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Electric potential is a measure of the electric potential energy per unit charge at a point in a field. You can think of it as the 'height' of an electrical landscape where a charge would gain or lose energy, much like how water flows downhill. The electric potential due to a point charge is determined using the formula:

• \[ V = \frac{kQ}{r} \]

where:

• \( V \) is the electric potential, measured in volts (V).
• \( k \) is Coulomb’s constant, approximately \( 8.99 \times 10^9 \, \mathrm{N\,m^2/C^2} \).
• \( Q \) is the charge in coulombs (C).
• \( r \) is the distance from the charge in meters (m).

The equation shows that the potential decreases as the distance from the charge increases, which is why at infinity, the potential is zero: it is far away from any charges.
In this problem, we start by using the potential at a distance to find the charge on the sphere, which then allows us to calculate the potential at any other point, like on the sphere's surface.

Surface charge density, denoted as \( \sigma \), refers to the amount of electric charge per unit area on the surface of a conductor. It's a crucial concept when dealing with charged conductors because it tells us how the charge is distributed on the surface.
The formula for surface charge density is:

• \[ \sigma = \frac{Q}{4\pi R^2} \]

Here:

• \( Q \) is the total charge on the surface of the sphere.
• \( R \) is the radius of the sphere.
• The denominator \( 4\pi R^2 \) is the surface area of the sphere.

This measure is useful when we need to comprehend how concentrated a charge is on a particular surface. A high surface charge density indicates a large amount of charge packed into a small area, whereas a low surface charge density shows that the charge is more spread out.
By calculating this value for the charged sphere, we gain insight into how the charge is distributed over the sphere's surface.

Conductors are materials that permit free movement of electric charge within them. When a conductor becomes charged, the charges reside solely on its outer surface. This happens because the charges repel each other and are free to move.
For a charged, conducting sphere, the electric field inside is zero because the charges move to the surface and reach a state of equilibrium, distributing themselves uniformly.

• Think of a conductor as a hollow hill: outside, the landscape varies; inside, it is flat.
• This property results in all points on the surface having the same potential, meaning the entire surface of the charged sphere is at the same electric potential, calculated using the point charge formula, but with the sphere's radius as the distance.

This knowledge is handy because it allows us to predict and calculate how a sphere or any other conducting object will affect its surroundings and where potential differences reside.

Coulomb's Law is a fundamental principle describing the force between two stationary charged particles. It states that:

• The force (\( F \)) between two charges is directly proportional to the product of their charges and inversely proportional to the square of the distance (\( r \)) between them.
• Mathematically, it is formulated as:\[ F = k \frac{|Q_1 Q_2|}{r^2} \]

where:

• \( F \) is the electrostatic force between the charges.
• \( Q_1 \) and \( Q_2 \) are the amounts of the two charges.
• \( k \) is Coulomb's constant (as previously mentioned).

Though not directly used in the exercise, Coulomb's Law underpins much of electrostatics. It's indispensable for understanding interactions between charges:

• It explains why oppositely charged objects attract and like charges repel.
• Helps predict how charges will distribute themselves across surfaces in a uniform manner when forces are balanced (as seen in conductors).

In many ways, it provides the foundation that makes calculations like those in the exercise possible.