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\$$times10^{16}$$−3.63×1016 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.

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 16c

Chapter 22, Problem 16c

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.

Verified step by step guidance

First, understand the concept of electric flux. Electric flux (Φ) is the measure of the electric field passing through a given area. It is given by the formula:

Φ = E A cos θ

, where E is the electric field, A is the area, and θ is the angle between the field lines and the normal to the surface.

Next, recall Gauss's Law, which relates the electric flux through a closed surface to the charge enclosed by that surface. Gauss's Law is expressed as:

Φ = \frac{Q}{ε}

, where Q is the total charge enclosed and ε is the permittivity of free space (

ε = 8.85 \times 10 ⁻ 12 C ² N ⁻ 1 m ⁻ 2

To find the charge density (σ), use the formula:

σ = \frac{Q}{A}

, where A is the surface area of Mars. The surface area of a sphere is given by:

A = 4 π r ²

, with r being the radius of Mars.

Combine Gauss's Law and the formula for charge density to express σ in terms of Φ and ε:

σ = \frac{Φ}{ε}

. This equation allows you to calculate the charge density using the given electric flux and known values.

Finally, substitute the given values into the equation. Use the electric flux value of

- 3.63 \times 10 ¹ ⁶

N·m²/C, the permittivity of free space, and the radius of Mars (approximately

3.39 \times 10 ⁶

m) to find the charge density.

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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, it helps quantify the electric field generated by Mars, which is crucial for determining the charge distribution on its surface.

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 essential for calculating the charge density on Mars by using the given electric flux value.

Charge Density

Charge density refers to the amount of electric charge per unit area on a surface. It is calculated by dividing the total charge by the surface area. Understanding charge density is crucial for determining how the electric charge is distributed across Mars's surface, assuming uniform distribution as stated in the problem.

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A hollow, conducting sphere with an outer radius of $0.250$ m and an inner radius of $0.200$ m has a uniform surface charge density of $+ 6.37 \times 10^{- 6}$ C/m². A charge of $negative 0.500$ $μ$ C is now introduced at the center of the cavity inside the sphere. What is the new charge density on the outside of the sphere?

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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

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

A hollow, conducting sphere with an outer radius of $0.250$ m and an inner radius of $0.200$ m has a uniform surface charge density of $+6.37$\times$10^{-6}$ C/m². A charge of $−0.500$ $\mu$ C is now introduced at the center of the cavity inside the sphere. Calculate the strength of the electric field just outside the sphere?

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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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