The relation between `B_H, B_v, and B` is

Options

A. `B=sqrt(B_(v)^(2)+B_(H)^(2))`
B. `B=B_(H).B_(V)`
C. `B=(B_(V))/(B_(H))`
D. `B=(B_(H))/(B_(V))`

Answer

To find the relation between the horizontal component of the Earth's magnetic field (B_H), the vertical component of the Earth's magnetic field (B_V), and the total Earth's magnetic field (B), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Components**:
- The total magnetic field (B) can be resolved into two components: the horizontal component (B_H) and the vertical component (B_V).
- The angle of dip (θ) is the angle between the total magnetic field and the horizontal plane.

2. **Defining the Components**:
- The horizontal component of the magnetic field is given by:
\[ B_H = B \cos \theta \]
- The vertical component of the magnetic field is given by:
\[ B_V = B \sin \theta \]

3. **Using the Pythagorean Theorem**:
- Since B_H and B_V are perpendicular to each other, we can use the Pythagorean theorem to relate them to the total magnetic field B:
\[ B^2 = B_H^2 + B_V^2 \]

4. **Substituting the Components**:
- Substituting the expressions for B_H and B_V into the equation:
\[ B^2 = (B \cos \theta)^2 + (B \sin \theta)^2 \]

5. **Simplifying the Equation**:
- Expanding both terms:
\[ B^2 = B^2 \cos^2 \theta + B^2 \sin^2 \theta \]
- Factor out B^2:
\[ B^2 = B^2 (\cos^2 \theta + \sin^2 \theta) \]

6. **Using the Trigonometric Identity**:
- We know from trigonometry that:
\[ \cos^2 \theta + \sin^2 \theta = 1 \]
- Therefore, we can simplify the equation to:
\[ B^2 = B^2 \cdot 1 \]
- This confirms that:
\[ B^2 = B^2 \]

7. **Final Relation**:
- Thus, the relation between the components is:
\[ B_H^2 + B_V^2 = B^2 \]

### Conclusion:
The relationship between the horizontal component (B_H), vertical component (B_V), and total magnetic field (B) is given by:
\[ B_H^2 + B_V^2 = B^2 \]