Problem 81 - University Physics with Modern Physics

Question

Strength of the Electric Force. Imagine two \(1.0-\mathrm{g}\) bags of protons, one at the earth's north pole and the other at the south pole. (a) How many protons are in each bag? (b) Calculate the gravitational attraction and the electrical repulsion that each bag exerts on the other. (c) Are the forces in part (b) large enough for you to feel if you were holding one of the bags?

Step-by-Step Solution

Answer

The bags contain about \(6.0 \times 10^{23}\) protons each. Gravitational attraction is negligible, but electrical repulsion is \(5.3 \times 10^{17} \, \mathrm{N}\), extremely large. Yes, the force is substantial.

Step 1: Determine the Number of Protons in Each Bag

First, find the number of protons in a 1.0-gram bag. The mass of a single proton is approximately \(1.67 \times 10^{-24} \, \mathrm{g}\). Therefore, the number of protons \(N\) can be calculated using the formula: \[ N = \frac{\text{mass of bag}}{\text{mass of one proton}} = \frac{1.0 \, \mathrm{g}}{1.67 \times 10^{-24} \, \mathrm{g/proton}} \approx 6.0 \times 10^{23} \]\protons.

Step 2: Calculate the Gravitational Attraction Between the Bags

The gravitational force \(F_g\) between two masses is given by Newton's law of gravitation: \[ F_g = \frac{G m_1 m_2}{r^2} \]where \(G = 6.674 \times 10^{-11} \, \mathrm{N(m/kg)^2}\), \(m_1 = m_2 = 0.001 \, \mathrm{kg}\), and \(r\) is the diameter of the Earth (approximately \(1.275 \times 10^7 \, \mathrm{m}\)). Substitute these values into the equation to find \(F_g\):\[ F_g = \frac{6.674 \times 10^{-11} \times 0.001 \times 0.001}{(1.275 \times 10^7)^2} \approx 2.75 \times 10^{-20} \, \mathrm{N}\].

Step 3: Calculate the Electrical Repulsion Between the Bags

The electrical force \(F_e\) is calculated using Coulomb's law: \[ F_e = \frac{k |q_1 q_2|}{r^2} \]where \(k = 8.99 \times 10^9 \, \mathrm{N(m^2/C^2)}\) and the charge \(q\) of a single proton is \(1.6 \times 10^{-19} \, \mathrm{C}\). The total charge in each bag \(q\) is \(N \times q = (6.0 \times 10^{23}) \times (1.6 \times 10^{-19} \, \mathrm{C}) = 9.6 \times 10^{4} \, \mathrm{C}\).Substitute these values into Coulomb's law:\[ F_e = \frac{8.99 \times 10^9 \times (9.6 \times 10^{4})^2}{(1.275 \times 10^7)^2} \approx 5.3 \times 10^{17} \, \mathrm{N}\].

Step 4: Compare the Forces and Determine Human Sensitivity

Humans can generally feel forces starting from a few tenths of a newton, but forces of the magnitude \(2.75 \times 10^{-20} \, \mathrm{N}\) (gravitational) are negligible to human perception. However, the electrical repulsion force of \(5.3 \times 10^{17} \, \mathrm{N}\) is extraordinarily large and certainly significant enough to be felt, or more accurately, it would be catastrophically impactful due to its immense magnitude.

Key Concepts

Gravitational Force

Gravitational force is a natural phenomenon by which all things with mass or energy are brought toward one another. In the exercise, we deal with two small masses, specifically 1 gram bags of protons, placed at opposite poles of the Earth. The gravitational force between them is calculated using Newton's law of gravitation. Despite seeming significant, the gravitational force between these bags is incredibly small, approximately \(2.75 \times 10^{-20} \, \text{N}\).

This tiny force arises because gravity depends on mass, and the mass of protons themselves is minuscule. Therefore, even when accumulated, proton masses don't give rise to a large force over such vast distances, like the diameter of the Earth. In real-life scenarios, gravitational forces are most noticeable when at least one of the objects has a substantial mass, such as a planet or star.

Gravitational forces are central in governing the orbits in our solar system and structuring galaxies, but they operate on a substantially larger scale than the one considered in this exercise.

Coulomb's Law

Coulomb's law describes how electric charges interact. It tells us that the electric force between charged objects is directly proportional to the product of their charges and inversely proportional to the square of their separation distance. In the exercise, we apply Coulomb's law to find the electric repulsion between bags filled with protons.

Each bag's total charge is calculated by multiplying the number of protons by the charge of one proton, yielding a substantial charge of \(9.6 \times 10^4 \, \text{C}\) per bag. Consequently, the electric force between the bags comes out extraordinarily large at \(5.3 \times 10^{17} \, \text{N}\).

This immense force starkly contrasts with the gravitational force and highlights how powerful electromagnetic interactions can be, especially relative to gravitational ones. On an atomic and subatomic level, these electric forces dominate and control interactions and bonding between particles, making them fundamental to understanding chemical reactions and the structure of matter.

Newton's Law of Gravitation

Newton's law of gravitation, an important principle, explains how gravitational forces are affected by mass and distance. It tells us that the gravitational force between two masses \(m_1\) and \(m_2\) is proportional to the product of these masses and inversely proportional to the square of the distance \(r\) separating them: \[ F_g = \frac{G m_1 m_2}{r^2} \].

In our exercise, the calculations show how, despite being essential on a cosmic scale, gravitational forces are nearly negligible when small masses are vast distances apart. The tiny value calculated for gravitational force illustrates why larger planetary masses are essential for noticeable gravitational interactions. This aspect of gravitational force emphasizes how dependent the force is on scale, becoming especially weak in contexts involving small, separated particles like the bags of protons described in the problem.

Understanding Newton's law of gravitation is key in fields ranging from astrophysics to engineering, affecting everything from satellite trajectories to understanding galaxy dynamics.

Protons

Protons are subatomic particles found in the nucleus of every atom, contributing to the atom's mass and positive charge. In the exercise, we find out how many protons are contained within a 1.0-gram bag, leading to about \(6.0 \times 10^{23}\) protons in each bag by dividing the total mass by the mass of a single proton.

Protons carry a positive charge of \(1.6 \times 10^{-19} \, \text{C}\) each, which contributes to the immense electric force when many are grouped together. Such charged particles are crucial in chemical interactions and also in technologies like nuclear reactors and accelerators.

The concept of protons is foundational when discussing elements and isotopes in chemistry as well as when exploring states and changes in matter energetically or structurally. The behavior of protons under electromagnetic forces, as explained by Coulomb's law, demonstrates their powerful role in both everyday phenomena and advanced scientific applications.