Ch 22: Gauss' Law

Young & Freedman Calc - University Physics 14th Edition

Young & Freedman Calc14th EditionUniversity PhysicsISBN: 9780321973610Not the one you use?Change textbook

Young & Freedman Calc 14th Edition

Ch 22: Gauss' Law

Problem 16a

Chapter 22, Problem 16a

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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Understand the concept of electric flux and its relation to electric charge. Electric flux (Φ) is the measure of the electric field passing through a surface. According to Gauss's Law, the electric flux through a closed surface is proportional to the charge enclosed by that surface.

Recall Gauss's Law, which states: Φ = Q / ε₀, where Φ is the electric flux, Q is the total electric charge enclosed, and ε₀ is the permittivity of free space (approximately 8.85 × 10⁻¹² C²/N·m²).

Rearrange Gauss's Law to solve for the total electric charge (Q) on the planet: Q = Φ × ε₀.

Substitute the given values into the equation. Use the provided electric flux value of -3.63 × 10¹⁶ N·m²/C and the known value of ε₀.

Perform the multiplication to find the total electric charge on Mars. Remember that the negative sign indicates the direction of the electric field relative to the surface.

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Key Concepts

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

Electric Flux

Electric flux is a measure of the electric field passing through a given surface. It is calculated as the product of the electric field and the area perpendicular to the field. In this context, the net electric flux is given, which helps in determining the total charge using Gauss's Law.

Gauss's Law

Gauss's Law relates the electric flux through a closed surface to the charge enclosed by that surface. It states that the total electric flux is equal to the enclosed charge divided by the permittivity of free space. This principle is crucial for calculating the total electric charge on Mars using the given electric flux.

Permittivity of Free Space

Permittivity of free space, denoted as ε₀, is a fundamental physical constant that characterizes the ability of a vacuum to permit electric field lines. It is essential in Gauss's Law, where it is used to relate electric flux to the enclosed charge. Its value is approximately 8.85 × 10⁻¹² C²/N·m².

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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. 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?

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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?

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

A very long uniform line of charge has charge per unit length $4.80$ $μ$ C/m and lies along the $x$ -axis. A second long uniform line of charge has charge per unit length $negative 2.40$ $μ$ C/m and is parallel to the $x$ -axis at $y equals 0.400$ m. What is the net electric field (magnitude and direction) at the following points on the $y$ -axis: (a) $y equals 0.200$ m and (b) $y equals 0.600$ m?

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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.

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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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 −3.63×1016-3.63\(\times\)10^{16} Nm2/C at the planet's surface. Calculate the total electric charge on the planet.

Verified video answer for a similar problem:

Key Concepts

Electric Flux

Gauss's Law

Permittivity of Free Space

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.