Lesson: Ampere's Law

Contributed by: VU Bioengineering RET Program, School of Engineering, Vanderbilt University

A photograph shows a shortwave loop antenna with attached tuner units, which looks like a 2-meter circle of metal on a pole with other small attached devices and wires.
Use Ampere's law to calculate the magnetic field around a current loop.
copyright
Copyright © 2008 Trixt, Wikimedia Commons https://commons.wikimedia.org/wiki/File:Loop_antenna.jpg

Summary

A class demo introduces students to the force between two current carrying loops, comparing the attraction and repulsion between the loops to that between two magnets. After a lecture on Ampere's law (including some sample cases and problems), students begin to use the concepts to calculate the magnetic field around a loop. This is applied to determine the magnetic field of a toroid, imagining a toroid as a looped solenoid. Students use Ampere's law to solve some homework problems.
This engineering curriculum meets Next Generation Science Standards (NGSS).

Engineering Connection

Understanding Ampere's law enables engineers to calculate the magnetic field around a loop, which is useful in studying the magnetic field produced by MRI magnets. In addition to calculating the magnetic field, engineers use Ampere's law to determine the amount of current and voltage needed to create a functioning circuit board to perform desired tasks. 

Learning Objectives

After this lesson, students should be able to

  • Apply Ampere's law to calculate magnetic fields in symmetric situations.
  • Describe the magnetic field of a toroid.

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Introduction/Motivation

In order to solve the MRI safety challenge (of this unit), we need to understand and measure the magnetic field given by current loops. Engineers continue to study magnetic fields, and they work with other engineers to aim to design more powerful MRI machines around 7 Tesla, compared to the conventional 1.5 Tesla used in most imaging facilities.

Class Demo: Force between Two Current Loops

Objective: The demo goal is to show the force between two current carrying loops. The attraction and repulsion between the loops closely resembles the force between magnets and can help students begin to visualize the force between current loops as related to a current within the solid or to the magnetic moment.

Materials:

  • 2 small-diameter coil
  • 2 30W or higher 5V DC power supply
  • 2 lab stands
  • 1 thin str aluminum foil
  • 4 wires with alligator clip leads
  • 1 roll tape

*This demo was tested with a 2-inch diameter coil of 100 turns using 20 gauge magnet wire held together with duct tape.

A drawing shows two ring stands next to each other, both with bars attached. A metal ring hangs from each one such that the rings are parallel and at the same height.
Figure A. Class demo setup.

Before presenting the demo, pose the problem of two short segments of wire aligned in a parallel fashion and have students determine the direction of the magnetic field in each segment produced by the other segment. Then have them determine the direction of the magnetic force on each segment. Then consider many of these segments forming a loop in which each segment of the loop is attracted to the corresponding segment on the other loop. Have the students determine the magnetic moment vector of each coil to conclude that loops with parallel magnetic moments will attract.

Set up the demonstration as shown Figure A. Connect each loop to its own power supply, but do not complete the circuit until you are ready to start the demonstration as the wires may overheat. Do not leave the circuits connected for more than a few seconds. Momentarily complete the circuit to both coils and watch them attract. Reverse the current in one of the loops to see them repel.

Lesson Background and Concepts for Teachers

Legacy Cycle Information

This lesson fits into the research and revise phase of the legacy cycle during which students are provided with additional information enabling them to revise their initial ideas for solving the challenge. The research aspect consists of a class demonstration on the force between two current loops and a lecture on Ampere's law and its applications.

Ampere's Law

Although the total magnetic flux through a closed surface must be zero, the sum of the magnetic field around a closed loop does not have to be zero. More specifically, if a closed loop is imagined in space, we can create a vector at each tiny segment called dl pointing in the direction of the loop with a magnitude equal to the length of the segment. Then we will look at the magnetic field at that point, and take the dot product

equation
and think of this as the magnetic field along that segment of the loop. The integral of these dot products around the loop,
equation
, is a mathematically precise way of considering the magnetic field around a closed loop (see Figure B).

A loop of wire has a battery and the current traveling counterclockwise. There is a magnetic field line pointing right and a ":dl" vector pointing down and to the right.
Figure B
copyright
Copyright © 2006 Eric Appelt, Vanderbilt University

As you might imagine, the magnetic field around a loop would be greater if a wire carrying current was passing through the interior of the loop. In fact, the exact value of the field around the loop is

equation
where I is the total current passing through the loop.

This result is named Ampere's law, after its discoverer.

Much like Gauss law, it is useful in determining the magnitude of the magnetic field in highly symmetric situations.

Example: Determine the magnetic field outside of an infinitely long straight wire.

Solution: Consider a closed circular loop of radius R around the wire centered on the axis of the wire, as shown in Figure C. Note that the diagram is rotationally symmetric, so that the magnetic field must have a constant magnitude around the loop.

A wire is horizontal on the page with current traveling to the right. A circular loop of wire is centered around the axis of the wire.
Figure C
copyright
Copyright © 2006 Eric Appelt, Vanderbilt University

Thus,

equation
.

Now by Ampere's law,

equation

This result is in agreement with the Biot-Savart law. Note that this would appear to work for a finite segment of wire and give the same result, contradicting the result from the Biot-Savart law saying that

equation B =

Experimentally, the Biot-Savart law has been shown to be correct. This discrepancy can be understood by realizing that the segment of wire must be part of a larger circuit, breaking the symmetry and invalidating the use of Ampere's law, or that the current must be a non-steady flow from one conductor to another. From this, we can surmise that Ampere's law must be valid only for steady-state currents.

Toroid

A line drawing of a donut-shaped loop of wire, with smaller loops making up the big loop.
Figure D

A toroid is basically a solenoid that is bent into a circle, as shown in Figure D. This doughnut-shaped figure has an inner radius a, an outer radius b, a current I, and a total number of turns N. To determine the magnetic field using amperes law, imagine a circular loop of radius r sharing a center with the toroid in the plane of the toroid.

Case 1: r < a

If r < a, then no current is passing through the loop, so

equation

Since the diagram possesses rotational symmetry, the magnetic field must have equal magnitude anywhere along the loop, so B = 0 everywhere inside the inner radius.

Case 2: a < r < b

In this case, each turn passes through the loop in one direction, as shown in the cross sectional view in Figure E.

Two identical circular loops are shown near each other with current traveling in opposite directions. Segment a is labeled as the distance from the inside of one  loop to the midpoint between the loops, b is the distance from the outside of a loop to the midpoint, and c is from the center of the loop to the midpoint.
Figure E
copyright
Copyright © 2006 Eric Appelt, Vanderbilt University

equation

equations

Case 3: r > b

In this case, for each inner turn passing current I through the loop in one direction, there is an outer turn passing current I through the loop in the other direction, so that the total current through the Amperian loop is zero, thus B = 0 everywhere outside the outer radius.

Vocabulary/Definitions

toroid: A surface generated by the revolution of any closed plane curve or contour about an axis lying in its plane.

Attachments

Assessment

Homework: Assign students to complete the Ampere's Law Homework questions to assess their progress in comprehending the concepts.

References

"MRI." Dictionary.com. Random House Unabridged Dictionary, Random House. Accessed June 23, 2008. http://dictionary.reference.com/browse/mri

Contributors

Eric Appelt

Copyright

© 2013 by Regents of the University of Colorado; original © 2006 Vanderbilt University

Supporting Program

VU Bioengineering RET Program, School of Engineering, Vanderbilt University

Acknowledgements

The contents of this digital library curriculum were developed under National Science Foundation RET grant nos. 0338092 and 0742871. However, these contents do not necessarily represent the policies of the NSF, and you should not assume endorsement by the federal government.

Last modified: July 20, 2017

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