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Chapter 3: Problem 26

Coulomb's law in electrical theory states that the force \(F\) of attraction between two oppositely charged particles varies directly as the product of the magnitudes \(Q_{1}\) and \(Q_{2}\) of the charges and inversely as the square of the distance \(d\) between the particles. (a) Find a formula for \(F\) in terms of \(Q_{1}, Q_{2}, d,\) and a constant of variation \(k\) (b) What is the effect of reducing the distance between the particles by a factor of one-fourth?

Short Answer

Expert verified
(a) \( F = k \frac{Q_1 Q_2}{d^2} \), (b) Force increases by a factor of 16.

Step by step solution

01

Understanding Coulomb's Law

Coulomb's Law expresses the force between two charged particles. The force \( F \) is directly proportional to the product of the magnitudes of the charges, \( Q_1 \) and \( Q_2 \), and inversely proportional to the square of the distance \( d \) between them. The formula can be written as \[ F = k \frac{Q_1 Q_2}{d^2} \] where \( k \) is a proportionality constant known as Coulomb's constant.
02

Effect of Distance Change on Force

When the distance \( d \) is reduced by a factor of one-fourth, the new distance becomes \( \frac{1}{4}d \). Substitute this new distance into Coulomb's Law to see the effect on the force: \[ F' = k \frac{Q_1 Q_2}{(\frac{1}{4}d)^2} \] Simplifying the denominator gives \[ F' = k \frac{Q_1 Q_2}{\frac{1}{16}d^2} = k \frac{Q_1 Q_2 \times 16}{d^2} = 16 \times F \]. Thus, reducing the distance by a factor of one-fourth increases the force by a factor of 16.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Electric Force
The term "electric force" refers to the force of attraction or repulsion between charged particles. It is one of the fundamental interactions in nature, playing a crucial role in the way electrons and protons interact in atoms. Electric force can be either attractive or repulsive:
  • Attractive when two particles have opposite charges.
  • Repulsive when the particles have like charges.
Coulomb's Law is used to calculate the magnitude of this force. The strength of the electric force is dependent on:
  • The magnitude of the charges involved.
  • The distance between them.
  • A constant of proportionality, known as Coulomb's constant.
This law helps us understand how charged particles will behave under various conditions.
Inverse Square Law
The inverse square law is a principle stating that a specified physical quantity is inversely proportional to the square of the distance from the source of that physical quantity. In the context of Coulomb's Law, this means:
  • The electric force between two charges decreases rapidly as the distance increases.
  • If the distance is doubled, the force becomes one-fourth of its original magnitude.
This concept is crucial in understanding how forces act over space, showing a drop-in force strength with the square of the increase in distance. This principle applies not only to electric forces but also to gravitational forces and light intensity.
Proportionality Constant
In Coulomb's Law, the proportionality constant is known as Coulomb's constant, denoted as \( k \). It is a crucial factor that allows us to determine the magnitude of the electric force accurately. The value of \( k \) is approximately \( 8.988 \times 10^9 \, \text{N m}^2/\text{C}^2 \).
This constant anchors the relationship described by Coulomb's Law, ensuring:
  • The force is calculated in the correct units based on the metric system.
  • It aligns with the empirical observations and measurements made in experiments.
Without \( k \), we wouldn't have the precision needed for practical applications in electrical engineering and physics.
Distance in Physics
Distance is a critical parameter in many physical laws, including Coulomb's Law. In physics, it usually refers to the separation between two points in space. Changes in this distance can greatly affect the physical interactions being observed or calculated. Here鈥檚 why it matters:
  • It determines how strong or weak the force between two particles will be.
  • Smaller distances lead to stronger forces, as indicated by the inverse square law.
  • In practical terms, this is why charges need to be in proximity to interact appreciably.
For example, in Coulomb's Law, if the distance between charged particles is reduced by a factor of four, the force increases by a factor of 16, illustrating the powerful effect distance has on electric forces.

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Most popular questions from this chapter

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Examine the expression for the given set of data points of the form \((x, y) .\) Find the constant of variation and a formula that describes how \(y\) varies with respect to \(x.\) $$x y,\\{(0.2,-26.5),(0.4,-13.25),(0.8,-6.625)\\}$$
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