The earth's magnetic field at a certain place has a horizontal component 0.3 Gauss and the total strength 0.5 Gauss. The angle of dip is

Options

A. `tan^(-1)((3)/(4))`
B. `sin^(-1)((3)/(4))`
C. `tan^(-1)((4)/(3))`
D. `sin^(-1)((4)/(3))`

Explanation

To find the angle of dip (θ) given the horizontal component (H) and the total magnetic field strength (B), we can use the following relationship:

1. **Understanding the Components**:
- The horizontal component of the Earth's magnetic field (H) = 0.3 Gauss
- The total magnetic field strength (B) = 0.5 Gauss

2. **Using the Relationship**:
The relationship between the total magnetic field (B), the horizontal component (H), and the vertical component (Z) is given by:
\[ B^2 = H^2 + Z^2 \]
We can also express the vertical component (Z) in terms of the angle of dip (θ) as:
\[ Z = B \sin(θ) \]
Therefore, we can rewrite the equation as:
\[ B^2 = H^2 + (B \sin(θ))^2 \]

3. **Calculating Vertical Component**:
To find the vertical component (Z), we can rearrange the equation:
\[ Z^2 = B^2 - H^2 \]
Substituting the known values:
\[ Z^2 = (0.5)^2 - (0.3)^2 \]
\[ Z^2 = 0.25 - 0.09 \]
\[ Z^2 = 0.16 \]
\[ Z = \sqrt{0.16} = 0.4 \text{ Gauss} \]

4. **Finding the Angle of Dip**:
Now, we can find the angle of dip using the tangent function:
\[ \tan(θ) = \frac{Z}{H} \]
Substituting the values:
\[ \tan(θ) = \frac{0.4}{0.3} = \frac{4}{3} \]

5. **Calculating the Angle**:
Now, we can find θ:
\[ θ = \tan^{-1}\left(\frac{4}{3}\right) \]

6. **Final Result**:
Using a calculator or trigonometric tables, we find:
\[ θ \approx 53.13^\circ \]

Thus, the angle of dip is approximately \( 53^\circ \).