At a place the true value of angle of dip is `60^(@)`. If dip circle is rotated by `phi^(@)` from magnetic meridian, the angle of dip is found to be `tan^(-1)(2)`. Then the value of `phi` is

Options

A. `45^(@)`
B. `15^(@)`
C. `60^(@)`
D. `30^(@)`

Answer

To solve the problem, we need to find the angle \(\phi\) given that the true angle of dip is \(60^\circ\) and the angle of dip after rotating the dip circle by \(\phi\) degrees is \(\tan^{-1}(2)\).

### Step-by-step Solution:

1. **Identify the True Angle of Dip**:
The true angle of dip, denoted as \(\delta\), is given as:
\[ \delta = 60^\circ \]

2. **Determine the New Angle of Dip**:
The new angle of dip after rotating the dip circle is given as:
\[ \delta' = \tan^{-1}(2) \]
This implies:
\[ \tan(\delta') = 2 \]

3. **Use the Relationship of Dip Angles**:
The relationship between the vertical and horizontal components of the magnetic field can be expressed as:
\[ \tan(\delta) = \frac{B_v}{B_h} \]
For the true angle of dip:
\[ \tan(60^\circ) = \sqrt{3} = \frac{B_v}{B_h} \]
Thus, we can express \(B_v\) in terms of \(B_h\):
\[ B_v = \sqrt{3} B_h \]

4. **Calculate the New Horizontal Component**:
When the dip circle is rotated by \(\phi\), the new horizontal magnetic field component \(B_h'\) can be expressed as:
\[ B_h' = B_h \cos(\phi) \]
The vertical component \(B_v\) remains the same.

5. **Set Up the Equation for the New Angle of Dip**:
For the new angle of dip:
\[ \tan(\delta') = \frac{B_v}{B_h'} = \frac{B_v}{B_h \cos(\phi)} \]
Substituting \(B_v = \sqrt{3} B_h\) into the equation gives:
\[ \tan(\delta') = \frac{\sqrt{3} B_h}{B_h \cos(\phi)} = \frac{\sqrt{3}}{\cos(\phi)} \]

6. **Substituting the Value of \(\tan(\delta')\)**:
Since \(\tan(\delta') = 2\), we can write:
\[ 2 = \frac{\sqrt{3}}{\cos(\phi)} \]

7. **Rearranging the Equation**:
Rearranging the equation gives:
\[ \cos(\phi) = \frac{\sqrt{3}}{2} \]

8. **Finding the Angle \(\phi\)**:
The value of \(\phi\) can be found by taking the inverse cosine:
\[ \phi = \cos^{-1}\left(\frac{\sqrt{3}}{2}\right) \]
This corresponds to:
\[ \phi = 30^\circ \]

### Final Answer:
Thus, the value of \(\phi\) is:
\[ \phi = 30^\circ \]