If earth's poles are reversed suddenly

[yeahhyeahyeah]

Homework Statement

Consider the Earth’s magnetic field as that of a dipole with the magnetic field around the equator having magnitude 30 µT. The radius of the Earth is 6.37 106 m.
a. Calculate the magnetic flux φB in low Earth orbit (r=6.5x10^6 m) at the equator, in units of Tm2.

b. Suppose the field (for some reason) suddenly reverses direction. What is the magnitude of the change of φB?

Homework Equations

flux = integral(B dA)

The Attempt at a Solution

The flux in low Earth orbit at the equator: I think this means the flux through the area of the ring surrounding the Earth from the surface up to r=6.5x10^6 m. The flux would then be the area of that ring times the given magnetic field, I think?
Then the flux, if the field reverses, is just going to be negative/opposite of whatever the last answer was, right? I do not know if I'm doing this wrong or not :X

 

[Redbelly98]

This is odd, they don't really define what area to use. The field is 30 µT at the equator, but inside the Earth it will be different. So you can't really calculate the flux through a circle given by the Earth's radius.

And yes, if the field reverses then the flux is just the opposite of what it was before.

 

[yeahhyeahyeah]

I looked up "low Earth orbit" and that's supposed to be the locus of points from the surface up to about 6.5x10^6. For my area, I used the area between the larger circle and the smaller, surface one. I used a uniform magnetic field though. I just got back from spring break though and I can't even remember how the magnetic field varies a distance from a dipole though.
Thanks for your reply
My professor writes these questions himself so that's why its not too clear/might have mistakes

 

[Redbelly98]

Okay, I didn't catch that there were two different radii given in the problem statement. Yes, using the area in between is probably what was intended.

Since the two radii are fairly close, B will not vary significantly throughout the area. So the 30 µT value will be fine.

For (b), not sure if this is clear: they are not asking for the flux after B changes direction. They are asking by how much the flux changes.

 

[LowlyPion]

Fwiw, my reading of the problem is that they may want the B field at the low Earth orbit distance, which you can calculate with the two different radii and ground level B at the equator. The |B| over a square meter then is flux/per square meter?

Surely they don't want total global flux?

 

[yeahhyeahyeah]

thanks! that makes sense

 

[Redbelly98]

a. Calculate the magnetic flux φB in low Earth orbit (r=6.5x10^6 m) at the equator, in units of Tm2.

Note the units here, T m² is specifically requested.

The |B| over a square meter then is flux/per square meter?

Surely they don't want total global flux?

But--flux per square meter is simply the magnetic field, 30 µT as given in the problem statement. I.e.,

(30 µT × 1 m²) / 1 m²

Hard to believe that's what they are after either, and it does not have the correct units of T m².

I'm inclined to go with the total global flux on this.

 

[LowlyPion]

But--flux per square meter is simply the magnetic field, 30 µT as given in the problem statement.

30 µT is ground level. They want it at orbital height don't they? So you have the ratio magic to work with the radii. I think that's the point of the problem. Then when it reverses the change is twice what's calculated?

Otherwise what closed surface would you take? I will freely admit that I think the point of the problem is confusing to me any way.

 

[Redbelly98]

30 µT is ground level. They want it at orbital height don't they? So you have the ratio magic to work with the radii.

That would make for a more meaningful problem. But then the statement about units is wrong.

Otherwise what closed surface would you take?

Good question. Taking the annular ring between r_Earth and r_orbit is pretty much a guess at this point.

I will freely admit that I think the point of the problem is confusing to me any way.

Agreed, same here. It seems like something is incorrect or incomplete about the problem statement.

 

[yeahhyeahyeah]

Err, oh well whatever, I just did
flux = B x area of annular ring

change in flux = 2 times above

And about the point of the problem, perhaps it would help to know that there is a final part that i didn't post because I thought I understood how to do it if I did the above correctly: it asks:

. If this field reversal occurs in a time ∆t, how small does ∆t have to be to cause an electric field of 1000 V/m in low Earth orbit (r=6.5 106m)?