If `B_H=4xx10^(-5)T and B_(V)=2xx10^(-5)T,` then the Earth's total field (in T) at the place is

Options

A. `6xx10^(-5)T`
B. `2sqrt(5)xx10^(-5)T`
C. `4xx10^(-5)T`
D. `3xx10^(-5)T`

Answer

To find the Earth's total magnetic field at a given place, we can use the Pythagorean theorem since the horizontal and vertical components of the magnetic field are perpendicular to each other.

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Horizontal magnetic field, \( B_H = 4 \times 10^{-5} \, T \)
- Vertical magnetic field, \( B_V = 2 \times 10^{-5} \, T \)

2. **Understand the Geometry:**
- The horizontal and vertical components are at right angles to each other, which means we can use the Pythagorean theorem to find the resultant magnetic field.

3. **Apply the Pythagorean Theorem:**
- The total magnetic field \( B \) can be calculated using the formula:
\[ B = \sqrt{B_H^2 + B_V^2} \]

4. **Substitute the Values:**
- Plugging in the values we have:
\[ B = \sqrt{(4 \times 10^{-5})^2 + (2 \times 10^{-5})^2} \]

5. **Calculate Each Component:**
- Calculate \( (4 \times 10^{-5})^2 \):
\[ (4 \times 10^{-5})^2 = 16 \times 10^{-10} \]
- Calculate \( (2 \times 10^{-5})^2 \):
\[ (2 \times 10^{-5})^2 = 4 \times 10^{-10} \]

6. **Add the Squares:**
- Now, add the two results:
\[ B_H^2 + B_V^2 = 16 \times 10^{-10} + 4 \times 10^{-10} = 20 \times 10^{-10} \]

7. **Take the Square Root:**
- Now, take the square root:
\[ B = \sqrt{20 \times 10^{-10}} = \sqrt{20} \times 10^{-5} \]

8. **Simplify \( \sqrt{20} \):**
- Since \( \sqrt{20} = 2\sqrt{5} \):
\[ B = 2\sqrt{5} \times 10^{-5} \, T \]

### Final Result:
The Earth's total magnetic field at the place is:
\[ B = 2\sqrt{5} \times 10^{-5} \, T \]