Problem 7 - University Physics with Modern Physics

Question

An average human weighs about 650 \(\mathrm{N} .\) If two such generic humans each carried 1.0 coulomb of excess charge, one positive and one negative, how far apart would they have to be for the electric attraction hetween them to equal their \(650-\mathrm{N}\) weight?

Step-by-Step Solution

Answer

\( r \approx 3719 \text{ meters} \).

Step 1: Understanding the Problem

We have two humans, each with a weight of 650 N, carrying charges of +1.0 C and -1.0 C. We need to find the distance between them such that the electric force between them equals their weight, which is 650 N.

Step 2: Coulomb's Law Formula Application

Coulomb's Law states that the electric force \( F \) between two charges is given by \( F = \frac{k |q_1 q_2|}{r^2} \), where \(k = 8.99 \times 10^9 \text{ N m}^2/ ext{C}^2\) is the Coulomb's constant, \( q_1 = 1.0 \text{ C} \) and \( q_2 = -1.0 \text{ C} \) are the charges, and \( r \) is the distance between the charges. The goal is to set this equal to 650 N.

Step 3: Set the Equation Equal to Human Weight

We equate the electric force from Coulomb's Law to the weight of a human: \( \frac{8.99 \times 10^9 \times |1.0 \times (-1.0)|}{r^2} = 650 \).

Step 4: Solve for Distance \( r \)

Rearranging the equation to solve for \( r \), we have \( r^2 = \frac{8.99 \times 10^9}{650} \). Calculate \( r^2 \) then take the square root to find \( r \).

Step 5: Calculate \( r \)

\( r^2 = \frac{8.99 \times 10^9}{650} \approx 1.383 \times 10^7 \); taking the square root, \( r \approx 3719 \text{ meters} \).

Key Concepts

Coulomb's Law

Coulomb's Law is a fundamental principle in physics that describes how electric forces act between two point charges. It explains that 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. Mathematically, it is expressed as: \[F = \frac{k |q_1 q_2|}{r^2}\]where:

• \( k \) is Coulomb's constant, approximately \( 8.99 \times 10^9 \, \text{N m}^2/\text{C}^2 \).
• \( q_1 \) and \( q_2 \) are the amounts of charge, measured in coulombs.
• \( r \) is the distance separating the charges.

Coulomb's Law is similar to Newton's law of universal gravitation, which also depends on distance but applies to masses instead. Both demonstrate an inverse-square law nature. This means as the distance (\( r \)) increases, the force (\( F \)) decreases rapidly. Unlike gravitational forces, electric forces can attract or repel, since charges can be positive or negative. This principle is crucial for understanding a variety of electrostatic phenomena and is widely utilized in both theoretical and practical applications across physics.

Electric Charge

Electric charge is a fundamental property of matter that causes it to experience a force within an electric field. There are two types of electric charges: positive and negative. Like charges repel each other, while opposite charges attract. This force of interaction is described by Coulomb's Law. The unit of electric charge in the International System of Units (SI) is the coulomb (\( ext{C} \)). Commonly, objects may hold charges in smaller magnitudes, such as microcoulombs (\( ext{μC} \)) or nanocoulombs (\( ext{nC} \)), given the large size of a single coulomb relative to common charges seen in experiments.

• **Positive Charge:** Typically associated with a deficiency of electrons.
• **Negative Charge:** Typically associated with an excess of electrons.

Materials can become charged through various methods, such as friction, conduction, and induction. When two materials rub together, electrons may be transferred from one to the other, charging both materials. Conductors, such as metals, allow charges to move freely across their surfaces, while insulators, such as plastic, do not readily permit this movement.Understanding electric charge is essential for studying electrostatics, as it underpins all interactions in this area, from simple static cling to complex plate capacitor phenomena.

Electrostatics

Electrostatics is the branch of physics that studies electric charges at rest. It encompasses the forces, fields, and potentials created by stationary charges, and is essential for understanding the principles behind electric forces and interactions. One of the key aspects of electrostatics is the interaction between charges as governed by Coulomb's Law. In electrostatic conditions, electric charges are fixed in place rather than moving, which distinguishes this area from other electric phenomena involving current.

• **Electric Fields:** Created by electric charges, these regions exert force on other charges within the field, influencing their position and movement.
• **Electric Potential Energy:** Relates to the work required to move a charge in an electric field. It's a crucial concept for understanding energy conservation in electric systems.
• **Capacitors and Insulators:** Devices that store charge and insulate electricity are governed by the principles of electrostatics, crucial for circuits.

Electrostatics has practical applications across many technologies, including photocopiers, inkjet printers, and capacitive touch screens. Moreover, understanding electrostatic principles helps explain natural phenomena, such as lightning or the behavior of dust particles in a charged environment. Overall, electrostatics provides the foundation for the vast field of electromagnetism, elucidating how and why stationary charges interact the way they do.