Problem 83 - 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? Batimate what the acceleration of each book would be if the books were 5.0 \(\mathrm{m}\) apart and there were no nonelectrical 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

(a) Net charge is approximately 107 C. (b) The force is about \(4.09 \times 10^{8}\,\text{N}\), repulsive; acceleration is \(4.09 \times 10^{8} \text{ m/s}^2\). (c) Stability indicates equal charges.

Step 1: Define the charge imbalance

The problem states that the charge of an electron is less than that of a proton by 0.00100\(\%\). This means each electron has a charge of \(q_{e} = (1 - 0.00100\%)e = 0.99999e\), where \(e\) is the elementary charge \(1.602 \times 10^{-19}\) C.

Step 2: Estimate number of atoms per textbook

Assume a textbook has a mass of about \(1\, \text{kg}\) and is mostly composed of water molecules (\(\text{H}_2\text{O}\)), which have 10 protons and electrons each. The molar mass of water is approximately \(18\, \text{g/mol}\), so there are roughly \(1000 \text{g} / 18 \text{g/mol} = 55.56\, \text{mol}\) of water. Thus, the approximate number of molecules is \(55.56\text{ mol} \times 6.022 \times 10^{23}\text{ molecules/mol} \), which equals approximately \(3.34 \times 10^{25}\) molecules.

Step 3: Calculate net charge of the textbook

Each water molecule has a net charge imbalance due to the difference in proton and electron charge: \(\Delta q = 2 \times (0.00100\% \times 1.602 \times 10^{-19} \text{ C}) \approx 3.204 \times 10^{-24}\, \text{C}\). Therefore, for the textbook with \(3.34 \times 10^{25}\) molecules, the net charge is \(Q = 3.34 \times 10^{25} \times 3.204 \times 10^{-24} \text{ C}\approx 107 \text{ C}\).

Step 4: Calculate electric force between two textbooks

Using Coulomb's Law, \(F = k \frac{|Q_1 Q_2|}{r^2}\), with \(k = 8.99 \times 10^9\, \text{N}\cdot \text{m}^2 / \text{C}^2\), \(r = 5\, \text{m}\), the force is \(F = 8.99 \times 10^9 \times \frac{(107)^2}{5^2}\approx 4.09 \times 10^{8}\,\text{N}\).

Step 5: Determine direction and type of force

Since both textbooks have a positive net charge, the force between them will be repulsive.

Step 6: Calculate acceleration

The acceleration \(a\) of each textbook is found using \(F = ma\). Assume each textbook has a mass of \(1 \text{kg}\), so \(a = F / m = 4.09 \times 10^{8} \text{ N} / 1 \text{ kg} = 4.09 \times 10^{8} \text{ m/s}^2\).

Step 7: Discuss stability of matter

If the charges of electrons and protons were not identical, there would be enormous forces between electrically neutral objects due to small imbalances, causing them to be unstable. The stability of matter implies that charges must be equal to an extremely high degree of accuracy.

Key Concepts

Charge Imbalance in Atoms

Atoms are typically electrically neutral. This is because they have the same number of protons and electrons, which have equal but opposite charges. When we talk about a charge imbalance in atoms, we mean that the charges of the electrons do not perfectly cancel out the charges of the protons.

Imagine if every electron had a charge that was slightly less than a proton’s, say by 0.00100\(\%\). This seemingly small difference would create a very noticeable net charge in atoms because with all the atoms in a material like a textbook, the discrepancies would add up quickly. For example, each water molecule in a textbook would have a minute net positive charge, resulting in a significant net charge for the entire textbook. In our case, this calculated to a massive 107 C of charge for an object usually neutral under normal circumstances.

This teaches us that even tiny changes at the atomic level can have extensive consequences on a larger scale. The perfect balance in charges ensures the neutral state of matter and highlights the precision nature enacts to maintain this balance.

Electric Force

Electric force is the interaction between charged particles due to their electric charges. It has a significant role in the fundamental forces of nature and determines how charged objects interact. This force can be calculated using Coulomb’s Law, which states the force between two point charges is equal to the Coulomb's constant times the product of the two charges divided by the square of the distance separating them.

In a scenario where textbooks have a net charge, as computed before, we can estimate the electric force between them. By using the net charge of 107 C and the known separation of 5 meters, Coulomb’s Law allows us to calculate an electrical force of approximately \(4.09 \times 10^8\) N. This force is repulsive because both textbooks have positive charges.

Such forces, when unopposed by other forces, could cause dramatic movement, evidenced by the calculated acceleration of \(4.09 \times 10^8\, \mathrm{m/s}^2\). Normally, electric forces maintain balance to stabilize structures at molecular and macroscopic levels, keeping matter intact and functional.

Stability of Matter

The stability of matter is a fundamental aspect of physical reality. It ensures that objects and substances maintain their shape and structure without spontaneously separating or attracting excessively. This stability is largely thanks to the precise balance of charges in atoms.

If electrons and protons did not have equal and opposite charges, matter would not be stable. The minor charge imbalance in a single atom would amplify as larger objects are formed. These disproportionate charges would lead to significant electric forces that might be strong enough to tear materials apart.

This natural precision—which ensures that protons and electrons neutralize each other's charges so effectively—underscores how delicate yet perfectly attuned are the forces that govern the material world. This stability allows chemistry, biology, and ultimately life itself to emerge and persist without disruptive electrical forces overwhelming stable structures.