91Ó°ÊÓ

The electric field is a crucial concept in electrostatics. It represents the force per unit charge exerted on a charged particle at any point in space. In this exercise, we're dealing with the electric field generated by a uniformly charged ring.
To understand how the electric field behaves along the axis of a ring, imagine the ring is made up of many small charge elements. Each of these tiny charges contributes to the total electric field at a point along the axis.
The net electric field at a distance "x" from the center of the ring can be calculated using the formula:

• \[E = \frac{kQx}{(x^2 + R^2)^{3/2}}\]

Here, "k" is Coulomb's constant, "Q" is the total charge on the ring, "R" is the radius of the ring, and "x" is the distance from the center.
This formula shows that the electric field decreases with the square of the distance from the center of the ring combined with the radius in the denominator. It means the farther you move away from the ring, the weaker the electric field becomes.

Coulomb’s Law is fundamental to understanding electric force interactions between charges. It states that the force between any two point charges is directly proportional to the product of their charges and inversely proportional to the square of the distance between their centers. This law can be represented by the equation:

• \[F = \frac{k \, |q_1 q_2|}{r^2}\]

Where "k" is Coulomb’s constant \(8.99 \times 10^9 \, ext{N m}^2/ ext{C}^2\), \(q_1\) and \(q_2\) are the magnitudes of the charges, and "r" is the distance between them.
In this specific problem, an electron which carries a negative charge is placed at a certain axial distance from a positively charged ring.
The force on the electron is attractive, pulling it towards the ring's center. This is due to opposite charges attracting each other, illustrating Coulomb's Law in action. Despite the electron not being a point charge, the formula still gives an approximation since the electron's size is negligible compared to the distances involved.

The principle of conservation of energy is pivotal in analyzing the motion of the electron in this scenario. It suggests that the total energy of an isolated system remains constant over time. In simpler terms, energy can't be created or destroyed, just transformed from one type to another.
For the electron moving towards the center of the ring, its total mechanical energy consists of kinetic and electric potential energy. Initially, the electron only has potential energy since it is released from rest.
As it moves, this potential energy turns into kinetic energy, giving the electron speed. By applying the conservation of energy, we relate the potential energy at the starting position to the kinetic energy at the center:

• \[\frac{1}{2}mv^2 = eE_0x_0\]

Here, "m" is the mass of the electron, "v" is its velocity at the center, "e" is the electron's charge, and \(E_0 x_0\) represents the initial electric potential energy.
By solving this equation, we can accurately obtain the electron's speed upon reaching the center of the ring, demonstrating the seamless conversion of energy states.