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Coulomb's Law is a fundamental principle that describes the interaction between two point charges. It states that the electric force between two charged objects is directly proportional to the product of their charges and inversely proportional to the square of the distance between their centers. The mathematical expression for Coulomb's Law is given by:\[ F = k \frac{|q_1q_2|}{r^2} \]where:

• \( F \) is the magnitude of the force between the charges,
• \( k \) is Coulomb's constant, approximately \(9 \times 10^9\, \text{Nm}^2/\text{C}^2\),
• \( q_1 \) and \( q_2 \) are the charges, and
• \( r \) is the distance between the charges.

In our exercise, Coulomb's Law helps in determining how the electric field at a certain point is influenced by surrounding charges. The law is useful for understanding the electric field (\( E \)), which can be derived from the force (\( F \)) divided by a test charge (\( q \)): \( E = \frac{F}{q} \). This forms the basis for calculating electric fields due to charged bodies like spheres.

Uniform Charge Distribution refers to a scenario where a charge is spread evenly over a volume, surface, or along a line. This concept is important when dealing with electric fields because it affects how the field strengths are calculated in various regions relative to the charged object.

For a sphere with uniformly distributed charge, the charge density (\( \rho \)) is constant. The electric field inside and outside the sphere is calculated differently:

• Inside the Sphere: The electric field at a distance \( r \) from the center is calculated using the volume charge density. Given by: \( E = \frac{kq}{R^3}r \), when \( r \leq R \).
• Outside the Sphere: The sphere can be treated as a point charge with total charge \( q \), and the electric field is calculated using: \( E = \frac{kq}{r^2} \).

In our task, point \( P \) is located inside sphere 1, so the electric field contribution from this sphere uses the inside sphere formula. Conversely, point \( P \) being outside sphere 2 means using the outside sphere formula. Understanding these two forms is crucial for solving related electric field problems.

The Ratio of Charges refers to the proportionality of the amounts of charge between two objects. In the given problem, we are tasked with finding the ratio \( \frac{q_2}{q_1} \) where the net electric field at a particular point is zero.

The process involves setting the electric fields produced by both spheres equal because their combination results in a net zero field at point \( P \). For uniform spheres, you would use the inside sphere formula for sphere 1 and the outside sphere formula for sphere 2. After equating the two expressions for electric fields:\[ \frac{k q_1}{R^3} \frac{R}{4} = \frac{k q_2}{\left(\frac{5R}{4}\right)^2} \]By simplifying and rearranging, the desired equation gives you:\[ \frac{q_2}{q_1} = \frac{16}{25} \]This calculated ratio tells us how much charge each sphere needs for the electric fields to cancel each other out at point \( P \). Understanding this concept is critical in electric field applications where balance or neutralization of forces is necessary.