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Maximizing torque

  1. May 17, 2014 #1
    1. The problem statement, all variables and given/known data

    Take a wire, and form it into a rectangular loop. It carries a current I.
    What should its dimensions be, and how should it be placed inside of a magnetic field to maximize the torque on it?

    2. Relevant equations

    tau = NIAB sin theta

    3. The attempt at a solution


    There's no answer to this in my book, but shouldn't the dimensions just be L/4 by L/4 (a square), and it's perpendicular to the field?
     
  2. jcsd
  3. May 17, 2014 #2
    How did you come up with that answer?
     
  4. May 17, 2014 #3
    since max(sin theta) = 1, when theta = pi/2, and the maximum area you can get is when the loop is a square.
    I think this should be right.


    @below: This is proved using simple geometry (the area of a square > area of rectangle with same perimeter)
     
    Last edited: May 17, 2014
  5. May 17, 2014 #4
    Can you prove it?
     
  6. May 17, 2014 #5
    I did what oneplusone did.
     
  7. May 17, 2014 #6
    Provide the proof.
     
  8. May 17, 2014 #7
    Can someone else please comment?
     
  9. May 17, 2014 #8

    haruspex

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    I assume paisiello2 is asking you to prove that a square gives the largest area for a rectangle with a given perimeter. It's not hard to prove - doesn't even require calculus.
     
  10. May 19, 2014 #9
    a,b are side lengths. 2a+2b= L, so a = b-L/2

    Maximize: b(b-L/2 = b^2 - bL/2. Using -b/2a we get L/4 =b=a
     
  11. May 21, 2014 #10

    Simon Bridge

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    So far so good - looks like you will benefit from a systematic approach to problem solving.

    Step 1,. spell out the theory - in words ... the torque on a rectangular wire loop sides a and b is proportional to [what?] and the sine of the angle between [what?] and [what?].

    Step 2: write down the equations describing the things you need to maximize.
    There are two.

    max tau occurs when function(angle)=something and function(rectangle dimensions)=something

    Step 3. evaluate the equations

    Step 4. summarize your results in a conclusion.
    (This should be one or two sentences that relate the question to the answer.)
     
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