## Applying Gauss’s Law

2.3 Applying Gauss’s Law

### Learning Objectives

By the end of this section, you will be able to:

- Explain what spherical, cylindrical, and planar symmetry are
- Recognize whether or not a given system possesses one of these symmetries
- Apply Gauss’s law to determine the electric field of a system with one of these symmetries

### Applying Gauss’s Law

This video provides examples of how we can apply Gauss’s Law to situations with symmetric charge distributions. It’s a longer than usual video (15 minutes) – just wanted to let you know.

### Strategy for Applying Gauss’s Law

*Identify the spatial symmetry of the charge distribution*. This is an important first step that allows us to choose the appropriate Gaussian surface. As examples, an isolated point charge has spherical symmetry, and an infinite line of charge has cylindrical symmetry.*Choose a Gaussian surface with the same symmetry as the charge distribution and identify its consequences*. With this choice, is easily determined over the Gaussian surface.*Evaluate the integral**over the Gaussian surface, that is, calculate the flux through the surface*. The symmetry of the Gaussian surface allows us to factor outside the integral.*Determine the amount of charge enclosed by the Gaussian surface*. This is an evaluation of the right-hand side of the equation representing Gauss’s law. It is often necessary to perform an integration to obtain the net enclosed charge.*Evaluate the electric field of the charge distribution*. The field may now be found using the results of steps 3 and 4.

**Practice!**

Practice 2.3.1 |
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A wire of length L is made of an insulating material. This wire holds a net charge of Q that is distributed uniformly along the length of the wire. What is the linear charge density, , of this charged wire? |

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Practice 2.3.2 |
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A plane sheet of length L and width W is made of an insulating material. This sheet holds a net charge of Q, which is distributed uniformly over the area of the sheet. What is the surface charge density, , of this charged sheet? |

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Practice 2.3.3 |
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A thin, spherical shell of radius R is made of an insulating material. This spherical shell holds a net charge of Q, which is distributed uniformly over the surface of the spherical shell. What is the surface charge density, , of this charged spherical shell? |

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Practice 2.3.4 |
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A solid cylinder of length L and radius R is made of an insulating material. This cylinder holds a net charge of Q that is evenly distributed throughout the cylindrical volume. What is the volume charge density, , of this charged cylinder? |

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Practice 2.3.5 |
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A very long (nearly infinite) wire is made of an insulating material. The wire carries a charge, Q, which is distributed evenly along the entire length of the wire, giving it a uniform linear charge density, . A Gaussian cylinder of length L and radius R is coaxial with the wire and encloses a portion of the charged wire. What is the charge enclosed by the Gaussian cylinder? |

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Practice 2.3.6 |
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A solid sphere of radius R is made of an insulating material. It holds a charge, Q, which is distributed evenly throughout the sphere and gives it a uniform volume charge density . How much charge is enclosed by a concentric Gaussian sphere of radius r where 0 < r < R, as shown in the figure? |

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Practice 2.3.7 |
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A very long (nearly infinite) wire is made of an insulating material. The wire carries a charge, Q, which is distributed evenly along the entire length of the wire, giving it a uniform linear charge density, . What is the magnitude of the electric field produced by the charged wire at a radial distance s from the wire? |

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Practice 2.3.8 |
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A solid sphere of radius R is made of an insulating material. It holds a charge, Q, which is distributed evenly throughout the sphere and gives it a uniform volume charge density . What is the magnitude of the electric field produced by the charged sphere inside the sphere at a radial distance r from the sphere’s center, where 0 < r < R? |

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**Discuss!**

Reflect on this question and take notes on how you would answer it. Then we will share these thoughts together in a class discussion.

Given an infinite sheet of charge as shown in the figure. You need to use Gauss’s Law to calculate the electric field near the sheet of charge.

Which of the given Gaussian surfaces (A – D) are best suited for this purpose? Note: you may choose more than one answer.

**Practice!**

Practice 2.3.9 |
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A very large (nearly infinite) plane sheet is made of an insulating material. The sheet carries a charge, Q, which is distributed evenly over the entire surface area, giving it a uniform surface charge density, . A Gaussian cylinder of length L and cross-sectional area A encloses a portion of the charged sheet, as shown in the figure. What is the charge enclosed by the Gaussian cylinder? |

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Practice 2.3.10 |
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A very large (nearly infinite) plane sheet is made of an insulating material. The sheet carries a charge, Q, which is distributed evenly over the entire surface area, giving it a uniform surface charge density, . A Gaussian cylinder of length L and cross-sectional area A encloses a portion of the charged sheet, as shown in the figure. What is the magnitude of the electric field produced by the surface charge at a distance s from the sheet? |

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