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

[theshonen8899]

Homework Statement

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

Homework Equations

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

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.

 

[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.

 

[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?

 

[TSny]

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

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?)

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.

So, you have an incorrect sign in your answer. See if you can find it.

 

[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

 

[TSny]

That looks correct. Good work!

 

[theshonen8899]

Awesome, thanks so much!