Problem 81 - University Physics with Modern Physics

Question

If Atoms Were Not Neutral... Because the charges on the electron and proton have the same absolute value, atoms are electrically neutral. Suppose this were not precisely true, and the absolute value of the charge of the electron were less than the charge of the proton by 0.00100\(\% .\) (a) Estimate what the net charge of this textbook would be under these circumstances. Make any assumptions you feel are justified, but state clearly what they are. (Hint: Most of the atoms in this textbook have equal numbers of electrons, protons, and neutrons.) (b) What would be the magnitude of the electric force between two textbooks placed 5.0 \(\mathrm{m}\) apart? Would this force be attractive or repulsive? Estimate what the acceleration of each book would be if the books were 5.0 \(\mathrm{m}\) apart and there were no non-electric forces on them. (c) Discuss how the fact that ordinary matter is stable shows that the absolute values of the charges on the electron and proton must be identical to a very high level of accuracy.

Step-by-Step Solution

Answer

Enormous forces show that electron and proton charges must be nearly identical for stability.

Step 1: Understanding the problem

We are asked to estimate the net charge of a textbook if the charge of an electron were slightly less than the charge of a proton by 0.001%. We assume that most of the atoms in the textbook have equal numbers of electrons and protons, so any net charge arises from this discrepancy. Additionally, we need to determine the electric force and acceleration between two such charged textbooks placed 5 meters apart, while also discussing the stability of ordinary matter.

Step 2: Determine net charge of the textbook

Assume an average textbook has about 1 kg mass. Approximately, a textbook primarily consists of carbon atoms, each weighing around 12 atomic mass units (amu). Using Avogadro's number, \[ N_A = 6.022 \times 10^{23} \text{ atoms/mole}, \] find the approximate number of atoms in the textbook: \[ \text{Number of carbon atoms} = \frac{1 \text{ kg}}{12\times10^{-3} \text{ kg/mole}} \times 6.022 \times 10^{23} \approx 5.02 \times 10^{25} \text{ atoms}. \] Considering one electron per atom, the excess positive charge is \[ \Delta q = (1.6 \times 10^{-19} \times 0.001 \%) \times 5.02 \times 10^{25} \approx 8.03 \times 10^{14}\,\mathrm{C}. \]

Step 3: Calculate electric force between textbooks

Use Coulomb's law to find the force between two textbooks. The formula is \[ F = \frac{{k |q_1 q_2|}}{{r^2}}, \] where \( q_1 = q_2 = 8.03 \times 10^{14} \mathrm{C} \), \( r = 5.0 \mathrm{m} \), and \( k = 8.99 \times 10^9 \mathrm{Nm^2/C^2} \). Compute \( F = \frac{(8.99 \times 10^9 \times (8.03 \times 10^{14})^2)}{5^2} \approx 2.31 \times 10^{40} \mathrm{N}. \) The force is repulsive because like charges repel.

Step 4: Estimate acceleration of each book

Newton's second law, \( F = ma \), gives acceleration \( a = \frac{F}{m} \). Assuming each textbook, \( m = 1\, \mathrm{kg}, \) the acceleration is \( a = \frac{2.31 \times 10^{40}}{1} = 2.31 \times 10^{40} \mathrm{m/s^2}. \)

Step 5: Discuss stability of ordinary matter

The fact that ordinary matter is stable suggests that the charges of the electron and proton must be nearly identical. Even a minuscule discrepancy could create enormous repulsive forces, causing atoms and molecules to disintegrate. Thus, the precision in the equivalence of the electron and proton charge is crucial for stability in matter.

Key Concepts

Atomic Structure

The atomic structure is the basic framework of all matter, consisting of three primary subatomic particles: protons, neutrons, and electrons. Protons, bearing a positive charge, reside in the nucleus of an atom along with neutrons, which have no charge. Surrounding this nucleus are electrons, negatively charged particles. The balance of these charges is what makes atoms electrically neutral: the positive charge from the protons exactly cancels out the negative charge from the electrons.

This delicate balance is a cornerstone of chemistry and physics. If, hypothetically, the charges on these particles were slightly different—for example, if electrons had a slightly lesser charge than protons—this neutrality disappears, leading to potentially massive impacts on how matter holds together. For instance, it would introduce net charges in everyday objects, dramatically changing how they interact with each other through electrical forces. This highlights the importance of charge balance in atomic structure, which ensures that under normal circumstances, atoms rest in their stable, neutral state.

Coulomb's Law

Coulomb's Law describes how two charged particles interact with each other electrically. It states that the electric force ( F ) between two charges is directly proportional to the product of the magnitudes of the two charges ( q_1 and q_2 ) and inversely proportional to the square of the distance ( r ) between them: \[ F = \frac{{k |q_1 q_2|}}{{r^2}} \]

Here, k is Coulomb's constant, approximately 8.99 \times 10^9 Nm²/C², which reflects the strength of the electric force in the vacuum. This law concludes that electric forces are incredibly strong, compared to gravitational forces, for example. This strength means even a small charge difference can produce significant forces, as we saw in the problem where books were assumed to have a very slight charge due to a hypothetical difference in electron and proton charges.

Also, because force is dependent on whether the charges are similar or opposite, Coulomb's law explains attraction and repulsion scenarios: like charges repel, while unlike charges attract.

Stability of Matter

The stability of matter is a fundamental concept in physics and chemistry, crucially dependent on the balance of electronegative and electropositive forces within an atom. If an atom has an equal number of protons and electrons, it achieves a state of electrical neutrality, contributing to the substance's overall stability.

When this neutrality is challenged, such as through unequal charges as considered in the exercise, it poses a significant threat to matter stability. Even a tiny imbalance, like a 0.001% difference between proton and electron charges, could make huge differences. This discrepancy could lead to potentially destructive forces between everyday objects.

• Disintegration: Molecules could not hold together.
• Repulsion: Massive repulsion forces could be triggered, preventing substances from coming close.

These are just a few manifestations of instability, reinforcing why perfect charge balance in atoms is essential for the stable and predictable world we observe.

Electric Force

Electric force is one of the four fundamental forces of nature, alongside gravity, strong nuclear, and weak nuclear forces. It acts between charged particles and can be either attractive or repulsive, depending on the charges involved. Through this force, particles with like charges repel one another, while those with opposite charges attract.

This concept is pivotal because it explains the behavior of particles at both atomic and macroscopic levels. In the given exercise, if textbooks had net charges due to charge discrepancies, they would experience electric forces in remarkably high magnitudes, despite being separated by a considerable distance like 5 meters.

Electric force governs many aspects of everyday phenomena, from the structure of atoms to the formation of molecules to the behavior of electric currents. Understanding electric force also unveils why matter prefers a neutral state: it minimizes disruptive forces, maintaining structural integrity and functional consistency in materials.