Problem 30 - University Physics with Modern Physics

Question

(a) What is the electric field of an iron nucleus at a distance of \(6.00 \times 10^{-10} \mathrm{m}\) from the nucleus? The atomic number of iron is \(26 .\) Assume that the nucleus may be treated as a point charge. (b) What is the electric field of a proton at a distance of \(5.29 \times 10^{-11} \mathrm{m}\) from the proton? (This is the radius of the electron orbit in the Bohr model for the ground state of the hydrogen atom.)

Step-by-Step Solution

Answer

The electric field at a distance of \(6.00 \times 10^{-10}\text{ m}\) from an iron nucleus is \(1.04 \times 10^{11} \text{ N/C}\), and at \(5.29 \times 10^{-11} \text{ m}\) from a proton is \(5.14 \times 10^{11} \text{ N/C}\).

Step 1: Identify the Formula for Electric Field

The electric field generated by a point charge can be calculated using the formula \( E = \frac{k \cdot |q|}{r^2} \), where \( E \) is the electric field, \( k \) is Coulomb's constant \( 8.99 \times 10^9 \text{ N m}^2/\text{C}^2 \), \( q \) is the charge, and \( r \) is the distance from the charge.

Step 2: Calculate Charge of Iron Nucleus

Since the atomic number of iron is 26, the nucleus has a charge of \( q = 26 \times e \), where \( e = 1.602 \times 10^{-19} \text{ C} \) is the elementary charge. Calculate: \( q = 26 \times 1.602 \times 10^{-19} \approx 4.165 \times 10^{-18} \text{ C} \).

Step 3: Calculate Electric Field of Iron Nucleus

Substitute the values into the electric field formula: \( E = \frac{8.99 \times 10^9 \times 4.165 \times 10^{-18}}{(6.00 \times 10^{-10})^2} \approx 1.04 \times 10^{11} \text{ N/C} \).

Step 4: Calculate Charge of Proton

A proton has a charge equal to the elementary charge \( q = e = 1.602 \times 10^{-19} \text{ C} \).

Step 5: Calculate Electric Field of Proton

Using the same formula, substitute the proton's charge and distance: \( E = \frac{8.99 \times 10^9 \times 1.602 \times 10^{-19}}{(5.29 \times 10^{-11})^2} \approx 5.14 \times 10^{11} \text{ N/C} \).

Key Concepts

Coulomb's Law

Coulomb's Law is a central concept in understanding electric fields. It describes the force between two charged objects. According to Coulomb's Law, the electric force between two point charges is directly proportional to the product of the magnitudes of the charges and inversely proportional to the square of the distance between them.
The mathematical expression is given by: - \( F = \frac{k \cdot |q_1 \, q_2|}{r^2} \) - Where:
- \( F \) is the force between the charges,
- \( k \) is Coulomb's constant (\( 8.99 \times 10^9 \, \text{N m}^2/\text{C}^2 \)),
- \( q_1 \) and \( q_2 \) are the amounts of the charges, - \( r \) is the distance separating the charges.
Understanding this law helps in predicting how charged objects will interact. By translating this into the electric field context, you can calculate the field generated by a single point charge using the formula: - \( E = \frac{k \cdot |q|}{r^2} \).
This means the electric field is strongest near the charge and decreases with the square of the distance.

Point Charge

The term 'point charge' is used in physics to denote a charge that is concentrated at a single point in space. It is an idealized model that simplifies calculations and helps to visualize electric fields.
Although real charged objects have some volume, for many purposes—especially at a distance—these can be treated as point charges.
For example, in the exercise discussed, the nucleus of an iron atom is considered a point charge. The benefit of this assumption is that it allows for the use of Coulomb's law to determine the electric field produced by the charge at any given distance.
The simplicity of the point charge model is why it's often employed to model atoms, where the nucleus and individual electrons can be treated as concentrated points of charge while studying electric fields in atomic systems.

Bohr Model

The Bohr model is an early model of atomic structure introduced by Niels Bohr in 1913. It describes the behavior of electrons in atoms.
According to the Bohr model, electrons travel in circular orbits around the nucleus, similar to how planets orbit the sun. The orbits correspond to different energy levels.
A key feature of the Bohr model is that an electron's orbit radii is quantized, meaning it can only occupy certain allowed distances from the nucleus. For instance, in the hydrogen atom's ground state, the electron orbits at a fixed radius of roughly \( 5.29 \times 10^{-11} \text{ m} \).
The Bohr model provides a simple framework to calculate the electric field experienced by electrons due to a proton in the nucleus by treating the proton as a point charge. Though modern physics describes atoms more accurately using quantum mechanics, the Bohr model remains a useful teaching tool.

Elementary Charge

The elementary charge is the fundamental unit of electric charge, symbolized as \( e \). It is a constant and one of the most critical parameters in physics. It represents the smallest unit of charge that is free in nature.
The value of the elementary charge is approximately \( 1.602 \times 10^{-19} \text{ C} \). This charge is carried by a proton, and an electron carries a charge of the same magnitude but opposite in sign.
In calculations, as seen in the exercise, the elementary charge helps in determining the total charge of an object or atom when multiplied by the number of charges (for instance, with the atomic number in an element). - Here, the iron nucleus with an atomic number of 26 means its charge is \( 26e \).
Understanding and using the elementary charge is fundamental for calculations involving electric phenomena.