Choose the correct option.

We need to determine the correct relationship between the true dip ($$\delta$$) and the apparent dip ($$\delta'$$) in Earth's magnetism.

1. Understand the Concepts of Dip

• Magnetic Meridian: The vertical plane passing through the magnetic axis of Earth. The actual magnetic field ($$\vec{B}$$) of Earth lies in this plane.
• True Dip ($$\delta$$): The angle made by the total magnetic field of Earth with the horizontal direction in the true magnetic meridian.
• Apparent Dip ($$\delta'$$): The angle of dip measured in any vertical plane other than the true magnetic meridian.

2. Set Up the Mathematical Relations

In the true magnetic meridian, let the horizontal component of Earth's magnetic field be $$B_H$$ and the vertical component be $$B_V$$. The true dip angle $$\delta$$ is given by:

$$\tan \delta = \frac{B_V}{B_H} \quad \text{--- (Equation 1)}$$

Now, consider another vertical plane that makes an angle $$\alpha$$ with the true magnetic meridian. In this new plane:

• The vertical component remains unchanged because it acts straight down: $$B_V' = B_V$$.
• The horizontal component is the projection of $$B_H$$ onto this new plane: $$B_H' = B_H \cos \alpha$$.

The apparent dip $$\delta'$$ measured in this plane is given by:

$$\tan \delta' = \frac{B_V'}{B_H'} = \frac{B_V}{B_H \cos \alpha}$$

3. Compare True Dip and Apparent Dip

Using the value from Equation 1 ($$\frac{B_V}{B_H} = \tan \delta$$), we can rewrite the equation for apparent dip as:

$$\tan \delta' = \frac{\tan \delta}{\cos \alpha}$$

$$\tan \delta = \tan \delta' \cdot \cos \alpha$$

For any plane other than the true magnetic meridian, the angle $$\alpha$$ is between $$0^\circ$$ and $$90^\circ$$ ($$0^\circ < \alpha < 90^\circ$$). In this range, the value of cosine is always a fraction less than $$1$$:

$$\cos \alpha < 1$$

Therefore, multiplying $$\tan \delta'$$ by a fraction less than $$1$$ makes it equal to $$\tan \delta$$:

$$\tan \delta < \tan \delta' \implies \delta < \delta'$$

4. Conclusion

Since the angle of true dip is smaller than the angle of apparent dip, it means that the True dip is less than apparent dip.

Final Answer: Option B (True dip is less than apparent dip.)