Earth's Magnetic Field Lab question

[lloyd21]

Homework Statement

Using the Logger Pro Program plot a graph with the angle on the horizontal axis and the measured field on the vertical axis. The curve should have its maximum value at (theta) = S . where
cos( theta - S) = cos(0) = 1. Use your graph to determine S, the declination at your location.

My graphs are uploaded, how do i solve this or did I already get the answer of 1.44 deg?

Homework Equations

The Attempt at a Solution

My graphs are uploaded, how do i solve this or did I already get the answer of 1.44 deg?

 

[haruspex]

I do not understand where you are getting 1.44 degrees from. At what x coordinate does you graph reach a maximum?

You will not get a very accurate answer just picking out the highest reading. The samples are too widely spaced. Your best bet is to fit a sine curve to all the points and see where that maximises.

 

[lloyd21]

the 1.44 is from my 20 degrees at 0.0144...for the maximum...if i find the declination on the web for the west side of germany it should be around 1.54...so I thought I may be close...do those numbers look alright from the magnetic sensor ?

 

[haruspex]

the 1.44 is from my 20 degrees at 0.0144...for the maximum...if i find the declination on the web for the west side of germany it should be around 1.54...so I thought I may be close...do those numbers look alright from the magnetic sensor ?

The 0.0144 is a field strength, not an angle. By what logic do you turn that into an angle of 1.44 degrees? The angle at which that field was measured is 20 degrees.

I think there is some confusion here between declination and inclination. Declination is the difference between magnetic north and true north. Yes, that might be about 1.5 degrees in Germany. But how are you going to measure it in a laboratory? You have no easy way to find true north.

The graph looks more like you are measuring inclination, i.e. the angle at which the field dips below horizontal. It should be roughly the same as your latitude. If you disagree, please describe the experiment in detail.

20 degrees is much too low for inclination in Germany, and looking at the graph it is clearly not where the true maximum lies. Draw any smooth curve through those points and the maximum will be closer to the 40 degree mark, which is more reasonable. As I posted, best bet is to fit the best sine curve to those points and read the maximum off that. Do you need help with that?

 

[lloyd21]

Yeah please I'd appreciate that big time. I don't really know what I'm doing here, and I'm still unsure if I measured the magnetic field with the sensor properly.

 

[haruspex]

Yeah please I'd appreciate that big time. I don't really know what I'm doing here, and I'm still unsure if I measured the magnetic field with the sensor properly.

Ok. You want to fit a curve of the form $y=c_1+c_2\sin(c_3x+c_4)$ to the data. In general that's hard, but here we know two things:
- the average field should be zero, so c₁=0.
- the cycle should repeat after 360 degrees, so if we measure x in radians then c₃=1.
So let's rewrite it more conveniently as $y=Asin(x+c)$.
We can expand the sine term as $\sin(x)\cos(c)+\cos(x)\sin(c)$. If we then write F=Asin(c) and G=Acos(c) it becomes
$y=G\sin(x)+F\cos(x)$.
This is now easy. It's just a linear regression of y as a function of two variables, sin(x) and cos(x).
See e.g. https://www.easycalculation.com/statistics/multiple-regression.php.
What tools do you have... matlab, excel...?

 

[haruspex]

The route I'm suggesting might be too advanced for you, so I've though up an easier way that should be better than trying to guess where those curves should hit max.
Instead, look at where the field is zero. Sketch a straight line through the points where the curve crosses the y-axis and read off the x coordinates at the two crossing points. You should have two angles approximately 180 degrees different. You can safely assume that halfway between these two is either a maximum or a minimum. What angle do you get for that?

 

[haruspex]

@lloyd21, please don't simply stop responding. How are you going with this?
A question: in your table of results, what direction is "0" degrees? Is that straight up, straight down, North or South? Which way did you rotate it from there?

Out of interest, I tried both methods that I proposed. The curve fitting gives me 28.6 degrees. If that is measured from vertically down, rotating to the north, it's reasonably accurate. The other method, drawing a straight line through the points near where they cross the y axis, gives me more like 26 degrees.

 

[lloyd21]

Sorry about the late reply, I'm 9 hours ahead of you! I'm just rotating it clockwise, the 0 degrees is pointed towards true north.

 

[lloyd21]

Also, on the graph, is there a way to find a curve fit in options? Not very familiar with this program

 

[lloyd21]

I'm not sure how you got 26 degrees with the second method? I don't understand how you did that haha!

 

[haruspex]

Sorry about the late reply, I'm 9 hours ahead of you! I'm just rotating it clockwise, the 0 degrees is pointed towards true north.

Then I'm still confused as to whether we are discussing inclination or declination.
How do you know where true north is in the lab?
Are you rotating the sensor in a horizontal plane or in a vertical plane?

My second method is this:
- Look at the two places where the curves are steepest and cross the y axis.
- Put straight lines through the points there, say two or three points each side of the crossing.
- Read off the x coordinates where these straight lines cross the y axis.
- These show the angles at which the field has minimum magnitude. The maximum magnitude should be half way between them, so take the average of these two x values.
- You can see this will give you the angle where the field is most negative. To get the angle where it will be most positive just subtract 180.
That procedure gave me roughly 26 degrees. If you get something very different, please post what you get for each step so that I can see where we diverge.

(If you are in Germany I'm 10 hours ahead of you.)

 

[lloyd21]

looking halfway between the points, I see an angle of 200 degrees?

 

[lloyd21]

true north is figured out by finding out the street I am located at and direction of true north

 

[lloyd21]

This is first part instructions.

 

[lloyd21]

So my first post is the data in the graph and table.

 

[haruspex]

looking halfway between the points, I see an angle of 200 degrees?

The two intercepts of the x-axis are about 100 and 310, so the midway is more like 205. As I posted, that gives an estimate of the location of the minimum, and looking at the graph around 200 that seems reasonable. You want the maximum, so subtract 180 degrees, giving 25. Again, eyeballing the graph, that does seem reasonable for the maximum.

From all the extra detail you have provided, I see that indeed you are trying to find the declination, but 25 degrees is far too much. In Western Germany it should be no more than 2 degrees. I suspect you have true North set wrongly. What map did you depend on to figure out the orientation of the street?

 

[lloyd21]

google maps?

 

[haruspex]

google maps?

Yes, that should be ok. Perhaps you made an error mapping it from street orientation to a direction in the lab? Or maybe there is another constant source of field within the lab that deflected it?

 

[lloyd21]

thats a possibility but I kept it away from anything magnetic that might have altered the sensor readings?

 

[haruspex]

thats a possibility but I kept it away from anything magnetic that might have altered the sensor readings?

Earth's field is not that strong, in the scheme of things. Steel framing/reinforcement in a building can be significant. The only way to be sure would be to repeat it in quite another part of the building.

A useful check would be to wander around the lab carrying a compass!

 

[lloyd21]

i did three times! haha

 

[haruspex]

i did three times! haha

Different parts of the building? Identical results?

 

[lloyd21]

yeah for sure, and made sure i started the calculation pointing towards true north. The data looks good I thought?

 

[haruspex]

yeah for sure, and made sure i started the calculation pointing towards true north. The data looks good I thought?

It's a reasonable looking curve, but the answer is flat wrong.

 

[lloyd21]

I agree...it should be around 1.54 degrees...

 

[lloyd21]

any suggestions on what I can do to fix it, because it will affect my total field big time. This is the second part of it.

 

[lloyd21]

 

[haruspex]

Ok, so here we are moving on to the inclination.
You mentioned Western Germany, so I plugged in Bonn at http://www.ngdc.noaa.gov/geomag-web/?useFullSite=true#igrfwmm. This says the peak horizontal field should be around 20 μT, whereas you see around 14.
If we assume all the discrepancies are the result of an interfering field then, armed with the results from your inclination experiment, and trusting the field data from the link, we could deduce the magnitude and direction of the interfering field. But there is no way, just based on the data you collect in the lab, to disentangle the interfering field from the Earth's field. The two simply add up fo a net local field, and that is all you can measure.

 

[lloyd21]

I used the same link, that's how I found the somewhat "true" declination to be 1.54 degrees.

 

[lloyd21]

I figured out the first part...I went and re did the lab again and got the max with completely different data. This second part though I am getting very off data and I think its because I'm not measuring right with the magnetic sensor...when measuring the magnetic inclination how should I hold the sensor?

 

[haruspex]

I figured out the first part...I went and re did the lab again and got the max with completely different data. This second part though I am getting very off data and I think its because I'm not measuring right with the magnetic sensor...when measuring the magnetic inclination how should I hold the sensor?

For the inclination, you need to start with the sensor horizontal and aligned with the max field from the first part, i.e. magnetic north. Then rotate the sensor in the vertical plane through that.

 

[lloyd21]

I did it three different times haha, all different ways, can you show me a picture of what it might look like?

 

[haruspex]

I did it three different times haha, all different ways, can you show me a picture of what it might look like?

3D pictures are difficult to draw well, and I'm not even good at 2D ones.
I thought my verbal description was easy enough to follow.
How about you post a picture of what you think I'm saying, or provide your own paraphrasis?

 

[lloyd21]

I actually think I figured it out...3 percent error seems reasonable too, thanks a lot for everything though!