Which Shape Maximizes the Magnetic Field at the Center: Circle or Square?

[stunner5000pt]

you are given a wire of length L and that carries a uniform current i through it.
the wire can bent into either a circle or a square
which shape gives the maximum magnetic field at its center?

for the circle
$$B = \frac{\mu_{0}}{4 \pi} \int \frac{i ds \cross r}{r^3}$$
im not qutie sure about the s part since ds = L, riught? so ds is constant value... and so is R so we should get
$$B = \frac{\mu_{0} i}{4 \pi R^2}$$
where $$R = \frac{L}{2 \pi}$$
so $$B = \frac{\mu_{0} i \pi}{L^2}$$

now for the squar

the problem is ds is a cosntant value, but r is not because it varies from s/2 to $\frac{s}{\sqrt{2}}$
so what do i do? Can i use the square as an Amperian loop and solve it like so
$$B (s^2) = \mu_{0} i$$
since 4S = L, s = L/4 so
$$B = \frac{16 \mu_{0} i}{L^2}$$
thus the magnetic field at the center due to th square loop is greater becasue 4 > pi?

Also i am a bit confused as to when i use the Ampere's law and Biot Savart Law, can the Ampere's law be used for the above described situations or not?

Please help! Your advice is greatly appreciated!

 

[mezarashi]

Quick reference to http://maxwell.byu.edu/~spencerr/websumm122/node70.html

shows that the solution to the Biot-savart integral for a loop is

$$B = \frac{\mu_o I}{2R}$$

Note that for the infinite wire, the solution is the same as that of Ampere's Law.

If you need to apply the integral to a square. You can do it once for one side then multiply by 4. I don't follow your method. Can you write out your integral limits more clearly.

 

[stunner5000pt]

well for the square can i just use the equation for the finite wire of length L where d is the distance between the point int eh center and wire, and multiply by 4?

the formula I am ean to use is $$B = \frac{\mu_{0} i}{4 \pi d} \frac{L}{\sqrt{\frac{L^2}{4} + d^2}}}$$
would this yile dthe required answer
of course L in the formula does not mean the length of the wire

 

[mezarashi]

That formula does not look very familiar. Can you show the derivation?

 

[stunner5000pt]

its in my textbook and it would be fine if i used it i believe