Problem 22 - Precalculus: Functions and Graphs

Question

Coulomb's law 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?

Step-by-Step Solution

Answer

(a) \( F = k \frac{Q_1 Q_2}{d^2} \). (b) Reducing distance to one-fourth increases the force 16 times.

Step 1: Determine Relationship with Coulomb's Law

Coulomb's law states that the force \( F \) between two charged particles is directly proportional to the product of the magnitudes of their charges \( Q_1 \) and \( Q_2 \), and inversely proportional to the square of the distance \( d \) between them. Expressed mathematically, this relationship is given by \( F \propto \frac{Q_1 Q_2}{d^2} \).

Step 2: Introduce Proportionality Constant

To convert the proportional relationship into an equation, introduce a constant of proportionality \( k \). The formula for the force becomes \( F = k \frac{Q_1 Q_2}{d^2} \). Thus, the force depends on the charges, the square of the distance, and the proportionality constant.

Step 3: Analyze Effect of Distance Reduction

To analyze the effect of reducing the distance \( d \) by a factor of one-fourth, substitute \( d \) with \( \frac{1}{4}d \) in the formula \( F = k \frac{Q_1 Q_2}{d^2} \). The new force will be \( F' = k \frac{Q_1 Q_2}{(\frac{1}{4}d)^2} = k \frac{Q_1 Q_2}{\frac{1}{16}d^2} = 16 \cdot k \frac{Q_1 Q_2}{d^2} = 16F \).

Step 4: Effect on Force when Distance is Reduced

When the distance between the particles is reduced by a factor of one-fourth, the force \( F \) is increased by a factor of 16. Therefore, the force becomes 16 times stronger.

Key Concepts

Electrical Charges

Electrical charges are fundamental to the study of electricity and magnetism. They are responsible for the interactions between particles that are either attractive or repulsive. There are two types of electrical charges: positive and negative. A positive charge occurs when there is a deficiency of electrons, while a negative charge is due to an excess of electrons. In nature, opposite charges attract one another, while like charges repel each other. This principle is fundamental in understanding forces between charged particles, as explored through Coulomb's Law.

Here are some key points about electrical charges:

• Measured in Coulombs (C).
• Like charges repel; opposite charges attract.
• Charge is quantized, existing in discrete quantities.

To fully understand forces between charges, we must also consider factors such as distance and the medium between the charges, which can affect the strength of the interaction.

Force of Attraction

The force of attraction is a type of electromagnetic force that occurs between particles with opposite electrical charges. If you have two particles with charges, one positive and one negative, the force acts to pull them toward each other. In Coulomb's Law, the force of attraction between two opposite charges can be calculated using the formula:

• This force is proportional to the product of the magnitudes of the charges.
• The greater the charges, the stronger the force of attraction.
• The force diminishes as the distance between the charges increases.

Understanding the role of the force of attraction is important in fields such as chemistry and physics, as it explains how molecules and atoms stick together to form more complex structures.

Inverse Square Law

The inverse square law is a fundamental principle that applies to various physical phenomena, including gravitational and electromagnetic forces. It states that the intensity of these forces is inversely proportional to the square of the distance from the source of the force. When applied to electrical charges, as in Coulomb's Law, the inverse square law dictates how the force between two charges diminishes as they move further apart.Mathematically, this law can be expressed as:- The force between two charges is inversely proportional to the square of the distance separating them, that is, \( F \propto \frac{1}{d^2} \).Some quick reminders regarding the inverse square law:

• Doubling the distance between two charges will make the force one-fourth as strong.
• Halving the distance will make the force four times as strong.
• This law helps describe how fields like gravity and electromagnetism behave over distance.

Understanding this concept is crucial for predicting how forces change with distance, a vital aspect of physics and engineering.

Proportionality Constant

In the equation derived from Coulomb's Law, the proportionality constant plays a critical role in converting the proportional relationship to an actual equation. This constant, often denoted as \( k \), provides a measure of how strongly two charges interact with each other. It is a fundamental constant in physics, specifically in electromagnetic equations.The formula that includes the proportionality constant is:

• \( F = k \frac{Q_1 Q_2}{d^2} \)
• Here, \( k \) is the Coulomb's constant, approximately \( 8.99 \times 10^9 \, \text{N} \, \text{m}^2 \text{/C}^2 \).
• The constant ensures that the force is calculated correctly and takes into account the units of measurement being used.

By understanding the role of the proportionality constant, you can accurately determine the force exerted by electrical charges, critical for practical applications in science and engineering.