91Ó°ÊÓ

An electric field is a region around a charged particle where other charged particles experience a force. The magnitude and direction of this force are determined by the properties of the source charge. The electric field (\textbf{E}) due to a uniformly charged infinite plane, like the metal plate in the exercise, is given by the formula: \[E = \frac{\sigma}{2\epsilon_{0}}\]Where

• \(\sigma\) is the surface charge density.
• \(\epsilon_{0}\) is the permittivity of free space \(8.85 \times 10^{-12} \, \textrm{C}^2/\left(\textrm{N} \cdot \textrm{m}^2\right)\).

In this case, the surface charge density is \(-2.0 \times 10^{-6} \, \textrm{C/m}^2\). Plugging in the values, the electric field is calculated to be: \[E = \frac{-2.0 \times 10^{-6}}{2 \times 8.85 \times 10^{-12}} = -1.13 \times 10^5 \, \textrm{N/C}.\]The negative sign indicates that the direction of the field is opposing the motion of the electron, which is consistent with the concept of electrostatic repulsion.

Kinetic energy is the energy that a body possesses due to its motion. For an electron, or any other particle with mass \(m\), moving with velocity \(v\), the kinetic energy \(KE\) is given by: \[KE = \frac{1}{2} mv^2.\]In this exercise, the initial kinetic energy of the electron is provided as \(1.60 \times 10^{-17} \, \textrm{J}\). This energy will be entirely converted into work done by the electrostatic force as the electron comes to a stop upon reaching the negatively charged plate. Since the electron stops completely, its final kinetic energy is zero. This conversion and energy conservation principle are instrumental to solving how far the electron travels.

Work done by a force is defined as the force applied over a distance. In the realm of electrostatics, work is done by the electric field when it moves a charge. The work done (\(W\)) by the electrostatic force is equal to the product of the charge (\(q\)), the electric field (\(E\)), and the distance (\(d\)) the charge moves. Mathematically: \[W = qEd.\]In the current exercise, the work done by the electrostatic force must be equal to the initial kinetic energy of the electron for it to come to rest. Given the initial kinetic energy (\(1.60 \times 10^{-17} \, \textrm{J}\)), and the electron charge \(e = 1.60 \times 10^{-19} \, \textrm{C}\), we use the relationship that \(\mathrm{KE} = W\) to determine the distance over which the force acts: \[1.60 \times 10^{-17} = 1.60 \times 10^{-19} \, \textrm{C} \times (-1.13 \times 10^5 \, \textrm{N/C}) \times d.\]This gives the distance \(d\approx 0.88 \, \textrm{mm}\) upon solving, indicating how far the electron must be shot from the plate.

Potential difference (often called voltage) between two points in an electric field is the work done to move a unit charge between those points. It reflects the energy needed to move charges against the field's force. The potential difference \(V\) between two points separated by a distance \(d\) in a uniform electric field \(E\) is: \[V = Ed.\]Thus, the potential difference is directly proportional to the displacement in the field and the strength of the electric field. In this exercise's context, knowing the electron's initial kinetic energy and the work done by the electrostatic force allowed us to link the potential difference with the distance \(d\) via \[W = eV\] or \[V=Ed.\] Understanding potential difference is critical, as it provides insights into how charged particles behave and are influenced by electric fields.