Maximizing Torque with a Wire Loop

[darksyesider]

Homework Statement

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?

Homework Equations

tau = NIAB sin theta

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?

 

[paisiello2]

How did you come up with that answer?

 

[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)

 

[paisiello2]

Can you prove it?

 

[darksyesider]

I did what oneplusone did.

 

[paisiello2]

Provide the proof.

 

[darksyesider]

Can someone else please comment?

 

[haruspex]

Can someone else please comment?

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.

 

[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

 

[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.)