Problem 79 - University Physics with Modern Physics

Question

cp Strength of the Electric Force. Imagine two 1.0 -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

Each bag contains about 6.0 × 10^{23} protons. The electrical repulsion is significantly stronger than the gravitational attraction and is enough to feel.

Step 1: Calculate Number of Protons in Each Bag

We know the mass of each bag is 1.0 g, and the mass of a single proton is approximately \(1.67 \times 10^{-24}\) g. To find the number of protons in each bag, divide the total mass of the bag by the mass of a single proton: \[\text{Number of protons} = \frac{1.0 \text{ g}}{1.67 \times 10^{-24} \text{ g/proton}}.\]This results in approximately \(6.0 \times 10^{23}\) protons per bag.

Step 2: Compute Gravitational Attraction between Bags

The gravitational force \(F_g\) between two objects is given by Newton's law of universal gravitation: \[F_g = G \frac{m_1 m_2}{r^2}\]where \(G\) is the gravitational constant (\(6.674 \times 10^{-11} \text{ N m}^2/\text{kg}^2 \)), \(m_1\) and \(m_2\) are the masses of the bags in kg, and \(r\) is the distance between them. Converting 1.0 g to kg gives 0.001 kg. The distance from pole to pole is the Earth's diameter, approximately 12,742 km or \(1.2742 \times 10^7\) m. \[F_g = \frac{6.674 \times 10^{-11} \times 0.001 \times 0.001}{(1.2742 \times 10^7)^2}\]Solving, \(F_g\) is approximately \(4.11 \times 10^{-16} \text{ N}\).

Step 3: Calculate Electrical Repulsion between Bags

The electrical force \(F_e\) due to the protons' charges can be calculated using Coulomb's law: \[F_e = k_e \frac{q_1 q_2}{r^2} \]where \(k_e\) is Coulomb's constant (\(8.988 \times 10^9 \text{ N m}^2/\text{C}^2\)), \(q_1\) and \(q_2\) are the total charges of the bags, and \(r\) is the same distance as above. The charge of a single proton is \(1.6 \times 10^{-19}\) C, so total charge per bag is approximately \(6.0 \times 10^{23} \times 1.6 \times 10^{-19} = 9.6 \times 10^4\) C.\[F_e = \frac{8.988 \times 10^9 \times 9.6 \times 10^4 \times 9.6 \times 10^4}{(1.2742 \times 10^7)^2}\]This results in \(F_e\) being approximately \(5.06 \times 10^{21} \text{ N}\).

Step 4: Compare Magnitude of Forces and Determine Sensibility

Comparing the magnitudes, \(F_e\) at \(5.06 \times 10^{21} \text{ N}\) is astronomically larger than \(F_g\) at \(4.11 \times 10^{-16} \text{ N}\). The electrical repulsion force is sufficient for one to feel if holding one of the bags, as it's many orders of magnitude larger than any typical forces a human can exert or resist.

Key Concepts

Electric Force

Electric force is a fundamental aspect of electromagnetism that describes how charged objects exert forces on each other. This force can be either attractive or repulsive.
For positively and negatively charged particles, the forces attract, while similar charges repel each other. This inherent characteristic not only governs small-scale interactions but also plays a role in larger, complex systems in fields such as electronics and chemistry.
Key characteristics of electric force include:

• Directionality: It exists along a line connecting two charged objects.
• Magnitude and distance: The force becomes stronger as charges increase or as distance decreases.

Electric force is crucial for understanding phenomena such as static electricity and the operation of electronic devices.
It helps explain how protons and electrons interact vastly differently from how objects experience gravitational forces.

Gravitational Force

Gravitational force is the attraction between objects due to their mass. It's one of the four fundamental forces of nature, impacting everything from apple falls to planet movements.
Unlike the electric force, gravitational force only attracts and never repels mass. This means that all masses in the universe are drawn toward each other, shaping the large-scale structure of the cosmos.
Properties of gravitational force include:

• Universality: Every object with mass exerts and experiences this force.
• Inverse square law: The strength of the force decreases with the square of the distance separating two masses.

This is the force you feel holding you down to Earth. Its effects are evident in tides due to the moon's pull or in the orbits of planets around the sun.
Despite its apparent strength on large scales, on a small scale, it is much weaker compared to other fundamental forces, including electromagnetism.

Coulomb's Law

Coulomb's law quantifies the electric force between two point charges. Formulated by Charles-Augustin de Coulomb in the 18th century, this law is foundational for electrostatics.
The law is expressed as:
\[ F_e = k_e \frac{q_1 q_2}{r^2} \]
where:

• \( F_e \) is the electric force in newtons (N),
• \( k_e \) is Coulomb's constant, approximately \(8.988 \times 10^9 \text{ N m}^2/\text{C}^2 \),
• \( q_1 \) and \( q_2 \) are the point charges in coulombs (C),
• \( r \) is the distance between the charges in meters (m).

Coulomb's Law demonstrates:

• The force increases with more charge or less distance.
• The force has the same field direction as the line through the charges.

Through this law, we can predict how charged particles interact, a fundamental insight for creating devices that manipulate or harness electrical charges.

Newton's Law of Universal Gravitation

Isaac Newton established the law of universal gravitation to explain how every particle attracts every other particle in the universe. This law provides a comprehensive framework for gravitational interactions on all scales.
The formula is given by:
\[ F_g = G \frac{m_1 m_2}{r^2} \]
where:

• \( F_g \) is the gravitational force in newtons (N),
• \( G \) is the universal gravitational constant, approximately \(6.674 \times 10^{-11} \text{ N m}^2/\text{kg}^2 \),
• \( m_1 \) and \( m_2 \) are the masses in kilograms (kg),
• \( r \) is the distance between the centers of masses in meters (m).

This describes how gravity operates ubiquitously, whether on the Earth’s surface or between distant galaxies.
Understanding this principle is crucial for considering the gravitational force as a long-range interaction that organizes universe-scale events. Yet, its effect in small scales can become negligible compared to electromagnetic interactions.