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 (b) just outside the paint layer;
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Ch 22: Gauss' Law
Chapter 22, Problem 16
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 N·m2/C at the planet's surface. Calculate: (a) the total electric charge on the planet;
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Identify the given electric flux, which is \\( -3.63 \times 10^{16} \, \text{N} \cdot \text{m}^2/\text{C} \\). The negative sign indicates the direction of the electric field.
Recall Gauss's Law, which relates the electric flux through a closed surface to the charge enclosed by that surface. The law is given by \\( \Phi_E = \frac{Q_{\text{enc}}}{\epsilon_0} \\) where \\( \Phi_E \\) is the electric flux, \\( Q_{\text{enc}} \\) is the total enclosed charge, and \\( \epsilon_0 \\) is the permittivity of free space (approximately \\( 8.85 \times 10^{-12} \, \text{C}^2/\text{N} \cdot \text{m}^2 \\)).
Rearrange Gauss's Law to solve for the total enclosed charge, \\( Q_{\text{enc}} \\). The formula becomes \\( Q_{\text{enc}} = \Phi_E \times \epsilon_0 \\).
Substitute the values of \\( \Phi_E \\) and \\( \epsilon_0 \\) into the equation to find \\( Q_{\text{enc}} \\).
The result from the calculation will give you the total electric charge on the planet Mars in coulombs (C).
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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 quantity of electric field lines passing through a given area. It is calculated as the product of the electric field strength and the area through which it passes, adjusted for the angle between the field lines and the normal to the surface. The unit of electric flux is Newton-meters squared per Coulomb (N·m²/C). Understanding electric flux is essential for calculating the total electric charge using Gauss's law.
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Electric Flux
Gauss's Law
Gauss's Law relates the electric flux through a closed surface to the charge enclosed by that surface. Mathematically, it states that the total electric flux through a closed surface is equal to the enclosed charge divided by the permittivity of free space. This principle is fundamental in electrostatics and allows for the calculation of electric fields and charges in symmetrical situations, such as spherical bodies like planets.
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Total Electric Charge
Total electric charge is a scalar quantity that represents the net amount of electric charge within a system. It can be positive or negative and is measured in Coulombs (C). In the context of the question, calculating the total electric charge on Mars involves using the electric flux value provided and applying Gauss's Law to determine how much charge is present on the planet's surface.
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Related Practice
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 7.4×10−15 m. (a) What is the electric field this nucleus produces just outside its surface?
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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 7.4×10−15 m. (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?
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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 −3.63×1016 N·m2/C at the planet's surface. Calculate: (b) 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 −3.63×1016 N·m2/C at the planet's surface. Calculate:(c) the charge density on Mars, assuming all the charge is uniformly distributed over the planet's surface.
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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 +6.37×10−6 C/m2. A charge of −0.500 μC is now introduced at the center of the cavity inside the sphere. (a) What is the new charge density on the outside of the sphere?
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