Ampere's Law

A much easier version of the Biot-Savart law that is useful for calculating the magnetic field within a closed path not a surface. This equation relates the magnetic field and the current to one another. Complete understanding of this topic makes other topics such as the nature of light easier to comprehend.

The Main Idea

A student is usually first taught how to find the magnetic field caused by the flow of individual charges and/ or currents. Ampere's law allows us to see a magnetic field and determine the currents that generate this field. Ampere’s law is a quantitative relationship between measurements of magnetic field a long a closed path and the amount and direction of the currents passing through that boundary. Ampere’s law is a quantitative relationship between measurements of magnetic field a long a closed path and the amount and direction of the currents passing through that boundary.

Here is a summary of the key steps in applying Ampere’s law:

1. Choose a (mathematical) closed path as a boundary.

2. Stretch an imaginary soap film over the boundary.

3. Walk around the boundary counterclockwise, integrating the field with the path.

4. Add up the positive and negative currents that pierce the soap film; this is ∑I inside path. Count as positive those currents that pierce the imaginary soap film coming out of the film (that is,in the direction of your right thumb with the fingers of your right hand curling around in the direction of your walk); currents that pierce the film going into the film are counted as negative.

5. Apply Ampere’s law, equating the two sums.

A Mathematical Model

[math]\displaystyle{ \oint_C \mathbf{B} \cdot \mathrm{d}\boldsymbol{\ell} = \mu_0 \iint_S \mathbf{J} \cdot \mathrm{d}\mathbf{S} = \mu_0I_\mathrm{enc} }[/math]

A Computational Model

How do we visualize or predict using this topic. Consider embedding some vpython code here Teach hands-on with GlowScript

<iframe src="https://trinket.io/embed/glowscript/7a9c557b3c" width="100%" height="356" frameborder="0" marginwidth="0" marginheight="0" allowfullscreen></iframe>

Examples

Be sure to show all steps in your solution and include diagrams whenever possible

Connectedness

1. How is this topic connected to something that you are interested in?

An application of Ampere's law in an area that interests me is Maglev trains. It's fascinating how magnetic fields are strong enough to suspend the huge body that is a train. These trains do not use the same motors that are in regular trains. Instead they use electromagnets and guide the trains over a guideway, raising it approximately 0.39 and 3.93 inches. Because they float on air, this eliminates friction and allows the trains to reach speeds getter than 300 miles per hour. Damn. This excerpt from How Stuff Works indicates more about how they work- "Once the train is levitated, power is supplied to the coils within the guideway walls to create a unique system of magnetic fields that pull and push the train along the guideway. The electric current supplied to the coils in the guideway walls is constantly alternating to change the polarity of the magnetized coils. This change in polarity causes the magnetic field in front of the train to pull the vehicle forward, while the magnetic field behind the train adds more forward thrust."

1. How is it connected to your major?

Because of the breadth of sub-fields and areas that make up mechanical engineering, there are many different applications of Ampere's laws. Every motor in the automotive industry makes use of magnetism. One of the most important components in the electric motors is the magnet. It supplies the rejection power to create circular or linear movement. cite http://www.magmamagnets.com/automotive

1. Is there an interesting industrial application? cite http://www.gizmag.com/ge-magnetocaloric-refrigerator/30835/

An interesting industrial application that I've seen is also in the field of HVAC and involves refrigeration. And no it's not just the magnetism that lets you stick stuff on your fridge... The modern day fridge even though it's come a long way in terms of energy efficiency, still remains as the biggest leach of electricity in a household. Incorporating magnetism actually can have the effect of making refrigerators up to 30% more efficient than what's currently out there. It all started when the magnetocaloric effect, https://en.wikipedia.org/wiki/Magnetic_refrigeration when certain materials change temperatures in the presence of a varying magnetic field, was first observed. Such technology has not yet been implemented because of issues in how bulky it is. Michael Benedict, design engineer at GE Appliances describes it as being "about the size of a cart." That being said, be on the lookout in 10 or so more years when refrigerators based on this effect hit the markets!

Link to youtube video to embed: https://www.youtube.com/watch?v=WlKKKMTA7XM

History

André-Marie Ampère, the founder of classical electromagnetism, was a French mathematician and physicist born into a merchant family. Due to his father’s strong beliefs, André was self-educated in his huge library. Fast forward about 30 years and André had become a well-established professor of mathematics, philosophy and astronomy at the University of Paris. In 1820, André had established what was later known as Ampere’s law. He was able to demonstrate that two parallel wires can be oriented, with different current flows, in a manner that let them either attract or repel one another. Your boy established a relationship between the length of a current carrying wire and the strength of their currents. In 1827-28, André was elected as a Foreign Member of the Royal Swedish Academy of Science and a foreign member of the Royal Swedish Academy of Science. In 1881, a while after his death in 1836, the ampere, a standard unit of electrical measurement, was named after him.

See also

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References

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Section 22.6 PATTERNS OF MAGNETIC FIELD: AMPERE'S LAW pg. 914- 920