To determine the distance from which a 100 eV electron should be projected so that it just fails to strike a metal plate with a surface charge density of -2 x 10^-6 C/m², we need to analyze the forces acting on the electron due to the electric field created by the charged plate. The goal is to find the point where the kinetic energy of the electron is exactly balanced by the potential energy due to the electric field, preventing it from reaching the plate.
Understanding the Electric Field
The surface charge density (σ) of the plate creates an electric field (E) in the space around it. For a large plate, the electric field can be calculated using the formula:
E = σ / (2ε₀)
where ε₀ (the permittivity of free space) is approximately 8.85 x 10^-12 C²/(N·m²).
Calculating the Electric Field
Given that σ = -2 x 10^-6 C/m², we can substitute this value into the equation:
E = (-2 x 10^-6 C/m²) / (2 x 8.85 x 10^-12 C²/(N·m²))
Calculating this gives:
E = -1.13 x 10^6 N/C
The negative sign indicates that the electric field points towards the plate, which is important for understanding the motion of the electron.
Energy Considerations
The kinetic energy (KE) of the electron when it is fired can be calculated using the formula:
KE = eV
where e is the charge of the electron (approximately 1.6 x 10^-19 C) and V is the energy in volts. For a 100 eV electron:
KE = 1.6 x 10^-19 C * 100 V = 1.6 x 10^-17 J
Potential Energy and Distance
The potential energy (PE) of the electron in the electric field at a distance (d) from the plate is given by:
PE = qEd
Here, q is the charge of the electron, and E is the magnitude of the electric field. Setting the kinetic energy equal to the potential energy gives:
KE = PE
1.6 x 10^-17 J = (1.6 x 10^-19 C)(1.13 x 10^6 N/C)(d)
Solving for Distance
Now we can solve for d:
d = (1.6 x 10^-17 J) / [(1.6 x 10^-19 C)(1.13 x 10^6 N/C)]
Calculating this gives:
d ≈ 0.088 m
Final Thoughts
Thus, the distance from which the 100 eV electron should be projected so that it just fails to strike the plate is approximately 0.088 meters, or 8.8 centimeters. This analysis illustrates the interplay between kinetic and potential energy in the context of electric fields, providing a clear example of how charged particles behave in the presence of electric forces.
