This document discusses the process of mineral prospecting using electrical resistivity measurements. Electrodes placed on the earth's surface are used to create an electric current that maps the subsurface resistivity distribution. The potentials are modeled using partial differential equations and Bessel functions to solve for different subsurface structures. Measurements in an electrolytic tank can effectively replace direct field measurements by simulating the subsurface resistivity conditions.

1. Presented by,
Pradeep Kumar Somasundaram
MTH5230
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2. Abstract
The process of finding the minerals under the earth’s crust
using the earthed electrodes. The current from the battery
conducted through the earth and the field of constant
current created on the surface of the earth are mapped. By
using Linear Partial Differential Equations the potentials are
determined and with help of Bessel functions and method of
separation of variable the prospecting is found in different
medium and found that electrolytic tank measurements
replace the direct measurements.
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3. Instrument
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4. Introduction
 Underground minerals, surface potentials
 Homogeneous medium satisfies Laplace equation
𝛻2
𝑉 = 0 −−−→ (1)
𝜕V
𝜕r
|z=0 = 0 −−−→ 2
 Considering a point electrode at point A
 Potential of the field
V =
Iϱ
2πR
−−−→ (3)
where,
R is the distance of the potential field from the source point A
ϱ is the specific resistance of the medium
I is the intensity of the current
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5. Potential field
 Potentials differ for an infinite medium
•---------•--r--•--r--•
A M O N
V M − V N =
𝜕V
𝜕r
∆r −−−−→ 4
V M − V(N)
∆r
≅
𝜕V
𝜕r
≅
Iϱ
2πr2 −−−−→ 5
where,
r is the distance between the point O to the points M and N.
O is the mid-point of the receiving circuit from the feeding electrode.
I is the current intensity of the feeding circuit which is known value.
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6. Homogeneous resistance
 Two layers
homogeneous resistance − ϱ0
homogeneous resistance -- ϱ1
thickness l
 The resistance can be represented as ϱ z =
ϱ0 where 0 ≤ z < l
ϱ1 where l < z
 r<<l the impedance will be ϱk = ϱ0
 r>>l the impedance will be ϱk = ϱ1
 Conditions of continuity
 V0 |z=l = V1 |z=l −−−−→ 6

1𝜕V0
ϱ0 𝜕r
|z=l =
1𝜕V1
ϱ1 𝜕r
|z=l −−−−→ 7
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7. Cylindrical symmetry

𝜕2V
𝜕r2 +
1
r
𝜕V
𝜕r
+
𝜕2V
𝜕z2 = 0 −−−−→ 8
 e±λzJ0 λr −−−−→ 9
where, J0 is the Bessel function of the zero order
λ is the separation parameter. The solutions will be of
 V0 r, z =
ϱ0I 1
2π (z2+r2)
+ 0
∞
(A0e−λz + B0eλz )J0 λr dλ −−−−→ 10
 V1 r, z = 0
∞
(A1e−λz + B1eλz )J0 λr dλ −−−−→ 11
 Find A0, B0, A1, B1 which are the functions of λ
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8. Special functions
For arbitrary r, A0 = B0
 For V1 the condition of the bounded nature as z∞; B1 = 0
V1 r, z = 0
∞
(A1e−λz
)J0 λr dλ
Formula found in the boundary value problem by the equations of special
functions

1
(z2+r2)
= 0
∞
J0 λr e−λz dλ
 𝑞 =
ϱ0I
2π
(12)
(13)
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9. By substituting the known values
By using the equations (6) and (7)
Derivation
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10. Solving equations (A) and (B)
Finding the values
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11. Contd.
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12. since |k|<1
The equation of V0 can be written as
Assuming z=0 we obtain the distribution of the potential on the earth’s
surface by solving the problem using the method of images.
Distribution of the potential
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13. Change of variables
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14. The limit of the nth term of the sum will be equal to Kn, from which it
follows that
To prove the impedance at infinity
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15. Conclusion
 Different conductivity profiles the impedances are also different.
 𝜌 𝑘 𝑟1
≠ 𝜌 𝑘(𝑟2
)
 Defects are determined by the presence of cavity under the surface.
 The cavity of the surface can be measured by placing a metallic piece
between the poles of a magnet and the magnetic field on the surface.
 Electrolytic tank.
 Replaces effectively the direct measurements of temperature, magnetic
and other fields.
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