Question

Two charged spheres are initially a distance \(d\) apart. The magnitude of the force on each sphere is \(F\). They are moved closer to each other such that the magnitude of the force on each of them is \(9 F\). By what factor has the difference between the two spheres changed?

Short answer

Based on the given problem, the factor by which the distance between the charged spheres changed when the force increased 9 times is \(\frac{1}{\sqrt{3}}\).

Step-by-step solution

Coulomb's Law

Coulomb's law states that the force exerted between two point charges is directly proportional to the product of the charges and inversely proportional to the square of the distance between them. Mathematically, this is expressed as: \(F=k\frac{|q_1q_2|}{r^2}\) where: \(F\) is the force between the charges, \(q_1\) and \(q_2\) are the charges of the spheres, \(r\) is the distance between the centers of the two spheres, \(k\) is the electrostatic constant. In this exercise, we know that the initial force between the spheres is \(F\) and it increases to \(9F\) when the distance between the spheres changes. Let's denote the initial distance between the spheres as \(d\) and the new distance as \(d'\).

Relating the initial and final forces

We are given that the force on each sphere is initially \(F\) and it increases to \(9F\). Therefore, we can write the following equations using Coulomb's Law for the initial and final forces: Initial force: \(F=k\frac{|q_1q_2|}{d^2}\) Final force: \(9F=k\frac{|q_1q_2|}{(d')^2}\)

Solving for the distance factor

From the equations in Step 2, we want to solve for the factor by which the distance between the spheres has changed, which is \(\frac{d'}{d}\). Divide the first equation by \(F\) and the second equation by \(9F\) to get: \(\frac{1}{d^2}=\frac{k|q_1q_2|}{F*d^2}\) \(\frac{1}{9(d')^2}=\frac{k|q_1q_2|}{9F*(d')^2}\) Now, we can divide the first equation by the second equation to eliminate \(k|q_1q_2|\) and obtain the factor by which the distance changed: \(\frac{1/d^2}{1/(9d'^2)}=\frac{F*d^2}{9F*(d')^2}\)

Solving for the ratio \(\frac{d'}{d}\)

After simplifying the equation in Step 3, we can solve for the ratio \(\frac{d'}{d}\): \(\frac{9(d')^2}{d^2}=\frac{F*d^2}{F*(d')^2}\) Since \(F\) cancels out, we obtain: \(\frac{9(d')^2}{d^2}=\frac{d^2}{(d')^2}\) Taking the square root of both sides, we get: \(\frac{3d'}{d}=\frac{d}{d'}\) By multiplying both sides by \(d'\) and \(d\), we finally obtain: \(3d'^2=d^2\) Thus, the factor by which the distance between the two spheres changed is \(\frac{d'}{d}=\frac{1}{\sqrt{3}}\).

Key concepts

Understanding Electrostatic Force

Understanding electrostatic force is crucial for comprehending interactions at a microscopic level. This force arises from the electric charges present on objects and operates over a distance, resulting in either attraction or repulsion between those objects. When we say that opposite charges attract and like charges repel, we're referencing the principles of electrostatic force in action.

This concept is not only pivotal in physics but also in chemistry, biology, and engineering, where it plays a role in the structure of atoms and molecules, the bonding and interactions between different substances, and even in the design of electronic devices and systems. Electrostatic force is involved anywhere charges are present and are interacting.

Formula for Electrostatic Force

As equation (1) reveals, the electrostatic force between two point charges, such as electrons or protons, can be calculated. The equation shows the relationship between the magnitude of the force and both the product of the charges and the square of the distance between them. The constant 'k' is Coulomb's constant, which provides a measure of the strength of the electric force. In a vacuum, its value is approximately 8.987 x 10⁹ Nm²/C².

Charge Interaction Principles

Charge interaction is the cornerstone of Coulomb's Law and explains how charged particles affect each other. This principle is vital in understanding the fundamentals of electromagnetism and the behavior of electrically charged objects. According to the law, two charges exert forces upon each other, and these forces are equal in magnitude but opposite in direction. This reciprocality is an example of Newton's third law of motion, where every action has an equal and opposite reaction.

The implication of charge interaction is extensive; ranging from the microscopic interactions within an atom to macroscopic effects such as the generation of electric current. Moreover, it's pivotal in the domain of electronics, influencing how components like capacitors and inductors operate.

Application in Exercise

In the exercise provided, the concept of charge interaction helps us determine the change in electrostatic force when the distance between two charged spheres is altered. The direct proportionality between force and charge, and the inverse proportionality between force and the square of distance, allows for the setup of equations that can be manipulated to find the new distance between the spheres when the force changes.

The Inverse Square Law

The inverse square law is a physical principle that describes the intensity of a physical quantity or force weakening as distance from the source increases. Specifically, it states that a specified physical quantity or intensity is inversely proportional to the square of the distance from the source of that physical quantity. This law applies not only to electrostatic forces but also to other phenomena such as gravity, light, and sound.

In the context of electrostatics, the inverse square law is deeply ingrained in Coulomb's Law, indicating that the force between two point charges diminishes rapidly as the distance between them increases. This rapid decrease in force with increasing distance makes the electrostatic force a short-range force, generally significant only at distances on the atomic scale.

Real-World Implications

The inverse square law has profound implications in everyday life and technology. For instance, it's essential in designing and understanding optical and acoustic devices, such as cameras and microphones. In medicine, it's utilized in radiology, as the intensity of radiation decreases with the square of the distance, which is crucial for safety and therapeutic applications.