Compute ΔB seen by a magnetometer flown on a satellite

[harlowz2]

1. Sample calculation - If a satellite carrying a magnetometer flies at 300km over a current that is flowing at 100km w/ a magnitude of 150 μA/m² (over a cross-section of ~5 km²), how big a ΔB will be seen by the magnetometer?
Assumptions (based on my understanding):
-Satellite altitude remains constant
-Current altitude and magnitude remains constant
-Speed of the satellite does not matter

Knowns:
Current density (Amps/m²) = J = 150
Distance between magnetometer and current = r = 200 km
Current area = Area = 5 km²



2. I have very limited experience with magnetic field calculations, but I believe the equation that should be used is as follows (note: instructor did not provide any equation):
A form of the Biot-Savart Law ⇔ B= (μ₀/4$\pi$) * $\int$(J x r)/r² d$\upsilon$



3. I do not have an attempt at a solution, since I have no idea where to start. Please help!

UPDATE: Here is what I have tried so far, and the units appear to check out.
Using a scalar form of the Biot-Savart Law, and assuming a unit depth for the volume integral, I get...
B = (μ₀/4$\pi$) * (J * Area)/r²

After plugging numbers in, I arrive at: ΔB = 1.1111*10^-12 T = 1.110 pT

Is this anywhere close, or even sound right? I have the tendency to pull ideas out of thin air.

Thanks,
Zack

 

[diazona]

First thought:

Using a scalar form of the Biot-Savart Law

Why do you do that?

 

[harlowz2]

I was assuming a scalar form because:
1) All values given to me in the problem statement are scalar, not vector.
2) I really have no idea how to do the problem. Doing that is my best attempt at the problem.

 

[diazona]

Sorry for the delayed response, I had some trouble getting to the site yesterday.

I was assuming a scalar form because:
1) All values given to me in the problem statement are scalar, not vector.

OK, well, it's a good thought, but that's not a correct assumption. Even if you're only given scalar values in the question, you will still generally have to use the vector form of the equation. The scalar version you used applies only under certain very specific conditions.

In this case, you'll have to use the geometry of the problem to determine the vectors you need to use. I would suggest starting by drawing a diagram, and if you can, upload it as an attachment here.

 

[asteriatic]

As a scientist, my response to this content would be to first clarify the assumptions and knowns stated. It is important to have a clear understanding of the parameters and variables involved in the problem in order to accurately solve it.

Next, I would use the Biot-Savart Law, which relates the magnetic field at a point to the current density and distance from the current. In this case, we are looking for the change in magnetic field (ΔB) at a specific location due to the current flowing below the satellite.

Using the given values, we can calculate the magnetic field at the location of the magnetometer by integrating the Biot-Savart Law over the area of the current. This will give us the total magnetic field at the location. Then, we can subtract the magnetic field without the current (background field) to get the change in magnetic field due to the current.

The final answer will depend on the specific units and constants used, but it should be in the range of nanotesla (nT) or picotesla (pT). It is always important to double check the units and make sure they are consistent and appropriate for the problem.

In conclusion, with the given assumptions and knowns, a scientist would use the Biot-Savart Law to calculate the change in magnetic field seen by the magnetometer flown on the satellite. This will provide a quantitative answer to the problem and can be used to further understand the effects of the current on the magnetic field.