91Ó°ÊÓ

In physics, charged spheres are objects that are either positively or negatively charged. Here, our task involves two spheres, each weighing 8.55 grams, which are initially neutrally charged. When electrons are added to the spheres, they become negatively charged.
By adding electrons to both spheres equally, you create like charges. According to the basic principles of electromagnetism, like charges repel each other. This repulsion force results from the charges and is observed as the interaction between two electric fields.
In this exercise, the spheres are brought 15.0 cm apart, allowing the charged spheres to exert an electric force on one another. This force can lead to noteworthy acceleration of the spheres when they are released.

Coulomb's constant, also known as the electrostatic constant, plays a vital role in calculating the electrostatic force between two charges. It is represented as \( k \) and holds a constant value of approximately \(8.99 \times 10^9 \, \text{N} \cdot \text{m}^2 / \text{C}^2\). Coulomb's constant originates from Coulomb's Law, formulated by Charles-Augustin de Coulomb.
Using this constant, you can determine the force between two charged objects by inserting it into the formula:

• \(F = k \frac{|q_1 q_2|}{r^2}\)

In this problem, the constant helps us calculate the force needed to achieve the acceleration of the spheres. By setting up the equation with the charges and rearranging, we find the necessary charge on each sphere to cause the desired separation force.

The electron charge is a fundamental property of electrons, which are subatomic particles carrying a negative charge. This charge is denoted by \( e \), and its known value is \( 1.602 \times 10^{-19} \, \text{C} \). Electrons are responsible for creating negative charges when added to particles like our charged spheres.
In this exercise, to achieve the required charge of \( 7.835 \times 10^{-5} \, \text{C} \) per sphere, you will determine how many electrons must be added. The process involves calculating the electron count using the formula:

• \( n = \frac{q}{e} \)

Here, \( n \) is the number of electrons. With the calculated charge \( q \), we find that approximately \( 4.89 \times 10^{14} \) electrons are needed. This allows each sphere to have an equal negative charge to cause the electrostatic repulsion.

Acceleration is a change in velocity caused by forces acting on an object. For this exercise, when the charged spheres are released, they accelerate at 25.0 times the standard gravitational acceleration \( g \). Standard \( g \) is roughly \( 9.8 \, \text{m/s}^2 \). Thus, the spheres' acceleration \( a \) is \( 25.0 \times g \), resulting in \( 245 \, \text{m/s}^2 \).
Forces between charged spheres arise due to electron repulsion. According to Newton's second law, \( F = ma \), the force \( F \) required can be calculated by multiplying mass \( m \) and acceleration \( a \). In the solution, this force equals approximately \( 2.09675 \, \text{N} \), setting the stage for the spheres' separation.
When released, the spheres will move away from each other because the repulsive force from like charges causes them to accelerate in opposite directions.