91Ó°ÊÓ

Coulomb's Law is a fundamental principle that helps us understand how electric charges interact with each other. It describes the force between two stationary, point charges. According to Coulomb's Law, the electric force (F) 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. This can be mathematically expressed as:\[F = k_e \frac{|Q_1Q_2|}{r^2}\]where:

• \(F\) is the force between the charges,
• \(Q_1\) and \(Q_2\) are the magnitudes of the charges,
• \(r\) is the distance between the charges,
• \(k_e\) is the electrostatic constant, approximately equal to \(8.99 \times 10^9 \, \text{N}·\text{m}^2/\text{C}^2\) .

In the context of a charged sphere, Coulomb's Law can be slightly modified since the sphere effectively behaves as a point charge at its center when calculating forces and fields at distances greater than or equal to its radius. This principle helps derive other properties, like the electric field, associated with the charged sphere.

A charged sphere, particularly when considered a solid metal sphere like copper, distributes charge uniformly across its surface. Inside a conductor, such as metal, the electric field is zero, and all excess charge resides on the surface.
The electric field (E) at the surface of a charged sphere can be determined using a form of Coulomb's Law. It follows:\[E = k_e \frac{Q}{r^2}\]Here:

• \(E\) is the electric field on the surface,
• \(Q\) is the total charge on the sphere,
• \(r\) is the radius of the sphere.

In this exercise, the electric field was given as \(3800 \, \text{N/C}\), directed inward, indicating a negatively charged sphere. The surface's electric field tells us about how the sphere's charge interacts with the space around it. Since our example assumes potential to be zero at infinity, the calculated electric potential at any point can give insight into the energy required to move a charge to that point through the sphere’s electric field.

The electric field is a vector field around a charged object where each point in the space has a vector associated with it that points in the direction that a positive test charge would move if placed at that point. The magnitude of the electric field tells us the force that a charge of \(1 \, \text{C}\) would experience at each point.
The formula for the electric field \(E\) around a point charge \(Q\) is given as:\[E = k_e \frac{Q}{r^2}\]

• \(E\) represents the electric field,
• \(k_e\) is the electrostatic constant,
• \(Q\) is the point charge,
• \(r\) is the distance from the charge.

For a charged sphere, such an electric field will be strong and directed towards or away from the sphere depending on the charge’s sign. In this problem, since the field is toward the center, it suggests a negative surface charge. Understanding the electric field is essential for computing the behavior of charges around the sphere, and ultimately, determining the potential at different points within and around the sphere.