Problem 57 - Fundamentals of Physics

Question

We know that the negative charge on the electron and the positive charge on the proton are equal. Suppose, however, that these magnitudes differ from each other by \(0.00010 \%\). With what force would two copper coins, placed \(1.0 \mathrm{~m}\) apart, repel each other? Assume that each coin contains \(3 \times 10^{22}\) copper atoms. (Hint: A neutral copper atom contains 29 protons and 29 electrons.) What do you conclude?

Step-by-Step Solution

Answer

Coins would repel each other with a force of \(1.742 \times 10^{8} \text{ N }\), indicating significant effects from small charge differences.

Step 1: Understanding the Charge Difference

First, note that we are given the electron and proton charges differ by \(0.00010 \%\). This implies a small unbalanced charge on each atom due to this difference.

Step 2: Calculating the Unbalanced Charge in a Copper Atom

In a neutral copper atom, there are 29 protons and 29 electrons. The actual charge of one proton (and one electron) is \(e = 1.6 \times 10^{-19}\) C. If the charge on the electron differs by \(0.00010 \%\), then the difference in charge per electron is: \[\Delta q = e \times \frac{0.00010}{100} = 1.6 \times 10^{-19} \times 0.000001 = 1.6 \times 10^{-25} \text{ C}\]Each copper atom thus has a net charge of 29 times this difference since there are 29 unbalanced charges.

Step 3: Total Unbalanced Charge in a Copper Coin

Each coin contains \(3 \times 10^{22}\) copper atoms, each having a net charge of \(q = 29 \times 1.6 \times 10^{-25} \text{ C} = 4.64 \times 10^{-24} \text{ C}\).Therefore, the total unbalanced charge on one coin is:\[Q = 3 \times 10^{22} \times 4.64 \times 10^{-24} = 1.392 \times 10^{-1} \text{ C}\]

Step 4: Calculating the Force of Repulsion

Using Coulomb's Law, the force between two charges is given by:\[F = \frac{k \cdot Q_1 \cdot Q_2}{r^2}\]Where:- \(k = 8.99 \times 10^9 \text{ N m}^2 \text{/C}^2\) is Coulomb's constant,- \(Q_1 = Q_2 = 1.392 \times 10^{-1} \text{ C}\) are the charges on each coin,- \(r = 1.0 \text{ m}\) is the separation distance. Substituting values:\[F = \frac{8.99 \times 10^9 \times 1.392 \times 10^{-1} \times 1.392 \times 10^{-1}}{1.0^2}\]\[F = 8.99 \times 10^9 \times 1.939264 \times 10^{-2}\]\[F = 1.742 \times 10^{8} \text{ N}\]

Step 5: Conclusion

Two copper coins would repel each other with a force of \(1.742 \times 10^{8} \text{ N }\), which is an enormous force for small objects like coins. Thus, even a small disparity in electric charges would cause macroscopic effects.

Key Concepts

Electric Charge

Electric charge is a fundamental property of particles that determines how they interact with other charged particles. Charges come in two types: positive and negative. Electrons carry a negative charge, while protons carry a positive charge.
These charges are measured in units called coulombs, with the basic charge of an electron or proton being approximately \(1.6 \times 10^{-19}\) C. In a stable atom, the number of electrons equals the number of protons, making the atom electrically neutral.
However, when there is an imbalance, even a tiny one, it can result in significant electrical forces as seen in the copper coin problem. A minuscule deviation in charges, such as \(0.00010\%\) different, leads to noticeable repulsion or attraction.

Force of Repulsion

The force of repulsion occurs when two like charges (either both positive or both negative) push away from each other. This force can be calculated using Coulomb's Law, which states that the force \( F \) between two charges \( Q_1 \) and \( Q_2 \) separated by a distance \( r \) is given by:

• \( F = \frac{k \cdot Q_1 \cdot Q_2}{r^2} \)
• Where \( k = 8.99 \times 10^9 \text{ N m}^2 / \text{C}^2 \) is the Coulomb's constant.

In the scenario with copper coins, even a small unbalanced charge results in a massive repulsive force due to the large number of charged particles involved. It highlights how sensitive the electric force is to changes in charge magnitude, even by a tiny percentage.

Atomic Structure

The atomic structure of an atom includes a nucleus containing protons and neutrons, surrounded by electrons in orbitals. Protons are positively charged, while electrons are negatively charged.
Neutrons have no charge and provide stability to the nucleus. In a neutral atom, the number of electrons and protons are equal, balancing the overall charge. However, when there's a tiny deviation in this balance, it results in an unbalanced charge per atom.
Understanding atomic structure is crucial in physics and chemistry as it explains how atoms interact, bond, and create matter as we observe in daily life. This structure determines many properties of materials, including electrical conductivity and reactivity.

Copper Atoms

Copper atoms are a key example when discussing electrical charge because they have 29 protons and 29 electrons under normal conditions, making them neutrally charged. Copper is highly conductive, which is why it is widely used in electrical wiring.
In the problem, each copper atom slightly shifts from neutral due to the charge imbalance, multiplying to a larger effect when considering the number of atoms in a small object like a coin. A neutral copper atom with 29 protons and electrons reinforces our understanding of how minuscule changes can scale up significantly in conductive materials.
Therefore, by examining copper, we recognize both its practical applications in technology and the immense power held in seemingly minor electrical imbalances.