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

Even without remembering the exact exponent in the horizontal-cylinder magnetic anomaly formula, the combination of \(\Delta k\), \(r\), and \(z\) that appears in \(\Delta B_z\) always has the form \(\Delta k\times f(r/z)\), i.e. everything about the size and depth of the body enters ONLY through the dimensionless ratio \(r/z\). So it is enough to compute \(\Delta k\cdot r^2/z^2\) as a numerical proxy for each model and simply compare the numbers:

P: \(0.02\times\dfrac{100^2}{200^2}=0.02\times0.25=0.0050\)
Q: \(0.01\times\dfrac{80^2}{160^2}=0.01\times0.25=0.0025\)
R: \(0.005\times\dfrac{200^2}{400^2}=0.005\times0.25=0.00125\)
S: \(0.02\times\dfrac{150^2}{300^2}=0.02\times0.25=0.0050\)

Reading down the list, P and S both come out to \(0.0050\), an exact match, while Q gives \(0.0025\) and R gives \(0.00125\) - both different from P/S and from each other. So regardless of the exact proportionality constant in front of the formula (which cancels when comparing like with like), P and S are the pair that give the same \(\Delta B_z\), confirming option \(\boxed{\text{(A) P, S}}\).