# Contribution to the magnetic field at the point by a thin wire

1. Feb 26, 2014

### theshonen8899

1. The problem statement, all variables and given/known data

A thin wire carries current along an arbitrary path, but when it passes through the origin, it is in the +y direction. Denote the magnitude of the current is 9.10A and we consider a point in space whose location is r = (-0.730m)*i + (0.390)*k.

Find the contribution to the magnetic field at the point being considered due only to the 0.500mm-long section of the wire centered at the origin.

The answer should be in the following format:
(dBx, dBy, dBz) = [ ] nT

2. Relevant equations

Biot-Savart Law:
(μo/4∏)*[I*(dL x r_hat)/(|r|^2)]

3. The attempt at a solution

So since there is no z-component in r, Bz should be 0. And since j x j = 0, By should also be 0.
For Bx, dl x r_hat should be r_vector/|r| -$\widehat{k}$ right?

I have a feeling I'm not doing the cross products correctly so I apologize in advance for my difficulty understanding this.

2. Feb 26, 2014

### TSny

Hello.

It does look as though you are having some difficulty with cross products.

See if this link helps. In particular, look at the example about halfway down that illustrates getting the components of the result.

Last edited: Feb 26, 2014
3. Feb 26, 2014

### theshonen8899

Thanks so much for the response! Okay so let me clarify, dl should be 0.500 mm j_hat right? Since it's on the y-axis.

|r| = √(0.73m^2)+(0.39m^2) = 0.828m

r_hat = (-0.730m/0.828m)*i + (0.390/0.828m)*k = -0.882m*i + 0.471m*k

dl x r_hat = (0.0005m*j) x (-0.882m*i + 0.471m*k)

x = aybz - azby = (0.0005m*j)(0.471m*k) - 0 = 0.0002355m*i
y = azbx - axbz = 0 - 0
z = axby - aybx = 0 - (0.0005m*j)(-0.882m*i) = -0.000441m*k << (from what I understand, j x i = -k right?)

Then plug these values into Biot Savart law:
(μo/4∏)*[I*(dL x r_hat)/(|r|^2)]

Bx = (μo/4∏)*[9.10A*(0.0002355m)/(0.828m^2)] = 3.12*10^-10
By = 0
Bz = (μo/4∏)*[9.10A*(-0.000441m)/(0.828m^2)] = -5.85*10^-10

Does that look right? Or at least better?

4. Feb 26, 2014

### TSny

That's close. But there's still a bit of a problem with how you are calculating the components in the cross product.

You should not be writing the unit vectors when evaluating the components like this.

Just multiply the numbers. These formulas have already taken account of the fact that j x i = -k, etc.

5. Feb 26, 2014

### theshonen8899

Ahh, that would make a lot of sense then. I kept having trouble with cross product signs because I didn't know they already took them into account. So then would it be:

x = aybz - azby = (0.0005m)(0.471m) - 0 = 0.0002355m
y = azbx - axbz = 0 - 0
z = axby - aybx = 0 - (0.0005m)(-0.882m) = 0.000441m

6. Feb 26, 2014

### TSny

That looks correct. Good work!

7. Feb 26, 2014

### theshonen8899

Awesome, thanks so much!