# Homework Help: Maximizing torque

1. May 17, 2014

### darksyesider

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. May 17, 2014

### paisiello2

How did you come up with that answer?

3. May 17, 2014

### oneplusone

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
4. May 17, 2014

### paisiello2

Can you prove it?

5. May 17, 2014

### darksyesider

I did what oneplusone did.

6. May 17, 2014

### paisiello2

Provide the proof.

7. May 17, 2014

### darksyesider

Can someone else please comment?

8. May 17, 2014

### haruspex

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.

9. May 19, 2014

### darksyesider

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

10. May 21, 2014

### Simon Bridge

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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