Question

\(\Delta B_z\) represents the maximum vertical magnetic anomaly along a profile due to a horizontal cylinder with susceptibility contrast \((\Delta k)\), and radius \((r)\) at a depth \((z)\) below the Earth's surface. Which combination(s) of \(\Delta k\), \(r\), and \(z\) labelled as P, Q, R and S given below, produces/produce the same \(\Delta B_z\)?
(P) \(\Delta k=0.02\), \(r=100\ m\), and \(z=200\ m\)
(Q) \(\Delta k=0.01\), \(r=80\ m\), and \(z=160\ m\)
(R) \(\Delta k=0.005\), \(r=200\ m\), and \(z=400\ m\)
(S) \(\Delta k=0.02\), \(r=150\ m\), and \(z=300\ m\)

• A. P, S
• B. P, Q
• C. R, S
• D. Q, R

Hint:

Note that \(z/r=2\) for all four models, so the shape factor cancels and \(\Delta B_z\) depends only on \(\Delta k\); look for the pair with equal \(\Delta k\).

Answer 1

The Correct Option is A

The maximum vertical magnetic anomaly of a long horizontal cylinder is proportional to the susceptibility contrast times a shape factor that depends only on the ratio of the radius to the depth:
\[\Delta B_z \propto \Delta k\cdot r^2/z^2\]
(the anomaly of a magnetized cylinder falls off with depth the way a line of dipoles does, and the size term always enters as the dimensionless combination \((r/z)^2\), scaled by \(\Delta k\)).

The key observation is to check the ratio \(z/r\) for each of the four models, because if that ratio is the same for all of them, the shape factor \((r/z)^2\) becomes a common constant and \(\Delta B_z\) then depends on \(\Delta k\) alone:
P: \(z/r=200/100=2\)
Q: \(z/r=160/80=2\)
R: \(z/r=400/200=2\)
S: \(z/r=300/150=2\)

All four models share exactly the same \(z/r=2\), so \((r/z)^2=1/4\) is identical in every case. This means \(\Delta B_z\) for each model is controlled purely by its \(\Delta k\) value:
P: \(\Delta k=0.02\)
Q: \(\Delta k=0.01\)
R: \(\Delta k=0.005\)
S: \(\Delta k=0.02\)

P and S both have \(\Delta k=0.02\), so they produce the same \(\Delta B_z\); Q (0.01) and R (0.005) are each different from the rest and from each other. Hence the pair that cannot be distinguished by their maximum vertical anomaly is P and S, option \(\boxed{\text{(A) P, S}}\).