Please answer the problem completely and explain briefly how the equations work.
ID: 107653 • Letter: P
Question
Please answer the problem completely and explain briefly how the equations work. This is my third time asking the same question so im asking whoever will answer this, please help me understand. Thanks.
Calculate the vertical, horizontal and total field magnetic anomaly profiles across a dipole which strikes in the direction of the magnetic meridian and dips to the south at 30° with the negative pole at the northern end 5m beneath the surface. The length of the dipole is 50m and the strength of each pole is 300Am. The local geomagnetic field dips to the north at 70°.
Explanation / Answer
The objective of magnetic surveys is to map the subsurface distribution of magnetization and from this infer the susceptibility and hence the magnetic mineral content of the rocks. The magnetization is detected by measuring the variations or anomalies in the Earth’s field caused by the fields produced by the subsurface magnetizations. The magnetized matter is made up of a sum or integral of the atomic or domain dipoles so the field of magnetized matter in a volume V is given by the integral of 3.1.1 over V:
B = meu0 / 4 pi integration [delta (M*V 9l/r))*daV
Where the elementary dipole moment has been replaced by the dipole moment per unit volume.
The problem with this deceptively simple formula is that the magnetization, M, depends on the field via M = H, and H is not known inside the body. We have an integral equation with the unknown inside and outside the integral. This is known as a Fredholm integral equation and its solution can usually only be obtained numerically. The general nature of the solution can be seen by looking at the solution for bodies of simple shape. For some bodies of revolution the problem can be solved analytically as a boundary value problem. The sphere is one such model and it is particularly useful because its anomaly can be used as a first order approximation for any subsurface body of compact shape. For this analysis it is important to note that the magnetization is uniform, and is in the direction of the inducing field. Further, the field outside is that of a dipole with a moment given by:
Dipole moment of sphere, m = H * (x2/ (1+x2/3))* Volume.
The moment is just the magnetization one would have calculated for the material of susceptibility x2, M = x2 H, times the volume, but reduced by the demagnetization factor of 1/(1+x2/3).
Hz is the only anomalous component of magnetic field from a uniformly magnetized half space. The anomalous field is = 2 pi M sin I. Note that the anomaly is independent of the height of the measurement point. This is not a very practical result because such half spaces don’t exist, but the result is useful to derive the anomaly of a layer and to show asymptotes for the anomaly of a vertical fault. The anomalous field of a layer of thickness h is the superposition (subtraction) of two half spaces a distance h apart vertically. It is consequently zero. The vertical contact between two media of different susceptibilities is the same as a quarter space of the difference in susceptibility. While there is no such thing as a magnetic pole, a long thin rod magnetized in the direction of its length has the basic response of a pole near one of its ends. The response is actually that of a very long dipole and the response of the far end is negligible. The vertical pipe is a practical model. The magnetization along the pipe is stronger than that transverse to the pipe so in northern latitudes the dominant effect is from the vertical magnetization which yields a pole - like response. The vertical field anomaly from a pole is symmetric and the depth is approximately 1.3 times the half width. The magnetic field given by:
B = - delta V(r) , the radial and tangential components of the magnetic field are:
Br = 2(meu0 m / 4 pi r3) sinlambda = 2(meu0 m / 4 pi r3) cos p
Btheta = (meu0 m / 4 pi r3) coslambda = (meu0 m / 4 pi r3) sin p
Where lambda is latitude and p is co-latitude
The total intensity of the field at a given latitude (colatitude) is given by:
B = integration [Br2 + Btheta2 = (meu0 m / 4 pi r3)*(4 cos2p + sin2 p) ½
And the inclination of the field at this location can be derived from the following: tan I = 2cot p = 2 tan lambda.
Isomagnetic maps: contour maps of equal
•declination -> isogonic
•inclination -> isoclinic
•magnetic field value -> isodynamic
Magnetic dip poles -> locations on the Earth's surface where I = 900
The magnetic dip poles presently located at:
•700N, 1010 W (North magnetic pole)
•300 S, 1430 E (South magnetic pole)
Diurnal variations -> daily associated with ionization of the upper atmosphere by solar radiation in sympathy with tidal effects of the Earth and moon; variation is generally 20 - 80 nT. Magnetic storms would be rapid and violent changes associated with sunspot activity; can exceed a 1000 nT of change over 24 hrs.