91Ó°ÊÓ

Coulomb's Law is a fundamental principle in electromagnetism, named after the French physicist Charles-Augustin de Coulomb. It provides a way to calculate the electric force between two point charges. According to Coulomb's Law, the force (\( F \)) between two charges is directly proportional to the product of their charges (\( q_1 \times q_2 \) ), and inversely proportional to the square of the distance (\( r^2 \) ) between them.

The equation is represented as \( F = k \times \frac{q_1 \times q_2}{r^2} \) , where \( k \) is Coulomb's constant (\( 9 \times 10^9 \, \mathrm{N \cdot m^2/C^2} \) ), illustrating the strength of the electrostatic interaction. In the context of our problem, Coulomb's Law helps to derive the formula for the electric field (\( E \) ) created by a linear charge density (\( \lambda \) ).

This foundational concept is critical for understanding electric fields, forces, and ultimately, how charges move in response to these forces, which directly connects to the behavior of the charged sphere in our textbook problem.

The electric field is a vector field around a charged object, representing the force that other charges would experience if placed within this field. The strength of the electric field is defined by the force felt per unit charge. Its units are newtons per coulomb (\( \mathrm{N/C} \) ).

For an infinite line of charge, the electric field (\( E \) ) at a distance \( r \) from the line of charge is described by \( E = \frac{{k\lambda}}{{r}} \) , where \( \lambda \) is the linear charge density and \( k \) is Coulomb's constant. Thus, the electric field diminishes as the distance from the line of charge increases. This directly informs our calculation of forces exerted on the charged sphere and ultimately its kinetic energy.

The work-energy theorem is a critical concept linking the work done by forces to changes in kinetic energy. It states that the net work done by all forces acting on an object will result in an equivalent change in the object’s kinetic energy. Mathematically, it's expressed as \( W = \Delta KE \) , where \( W \) is the work and \( \Delta KE \) is the change in kinetic energy.

In the case of our charged sphere, as it accelerates from rest under the effect of the electric field, all the work done on the sphere by the electric force converts into its kinetic energy. This connection allows us to use the calculated work to find the final kinetic energy of the sphere once it has moved a certain distance away from the line of charge.

Linear charge density (\( \lambda \) ) is essentially a measure of electric charge per unit length. In equations, it's denoted as \( \lambda = \frac{Q}{L} \) , where \( Q \) is the total charge, and \( L \) is the length over which the charge is distributed.

For continuous charge distributions, such as the long line of charge in our problem, it is a valuable concept to quantify the charge distribution. Knowing the linear charge density, \( \lambda \) , allows us to calculate the electric field, and subsequently the force that acts on the charged sphere. The uniformity of \( \lambda \) greatly simplifies the force and energy calculations that are essential to solving this problem regarding the kinetic energy of the sphere.