At a given place on the earth's surface, the horizontal component of earth's magnetic field is `2 xx 10^(-9)T` and resultant magnetic field is `4 xx 10^(-5) T`. The angle of dip (in degree) at this place is

Answer

To find the angle of dip (δ) at a given place on the Earth's surface, we can use the relationship between the horizontal component (Bh) and the resultant magnetic field (B). The angle of dip can be calculated using the formula:

\[ \tan(δ) = \frac{B_v}{B_h} \]

Where:
- \(B_v\) is the vertical component of the Earth's magnetic field.
- \(B_h\) is the horizontal component of the Earth's magnetic field.

Given:
- Horizontal component, \(B_h = 2 \times 10^{-9} \, T\)
- Resultant magnetic field, \(B = 4 \times 10^{-5} \, T\)

### Step 1: Calculate the vertical component of the magnetic field
Using the Pythagorean theorem, we can relate the horizontal and vertical components to the resultant magnetic field:

\[ B^2 = B_h^2 + B_v^2 \]

Rearranging this gives:

\[ B_v^2 = B^2 - B_h^2 \]

Substituting the values:

\[ B_v^2 = (4 \times 10^{-5})^2 - (2 \times 10^{-9})^2 \]

Calculating each term:

\[ (4 \times 10^{-5})^2 = 16 \times 10^{-10} = 1.6 \times 10^{-9} \]
\[ (2 \times 10^{-9})^2 = 4 \times 10^{-18} \]

Now, substituting these values into the equation:

\[ B_v^2 = 1.6 \times 10^{-9} - 4 \times 10^{-18} \]

Since \(4 \times 10^{-18}\) is negligible compared to \(1.6 \times 10^{-9}\), we can approximate:

\[ B_v^2 \approx 1.6 \times 10^{-9} \]

Taking the square root gives:

\[ B_v \approx \sqrt{1.6 \times 10^{-9}} \approx 4.0 \times 10^{-5} \, T \]

### Step 2: Calculate the angle of dip (δ)
Now we can use the tangent function to find the angle of dip:

\[ \tan(δ) = \frac{B_v}{B_h} \]

Substituting the values we have:

\[ \tan(δ) = \frac{4.0 \times 10^{-5}}{2 \times 10^{-9}} \]

Calculating this gives:

\[ \tan(δ) = \frac{4.0 \times 10^{-5}}{2 \times 10^{-9}} = 20000 \]

### Step 3: Finding the angle δ
To find δ, we take the arctan:

\[ δ = \tan^{-1}(20000) \]

Using a calculator:

\[ δ \approx 89.995 \, degrees \]

### Conclusion
The angle of dip at this place is approximately \(90\) degrees.

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