Interpreting Dip Circle Measurements: True or Apparent Dip?

[Jahnavi]

Homework Statement

Homework Equations

The Attempt at a Solution

I know how to solve this problem . I just need help in interpreting the language of the problem as to what is given and what is required to calculate .

1) Is 30° the true dip and we are asked to calculate apparent dip
OR
2) Is 30° the apparent dip and we are asked to calculate true dip

Thanks

 

[Charles Link]

Hello again. :) I think I figured out how to do this one, but I needed to google the instrument to learn what they were talking about. $\\$ If you define a coordinate system with $\hat{z}$ upward , $\hat{y}$ pointing north, and $\hat{x}$ pointing eastward, you can determine the vector components of the unit vector $\hat{a}$ for the direction of the magnetic field $\vec{B}=B_o \hat{a}$. Call that vector $\hat{a}=a_x \hat{i}+a_y \hat{j} +a_z \hat{k}$. The $a_z$ component is the easiest one to calculate. Since the angle (dip angle) the magnetic field makes with the horizon is 30 degrees, (and this means that $a_z$ is negative but that isn't important), you can compute the $a_z$ component. (Note that this vector has unit length along the hypotenuse). $\\$ The 45 degrees declination means the magnetic field is in the northeast direction rather than north. Would you agree that $a_x=a_y$? You already computed $a_z$ without computing $a_x$ or $a_y$. Since $\hat{a}$ is a unit vector, you should now be able to compute $a_x$ and $a_y$. $\\$ The next part should get you to the answer: This instrument is set up along the geographic meridian, so that it will apparently only be affected by the $y$ and $z$ components of the magnetic field. Essentially the magnetic field of relevance is $\vec{B}_{relevant}=B_o(a_y \hat{j}+a_z \hat{k})$. (The instrument apparently can't respond to the x component when you line it up along the north (y) direction). You should now be able to compute how the instrument will point in response to the portion of the magnetic field that lies in the y-z plane, ignoring the x-component. $\\$ (Note: Had the angle been some arbitrary angle $\theta$ , in general $\frac{a_x}{a_y}=\tan{\theta}$. For $\theta=45$ degrees, it made it simple). $\\$ See also: https://www.britannica.com/technology/dip-circle They specified geographic meridian here and not magnetic meridian, as in the article. In this case, they apparently lined up the wheel of the instrument, to which a needle is attached, in the y-z plane. The wheel is free to spin in the y-z plane, (with its axle along the x-axis). The compass needle is attached to the wheel, and will point along the magnetic field direction that is in the y-z plane. (It might interest you, the compass needle is a long permanent magnet, and when placed in a magnetic field, the needle aligns with the magnetic field, with the" +" pole of the needle in the direction of the magnetic field, and the "-" pole opposite the magnetic field direction).$\\$ And yes, I get an answer that is listed. See if you get the same answer. Also, it might be worthwhile to compare this angle, (plug $\tan^{-1}x$ into the calculator), to the "true" dip angle=the angle they gave you. $\\$ Additional question for you: What would be the tangent of the dip angle that this instrument measures if you line up the plane of the wheel so that it now is in the direction of the magnetic meridian? (i.e. you use a compass to line up the wheel to get the "true" dip reading).

 

[Jahnavi]

Hi Charles ,

Thank you so much for your response .

I am extremely sorry for replying so late . As I said in the OP , the issue I have in this question is in understanding as to what is given in the problem statement . Is the given angle 30° the true dip or the apparent dip ?

the "true" dip angle=the angle they gave you.

OK . You are taking 30° as the true dip .

But don't you think the "where" in the problem statement makes 30° "apparent dip" and we are supposed to find the true dip ? I think instead if "where" in the problem statement is replaced by "when" then we could interpret 30° as true dip .

Nonetheless , if I take 30° as the true dip then I get option 2) .

But , if I take 30° as the apparent dip ( which I believe is the case ) then I get option 1) .

What do you think ?

 

[Charles Link]

When it says "the dip and declination are known to be... " it is referring to "true dip". And that would mean answer (2) is correct. $\\$ The interesting thing is that this instrument measures a dip of $\theta=\tan^{-1}(\sqrt{\frac{2}{3}}) \approx 39 \,$ degrees if you (incorrectly) line up the wheel in the geographic north meridian= with the true dip being 30 degrees when the declination is 45 degrees. It actually reads too high when not properly aligned with the magnetic meridian= which is "north" according to a compass. $\\$ When lined up properly with the magnetic meridian, it reads $\theta=$ 30 degrees $(=\tan^{-1}(\frac{1}{\sqrt{3}} ))$. $\\$ The "link" I gave in post 2 describes the use of this instrument to get a proper reading fairly clearly. (I was totally unfamiliar with this type of instrument before I googled it). $\\$ And one more question for you: If the declination was +45 degrees, so that the magnetic field pointed northeast, (just like in the problem), and had a true dip reading of 30 degrees, what (apparent) dip reading would you get if you lined this instrument up along the northwest direction, i.e. so that there is no horizontal component of the magnetic field in the direction that you point the wheel of the instrument? (i.e. which direction does the magnetic needle point that is attached to the wheel?) Also, what if the (true) dip angle was only 15 degrees=what would the instrument read when the wheel is turned to point northwest?

 

[Jahnavi]

When it says "the dip and declination are known to be... " it is referring to "true dip".

OK .

what (apparent) dip reading would you get if you lined this instrument up along the northwest direction, i.e. so that there is no horizontal component of the magnetic field in the direction that you point the wheel of the instrument? (i.e. which direction does the magnetic needle point that is attached to the wheel?)

Apparent dip would be 90° . Needle would point straight down .

 

[Charles Link]

Apparent dip would be 90° . Needle would point straight down .

Very good ! That is correct. :) :)

 

[Jahnavi]

Thanks !