A dip needle arranged to move freely in the magnetic meridian dips by an angle `theta`. The vertical plane in which the needle moves is now rotated through an angle `alpha` from the magnetic meridian. Through what angle the needle will dip in the new position?

Options

A. `theta`
B. `alpha`
C. more than `theta`
D. less than `theta`

Answer

To solve the problem, we need to analyze the situation step by step.

### Step 1: Understand the Initial Setup
Initially, the dip needle is in the magnetic meridian and dips at an angle \( \theta \). This means that the magnetic field has both a vertical and a horizontal component. The angle \( \theta \) is defined by the tangent of the angle formed by these components.

### Step 2: Analyze the Components
In the magnetic meridian, we can express the tangent of the angle \( \theta \) as:
\[ \tan \theta = \frac{B_v}{B_h} \]
where \( B_v \) is the vertical component of the magnetic field and \( B_h \) is the horizontal component.

### Step 3: Rotate the Vertical Plane
Now, the vertical plane in which the needle moves is rotated through an angle \( \alpha \). This rotation affects the horizontal component of the magnetic field but does not change the vertical component.

### Step 4: New Tangent Expression
After the rotation, the new angle of dip \( \theta' \) can be expressed as:
\[ \tan \theta' = \frac{B_v}{B_h'} \]
where \( B_h' \) is the new horizontal component after the rotation. The relationship between the new horizontal component and the original horizontal component can be expressed as:
\[ B_h' = B_h \cos \alpha \]

### Step 5: Substitute into the Tangent Expression
Substituting \( B_h' \) into the equation for \( \tan \theta' \):
\[ \tan \theta' = \frac{B_v}{B_h \cos \alpha} \]

### Step 6: Compare the Two Angles
Now we can compare \( \tan \theta' \) and \( \tan \theta \):
\[ \tan \theta' = \frac{B_v}{B_h \cos \alpha} \quad \text{and} \quad \tan \theta = \frac{B_v}{B_h} \]
Since \( \cos \alpha < 1 \), it follows that:
\[ \tan \theta' > \tan \theta \]
This implies that:
\[ \theta' > \theta \]

### Conclusion
Thus, when the vertical plane is rotated through angle \( \alpha \), the angle of dip \( \theta' \) will be greater than the original angle \( \theta \).

### Final Answer
The angle through which the needle will dip in the new position is greater than \( \theta \).

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