If θ1 and θ2 be the apparent angles of dip observed in two vertical planes at right angles to each other, then the true angle of dip θ is given by
medium
Motion in a Plane
2017
physics
tan2θ=tan2θ1+tan2θ2
Explanation
To solve this problem, we need to find the true angle of dip θ given the apparent angles of dip θ1 and θ2 in two vertical planes at right angles to each other.
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The relationship between the true angle of dip and the apparent angles of dip is given by the formula:
cot2θ=cot2θ1+cot2θ2
Let's derive this formula step-by-step:
1. Consider the magnetic field components in the two planes:
• Let
H
be the horizontal component of the Earth's magnetic field.
• Let
V
be the vertical component of the Earth's magnetic field.
2. The apparent angles of dip
θ1
and
θ2
are observed in two vertical planes at right angles.
3. The tangent of the apparent angle of dip is given by:
tanθ1=H1V
and
tanθ2=H2V
4. The horizontal components
H1
and
H2
are related to the true horizontal component
H
by:
H1=Hcosα
and
H2=Hsinα
5. The true angle of dip
θ
is given by:
tanθ=HV
6. Using the identities for cotangent:
cotθ=VH,cotθ1=VH1,cotθ2=VH2
7. Substitute the expressions for
H1
and
H2:cotθ1=VHcosα,cotθ2=VHsinα
8. The true angle of dip is related to the apparent angles by:
cot2θ=(VH)2=(VHcosα)2+(VHsinα)2
9. Simplify the expression:
cot2θ=cot2θ1+cot2θ2
Therefore, the correct option is Option 4:
cot2θ=cot2θ1+cot2θ2
This concludes the derivation and confirms that the true angle of dip is given by the sum of the squares of the cotangents of the apparent angles of dip.