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Coulomb's law is a fundamental principle that describes how electric charges interact. It states that the force between two charges is directly proportional to the product of the magnitudes of the charges and inversely proportional to the square of the distance between them. Mathematically, Coulomb's law can be expressed as:

• \[ F = k \frac{{|q_1 q_2|}}{{r^2}} \]

Here, \( F \) represents the force between the charges, \( q_1 \) and \( q_2 \) are the magnitudes of the two charges, \( r \) is the distance between the centers of the two charges, and \( k \) is Coulomb's constant (\(8.99 \times 10^9 \text{ N m}^2/\text{C}^2\)). This law helps us understand how electric forces can either pull charges together or push them apart.
Understanding Coulomb's law is essential in calculating the potential of charged objects, as it helps to determine the electric field around them. This is particularly important in scenarios involving insulators and conductors where charge distribution impacts electric potential.

Electric potential, often referred to as voltage, is a measure of the potential energy per unit charge at a point in an electric field. It tells us how much potential energy a charge would have if placed at that point. The potential energy is influenced by both the amount of charge and the configuration of the electric field.
The electric potential \( V \) at a point due to a point charge \( Q \) is given by the formula:

• \[ V = \frac{kQ}{r} \]

where \( k \) is Coulomb's constant, \( Q \) is the charge, and \( r \) is the distance from the charge to the point of interest.
In the context of a sphere, as mentioned in the problem statement, this formula explains how the potential changes based on the accumulated charge and the size of the sphere (determined by the radius). When the sphere releases \(\beta\)-particles, each adding charge equivalent to an electron, the potential of the sphere increases, impacting the time required for the potential to reach a certain level, such as 1 V. This concept is crucial for understanding how devices store and utilize energy through electric charges.

The charge of an electron is one of the fundamental constants in nature, crucial to understanding atomic and subatomic interactions. It is denoted as \( e \) and has a value of approximately \( 1.6 \times 10^{-19} \, \text{C} \) (Coulombs).

• Electrons are negatively charged particles, and their charge is equal and opposite to that of protons.
• This tiny charge value underlies the interactions in most electrical phenomena we experience, including the operation of electronics and chemical bonding.

In the exercise, knowing the charge of an electron allows us to calculate the rate of charge accumulation on a surface when a certain number of \( \beta\)-particles, which have the same charge as electrons, are emitted per second.
This information is then used to determine how long it takes for the potential of an object, such as a metal sphere, to reach a specified level. Understanding the charge of an electron is therefore pivotal for understanding how charges build up in various systems.