, use the equation σ = 2ε₀e/(vr²) where σ is the surface charge density, ε₀ is the permittivity of free space, e is the charge of an electron, v is the velocity of the electron, and r is the closest distance.

Step-by-step explanation:

To find the plate's surface charge density, we need to determine the electric field created by the plate. We can use the equation for the electric field due to a uniformly charged plate, E = σ/2ε₀, where σ is the surface charge density and ε₀ is the permittivity of free space.

Given that the electron reaches a closest distance of 15 cm and is repelled, we can equate the electric force F = eE to the centripetal force F = mv²/r, where e is the charge of an electron, E is the electric field, m is the mass of an electron, v is the velocity of the electron, and r is the closest distance.

Simplifying the equations and solving for σ, we find the surface charge density to be σ = 2ε₀e/(vr²).

Answer 2

Answer: 1.85*10^-27 C/m^2

Explanation: In order to explain this problem, firstly we have to consider the kinematic equations given by:

xf=vo*t-(a*t^2)/2 and we considerer xo=0 and xf= 15 cm the distance that the electron moves before changes its direction of movenet due to the influence of the electric field from the negative charged plate.

and

vf=vo-a*t vf=0 when the electron starts to be repelled.

From these equations we have:

t=vo/a put it in the distante equation we have:

xf=vo*vo/a-(a/2)*(vo/a)^2 then xf=vo^2/a-(vo^2/2*a)= vo^2/(2*a)=

0.15 m=(8.3*10^6)^2/(2*a)

we have used that: vo=8300 Km/s= 8300 km/s*1000 m/km=8.3*10^6m/s

a= (8.3*10^6)^2/(2*0.15)=5.74*10^13 m/s^2

Secondly, we use the second Newton law:

F=m*a where the force from the electric field is given by:

Fe=σ/εo where σ is the surface charge density and the vaccum pernitivity is εo=8.85*10^-12 C^2/m^2*N

Finally, we have

Fe= σ/εo = m*a

σ= m*a *εo= 9.1 *10^-31* 5.74*10^13 *8.85*10^-12=1.85*10^-27 C^2/m^2