an electron is shot directly towards the center of large metal plate that has negative charge of density 2 x 1/1000000 c/m . If the intial kinectic energy of electron is 100ev and if the electron is to stop give as it reached the plate , how far from the plate mustit be shot

To determine how far the electron must be shot from the negatively charged metal plate to come to a stop just as it reaches the plate, we need to analyze the forces acting on the electron and the energy transformations involved. Let's break this down step by step.

Understanding the Problem

We have an electron with an initial kinetic energy of 100 eV, which is equivalent to a certain amount of joules. The plate has a negative charge density of 2 x 10^-6 C/m². The goal is to find the distance from the plate where the electron should be shot so that it stops right at the plate.

Converting Energy Units

First, we need to convert the kinetic energy of the electron from electronvolts to joules. The conversion factor is:

• 1 eV = 1.6 x 10^-19 J

Thus, the initial kinetic energy (KE) in joules is:

KE = 100 eV × 1.6 x 10^-19 J/eV = 1.6 x 10^-17 J

Electric Field Due to the Charged Plate

The electric field (E) created by a uniformly charged infinite plane sheet can be calculated using the formula:

E = σ / (2ε₀)

where:

• σ = charge density = 2 x 10^-6 C/m²
• ε₀ = permittivity of free space = 8.85 x 10^-12 C²/(N·m²)

Substituting the values:

E = (2 x 10^-6 C/m²) / (2 x 8.85 x 10^-12 C²/(N·m²)) = 1.13 x 10^5 N/C

Force on the Electron

The force (F) acting on the electron due to the electric field is given by:

F = qE

where q is the charge of the electron (approximately -1.6 x 10^-19 C). Therefore:

F = (-1.6 x 10^-19 C)(1.13 x 10^5 N/C) = -1.81 x 10^-14 N

The negative sign indicates that the force is directed opposite to the motion of the electron.

Finding the Stopping Distance

To find the distance (d) the electron must travel before stopping, we can use the work-energy principle. The work done by the electric field on the electron will equal the initial kinetic energy:

Work = F × d = KE

Substituting the values we have:

1.81 x 10^-14 N × d = 1.6 x 10^-17 J

Solving for d gives:

d = (1.6 x 10^-17 J) / (1.81 x 10^-14 N) ≈ 0.00088 m or 0.88 mm

Final Thoughts

In summary, the electron must be shot from a distance of approximately 0.88 mm away from the negatively charged plate to ensure it stops right at the plate. This calculation illustrates the interplay between kinetic energy and electric forces, showcasing how charged particles behave in electric fields.