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Binary Systems - Magnetic moments add together just as...

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Binary Systems Let us suppose that we have a system consisting of N magnets, each of which is localized and attached to a separate site. Each magnet has a magnetic moment whose magnitude is m . Think of each magnet as a vector of magnitude m . We won't focus on the details of the Electromagnetism here but on the statistics that rule our system. Calling the system binary means that each magnet can be oriented either in the "up" position or the "down" position, and no other. If a magnet is in the down position, then we say that its magnetic moment is - m , if up, it is + m . The magnets do not interact with each other; i.e. the position of a magnet's neighbors does not influence its position. A sample collection of such magnets can be seen in . Figure %: Binary System of Magnets
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Unformatted text preview: Magnetic moments add together just as vectors do. Therefore, we can ask, how many ways are there to have a total magnetic moment M of M = Nm ? Such a state would require all of the magnets to be in the up position, so there is only one way to achieve this state. How many ways are there to have a total magnetic moment of M = ( N- 2) m ? Such a state requires one magnet to be in the down position. Since there are N magnets, there are N such ways. Letting C represent the up position and D represent the down, we can use a shorthand notation for representing all of the possible states of the system: ( C + D ) N Using a binomial expansion , and writing in summation notation, we can write: ( C + D ) N = C N-i D i...
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