Problem 14 - University Physics with Modern Physics

Question

22\. 14. Electric Fields in an Atom. The nuclei of large atoms, such as uranium, with 92 protons, can be modeled as spherically symmetric spheres of charge. The radius of the uranium nucleus is approximately \(7.4 \times 10^{-15} \mathrm{m}\) (a) What is the electric field this nucleus produces just outside its surface? (b) What magnitude of electric field does it produce at the distance of the electrons, which is about \(1.0 \times 10^{-10} \mathrm{m} ?(\mathrm{c})\) The electrons can be modeled as forming a uniform shell of negative charge. What net electric field do they produce at the location of the nucleus?

Step-by-Step Solution

Answer

The electric field at the nucleus is zero due to the electrons, strong just outside the nucleus, and weaker at the electrons' distance.

Step 1: Calculate the Charge of the Nucleus

First, calculate the total charge of the uranium nucleus. A uranium nucleus contains 92 protons, and each proton has a charge of \( e = 1.602 \times 10^{-19} \text{C} \). The total charge \( Q \) is given by \( Q = 92 \times e = 92 \times 1.602 \times 10^{-19} \text{C} \).

Step 2: Determine the Electric Field Just Outside the Nucleus

The electric field just outside the surface of a spherically symmetric charged sphere is given by the formula: \[ E = \frac{1}{4\pi\varepsilon_0} \times \frac{Q}{r^2} \]where \( r = 7.4 \times 10^{-15} \text{ m} \) is the radius of the nucleus, and \( \varepsilon_0 \) is the vacuum permittivity \( = 8.85 \times 10^{-12} \text{ C}^2/\text{N} \cdot \text{m}^2 \). Substitute the known values to find \( E \).

Step 3: Calculate the Field at Electron Distance

Use the same formula for the electric field with a different \( r \) to find the electric field at the distance of the electrons: \[ E = \frac{1}{4\pi\varepsilon_0} \times \frac{Q}{r^2} \]where \( r = 1.0 \times 10^{-10} \text{ m} \). Using the previously calculated \( Q \), substitute \( r \) with the new distance.

Step 4: Calculate the Net Electric Field at the Nucleus Due to Electrons

The electrons surrounding a nucleus can be approximated as a uniform shell of charge. By Gauss's law, a charge shell produces no electric field inside itself. Therefore, at the location of the nucleus, the electric field due to the electrons is zero.

Key Concepts

Uranium Nucleus

The uranium nucleus is a fascinating structure in the field of nuclear physics. It is composed of protons and neutrons and houses 92 protons. Each proton carries a positive charge of approximately \(1.602 \times 10^{-19} \text{ C}\). Thus, the total charge of the uranium nucleus can be calculated by multiplying this charge with the number of protons, leading to a significant positive charge.
The nucleus is extremely small, with a radius of about \(7.4 \times 10^{-15} \text{ m}\). Despite its tiny size, it is packed with energy, due to both the mass of the protons and neutrons, and the electric potential energy generated by the charges.Being a primary source of charge, a uranium nucleus generates an electric field just outside its surface. Understanding this field is essential for grasping the interactions within an atom and the forces that hold the nucleus together.

Spherically Symmetric Charge

When we say the uranium nucleus is a "spherically symmetric charge," we imply that the charge density is evenly distributed across the spherical volume of the nucleus. This symmetry simplifies the calculations of electric fields around it.
By considering the charge as spherically symmetric, we use the formula for scalar electric fields, avoiding complex vector calculations. This symmetry allows us to treat the nucleus as if all of its charge were concentrated at its center when determining the electric field at points outside the nucleus.

• The symmetry results in identical electric field magnitudes at any point equidistant from the nucleus.
• This uniformity implies a predictable electric potential energy surrounding the nucleus, essential for understanding atomic behavior and reactions.

Such considerations help predict forces and energies within atoms, crucial for imagining nuclear reactions and bonding scenarios in chemistry and physics.

Gauss's Law

Gauss's law is a pivotal concept when dealing with electric fields emanating from symmetric charge distributions, like the uranium nucleus. It states that the net electric flux through a closed surface is proportional to the charge enclosed by that surface. Mathematically, it is expressed as: \[ \Phi = \oint \mathbf{E} \cdot d\mathbf{A} = \frac{Q_{\text{enc}}}{\varepsilon_0} \]where \(\Phi\) is the electric flux, \(\mathbf{E}\) is the electric field, \(d\mathbf{A}\) is a differential area of the closed surface, \(Q_{\text{enc}}\) is the total charge enclosed within the surface, and \(\varepsilon_0\) is the vacuum permittivity.
In the case of the uranium nucleus, Gauss’s law helps to confirm that a spherically symmetric charge produces an electric field that remains unchanged within a hollow symmetric shell. This means that if you are inside a uniformly charged spherical shell, the net electric field felt is zero, highlighting the nature of electric shielding.

Vacuum Permittivity

Vacuum permittivity, symbolized as \(\varepsilon_0\), is a fundamental physical constant crucial in the computation of electric fields. It characterizes the capacity of the vacuum to permit electric field lines. The value of vacuum permittivity is approximately \(8.85 \times 10^{-12} \text{ C}^2/\text{N} \cdot \text{m}^2\).
This constant appears in Gauss's law and Coulomb's law, linking electric field strength to the charge that generates it. It provides a scale to measure how much electric field can pass through a medium, with vacuum being the least resistive medium possible.
Understanding vacuum permittivity is essential for grasping electric interactions in a vacuum. It frames the scale and strength of interactions between charged particles, making it a vital element in fields like electromagnetism, atomic physics, and electrical engineering. By standardizing the interaction magnitude, \(\varepsilon_0\) allows consistent calculation and comparison of electric fields across different scenarios.