The nuclei of large atoms, such as uranium, with 929292 protons, can be modeled as spherically symmetric spheres of charge. The radius of the uranium nucleus is approximately 7.4×10−157.4\$$times10^{-15}7.4×10$$−15 m. 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?

Ch 22: Gauss' Law

Young & Freedman Calc - University Physics 14th Edition

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Young & Freedman Calc 14th Edition

Ch 22: Gauss' Law

Problem 12c

Chapter 22, Problem 12c

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$10^{-15}$ m. 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?

Verified step by step guidance

Understand that the problem involves calculating the net electric field produced by a shell of negative charge (electrons) at the location of the nucleus.

Recall that according to Gauss's Law, the electric field inside a uniformly charged shell is zero. This is because the electric field contributions from all parts of the shell cancel out at any point inside the shell.

Since the nucleus is located at the center of the electron shell, it is inside the shell. Therefore, the net electric field at the location of the nucleus due to the electron shell is zero.

Express this concept mathematically using Gauss's Law: $ E = \frac{Q_{enc}}{\varepsilon_0 A} $, where $ Q_{enc} $ is the charge enclosed by the Gaussian surface. For a point inside a shell, $ Q_{enc} = 0 $, leading to $ E = 0 $.

Conclude that the net electric field at the location of the nucleus, due to the electrons modeled as a uniform shell of negative charge, is zero.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Electric Field

The electric field is a vector field around a charged particle that represents the force exerted per unit charge at any point in space. It is defined as E = F/q, where F is the force experienced by a small positive test charge q. The direction of the electric field is the direction of the force it would exert on a positive charge.

Gauss's Law

Gauss's Law relates the electric flux through a closed surface to the charge enclosed by that surface. Mathematically, it is expressed as Φ = ∮E·dA = Q_enclosed/ε₀, where Φ is the electric flux, E is the electric field, dA is a differential area on the closed surface, and ε₀ is the permittivity of free space. This law is particularly useful for calculating electric fields with high symmetry.

Shell Theorem

The Shell Theorem states that a spherically symmetric shell of charge exerts no net electric force on a charge located inside the shell. This implies that the electric field inside a uniformly charged spherical shell is zero. This concept is crucial for understanding the electric field produced by the electrons modeled as a shell around the nucleus.

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Textbook Question

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 $7.4 \times 10^{- 15}$ m. What is the electric field this nucleus produces just outside its surface?

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Textbook Question

Some planetary scientists have suggested that the planet Mars has an electric field somewhat similar to that of the earth, producing a net electric flux of $-3.63$\times$10^{16}$ Nm²/C at the planet's surface. Calculate the electric field at the planet's surface (refer to the astronomical data inside the back cover).

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Textbook Question

Some planetary scientists have suggested that the planet Mars has an electric field somewhat similar to that of the earth, producing a net electric flux of $-3.63$\times$10^{16}$ Nm²/C at the planet's surface. Calculate the charge density on Mars, assuming all the charge is uniformly distributed over the planet's surface.

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Textbook Question

A charged paint is spread in a very thin uniform layer over the surface of a plastic sphere of diameter $12.0$ cm, giving it a charge of $−49.0$ $μ$ C. Find the electric field just outside the paint layer;

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Textbook Question

Some planetary scientists have suggested that the planet Mars has an electric field somewhat similar to that of the earth, producing a net electric flux of $- 3.63 \times 10^{16}$ Nm²/C at the planet's surface. Calculate the total electric charge on the planet.

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Textbook Question

A charged paint is spread in a very thin uniform layer over the surface of a plastic sphere of diameter $12.0$ cm, giving it a charge of $negative 49.0$ $μ$ C. Find the electric field just inside the paint layer.

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