Question

Use Coulomb's law, \(F=k Q_{1} Q_{2} / d^{2}\), to calculate the electric force on an electron \(\left(Q=-1.6 \times 10^{-19} \mathrm{C}\right)\) exerted by a single proton if the particles are \(0.53 \times 10^{-10} \mathrm{~m}\) apart. The constant \(k\) in Coulomb's law is \(9.0 \times 10^{9} \mathrm{~N} \cdot \mathrm{m}^{2} / \mathrm{C}^{2}\). (The unit abbreviated \(\mathrm{N}\) is the Newton, the SI unit of force.)

Short answer

The electric force on the electron exerted by a single proton is approximately \( -8.21 \times 10^{-8} \mathrm{N} \). The negative sign indicates that the force is attractive, as the electron and proton have opposite charges.

Step-by-step solution

List the given values

We are given the following values: - Charge of electron (Q1): -1.6 × 10^{-19} C - Charge of proton (Q2): +1.6 × 10^{-19} C (equal and opposite of the electron's charge) - Distance between the charges (d): 0.53 × 10^{-10} m - Coulomb's constant (k): 9.0 × 10^9 N·m^2 / C^2

Apply Coulomb's law formula

Using the formula for Coulomb's law, we can find the electric force (F): \[F=k \frac{Q1 \cdot Q2}{d^{2}}\]

Substitute the given values into the formula

Now, we can substitute the given values into the formula: \[F= (9.0 \times 10^{9}\ \mathrm{N \cdot m^{2} / C^{2}}) \frac{(-1.6 \times 10^{-19}\ \mathrm{C}) \cdot (1.6 \times 10^{-19}\ \mathrm{C})}{(0.53 \times 10^{-10}\ \mathrm{m})^{2}}\]

Calculate the electric force

Perform the calculation to find the electric force F: \[F= (9.0 \times 10^{9}) \cdot (-1.6 \times 10^{-19}) \cdot (1.6 \times 10^{-19}) / (0.53 \times 10^{-10})^{2}\] \[F= (-2.304 \times 10^{-28})/(2.809 \times 10^{-21})\] \[F= -8.21 \times 10^{-8} \mathrm{N}\] The electric force on the electron exerted by a single proton is approximately -8.21 × 10^{-8} N. The negative sign indicates that the force is attractive, which makes sense because the electron and proton have opposite charges.

Key concepts

Electric Force Calculation

Coulomb's Law plays a critical role in physics by allowing us to calculate the electric force between two charged particles, such as an electron and a proton. This law is expressed by the formula: \[ F = k \frac{Q_1 Q_2}{d^2} \] where:

• F is the magnitude of the electric force between the charges.
• k is Coulomb's constant \((9.0 \times 10^9 \mathrm{N \cdot m^2 / C^2})\).
• Q1 and Q2 represent the charges of the two particles (in Coulombs).
• d is the distance separating them (in meters).

By plugging in the values from the exercise, the electric force F is computed to be approximately \(-8.21 \times 10^{-8} \; \mathrm{N}\). It is key to note that the negative sign in the result denotes the attractive nature of the force, as opposite charges attract each other.

Electron and Proton Interaction

The interaction between an electron and a proton is governed by their charges. An electron carries a negative charge of \(-1.6 \times 10^{-19} \; \mathrm{C}\), while a proton has a positive charge of \(+1.6 \times 10^{-19} \; \mathrm{C}\). This opposite charge pairing induces an attractive force. When you calculate the electric force using Coulomb's Law, the negative result is a key indicator of this attraction. The force is not just a result of the magnitude of these charges, but also of the inherent property that opposite charges exert a force towards each other. This fundamental electrostatic interaction is a cornerstone in understanding atomic structure, as it describes why electrons remain in proximity to protons, like those in the nucleus of an atom.

Charge and Distance Relationship

The force exerted between two charged particles depends heavily on both the magnitude of their charges and the distance separating them. According to Coulomb’s Law, the force is inversely proportional to the square of the distance. This means:

• As the distance \(d\) between charges increases, the electric force \(F\) decreases dramatically.
• Conversely, reducing the distance leads to a significant increase in force.

The formula \( F = k \frac{Q_1 Q_2}{d^2} \) highlights that even slight changes in distance have a pronounced impact on the force. In atomic-scale interactions, such as the electron and proton example, the small distances involved mean that forces could be substantial despite the minimal magnitudes of the charges. Understanding this relationship is pivotal in fields like chemistry and physics, where it helps explain phenomena like molecular bonding and the structure of matter at the atomic level.