Magnetic field due to current loop

[exitwound]

Homework Statement

Homework Equations

$$\vec B = \frac{\mu i \phi}{4 R}$$

The Attempt at a Solution

The field due to the curve R_1 is:

$$\vec B = \frac{\mu i \phi}{4 R}$$

$$\vec B = \frac{(1.26x10^-6)(.281)(\pi)}{4 (.023)} = -1.21x10^-5 \hat k$$

The field due to the curve R_2 is:

$$\vec B = \frac{(1.26x10^-6)(.281)(\pi)}{4 (.066)} = 4.21x10^-6 \hat k$$

Add 'em up and you get $-7.88x10^-6 k$ but apparently it's wrong.

What the heck am I doing wrong?? I even derived the formula separate from the book to ensure I knew what was going on.

 

[Troels]

First of all, what's the $$\phi$$ in the formula? The fraction of the circle? Note that the formula for a full loop is

$$\vec B = \frac{\mu_0 i}{2 R} \hat z$$

so you have already taken into account that you are dealing with only half a loop by multiplying with 1/2, getting a 4 in the denominator.

Now, the field due to the outer loop is:

$$\vec B_2 = \frac{\mu_0 i}{4 R_2} \hat z$$

and the inner loop:

$$\vec B_1 = -\frac{\mu_0 i}{4 R_1} \hat z$$

Adding:

$$\vec B=\frac{\mu_0 i}{4 R_2} \hat z +\left( -\frac{\mu_0 i}{4 R_1} \hat z\right)=-\frac{\mu i}{4}\left(R_1^{-1}-R_2^{-1} \right) \hat z$$

Inserting the numericals I get:
(EDIT: The radii should of course be in meters!)
$$\vec B=-\frac{4\pi\times10^{-7}\cdot 0.281}{4}\left(0.023^{-1}-0.066^{-1}\right)=-2.5\times 10^{-6}\hat z\,\mathrm{T}$$
(with a little help from Wolfram Alpha)

So, you must have punched wrong on the calculatur somewhere :) Try to construct a complete expression before inserting numericals, that greatly reduces that risk

 

[exitwound]

Phi is the partial angle of the circle. In this case, it's pi.

In your calculations, there's no variable for how far around the circle the loop goes. You need to have a phi to calculate whether or not you're going 360 degrees around the point, or 180, or 90 etc.

 

[Troels]

In your calculations, there's no variable for how far around the circle the loop goes.

Oh but there is - it's the very first paragraph of my reply :)

You need to have a phi to calculate whether or not you're going 360 degrees around the point, or 180, or 90 etc.

No.

You need to multiply with the fraction of the circle you are dealing with. A full circle gives one value, a half circle gives half that value. Straight forward

 

[exitwound]

The equation should be:

$$\vec B = \frac{\mu i \phi}{4\pi R}$$

So when $\pi$ is put into Phi, the equation comes out as

$$\vec B = \frac{\mu_0 i}{4 R} \hat z$$

which is one half the full circle equation that you posted.

(you edited your post, and your first paragraph changed)

 

[Troels]

The equation should be:

$$\vec B = \frac{\mu i \phi}{4\pi R}$$

So when $\pi$ is put into Phi, the equation comes out as

$$\vec B = \frac{\mu_0 i}{4 R} \hat z$$

which is one half the full circle equation that you posted.

(you edited your post, and your first paragraph changed)

Oh yes, and I have to edit it again I'm affraid - seems I messed up my units on the radii - sorry :) :)

 

[exitwound]

Guess what, I submitted -2.5e-6 and it's wrong too.

 

[Troels]

Guess what, I submitted -2.5e-6 and it's wrong too.

Come to think of it, it says "magnitude" so a personal guess of mine is you should prob. omit the minus...

 

[exitwound]

Thats the correct answer. No negative. This is why I HATE HATE HATE this crappy online submission system. If that was handed into a homework grader, I might have gotten 9/10 on it. But online would give me 0/10.

I complained to the dean about it, but he said they don't have enough money to pay staff to check 900 students' homeworks. They have enough money to constantly upgrade the football stadium though (Penn state). Did I Mention i HATE this format of homework?