# 20. Ampere's Law

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## Learning Objectives

• To review the difference between an open surface and a closed surface.
• To review what is meant by the line integral of a vector around the bounding contour of an open surface.
• To comprehend the meaning of Ampere's Law.
• To be able to explain the meaning of surface current.
• To use Ampere's Law to calculate magnetic fields for an infinite wire, and for an infinite plane.

## Learning Objectives

Preparation

### Course Notes

Read through the course notes before watching the video.  The course note files may also contain links to associated animations or interactive simulations.

Read sections 9.3 through 9.13:
Sources of Magnetic Fields (PDF - 1.9MB)

## Lecture Video

### Video Excerpts

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Download the complete lectures from this course:

## Learning Activities

### Guided Activities

Read through the class slides. They explain all of the concepts from the module.

Slides (PDF - 1.7MB)

### Self-Assessment

Do the Concept Questions first to make sure you understand the main concepts from this module. Then, when you are ready, try the Challenge Problems.

### Concept Questions

Concept Questions (PDF)

Solutions (PDF)

### Challenge Problems

Challenge Problems (PDF)

Solutions (PDF)

## Problem Solving Help

Watch the Problem Solving Help videos for insights on how to approach and solve problems related to the concepts in this module.

### Problem 1: Using Ampere's Law to Find the Field of a Metallic Cylinder and Cylindrical Shell

A solid cylindrical metal conductor has radius a. It is surrounded by a cylindrical conducting metal shell with inner radius b and outer radius c, with a < b < c. The inner conductor carries a current I1 and the outer cylindrical shell carries a current I2 in the same direction. The currents are uniformly distributed over the conductors through which they flow. What is the magnetic field is everywhere in space?

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» iTunes U (MP4 - 21MB)
» Internet Archive (MP4 - 21MB)

### Problem 2: Magnetic Field

An infinite wire carrying current I is bent into two lengths perpendicular to each other. One segment runs along the negative x-axis up to the origin, and the other segment extends from the origin down the negative y-axis. What is the magnetic field on the x-axis a distance a from the origin?

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» iTunes U (MP4 - 5MB)
» Internet Archive (MP4 - 5MB)

### Problem 3: Wire with Varying Current Density

A long wire of radius R carries a total current I, with a non-uniform current density J=αr. What is the magnetic field everywhere?

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» iTunes U (MP4 - 17MB)
» Internet Archive (MP4 - 17MB)

### Problem 4: Magnetic Field from Moving Sheets of Charge, and from a Rotating Cylindrical Shell of Charge

A large planar sheet has a charge per unit area σ and moves at speed v. Using Ampere's Law, find the magnetic field everywhere. Suppose there is another sheet parallel to the first one, a distance d away, also carrying charge per unit area σ, but moving at speed -v. Now what is the magnetic field everywhere?

In a separate but related problem, consider a long cylinder of radius a carrying a fixed surface charge σ. The cylinder rotates about its axis at an angular speed ω, in radians per second. What is the magnetic field everywhere?

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