At a certain place, the horizontal component `B_(0)` and the vertical component `V_(0)` of the earth's magnetic field are equal in magnidude. The total intensity at the place will be

Options

A. `B_0`
B. 2`B_0`
C. 3`B_0`
D. `2^(1/2)B_0`

Answer

To find the total intensity of the Earth's magnetic field at a certain place where the horizontal component \( B_0 \) and the vertical component \( V_0 \) are equal in magnitude, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Components**:
We are given that the horizontal component \( B_0 \) and the vertical component \( V_0 \) of the Earth's magnetic field are equal in magnitude. Thus, we can denote:
\[ B_0 = V_0 \]

2. **Use the Pythagorean Theorem**:
The total intensity \( B \) of the magnetic field can be found using the Pythagorean theorem, since the horizontal and vertical components are perpendicular to each other. The formula for the resultant of two perpendicular vectors is:
\[ B = \sqrt{B_0^2 + V_0^2} \]

3. **Substitute the Values**:
Since \( B_0 = V_0 \), we can substitute \( V_0 \) with \( B_0 \) in the equation:
\[ B = \sqrt{B_0^2 + B_0^2} \]

4. **Simplify the Expression**:
This simplifies to:
\[ B = \sqrt{2B_0^2} \]

5. **Final Calculation**:
Taking the square root gives:
\[ B = B_0 \sqrt{2} \]

6. **Conclusion**:
Therefore, the total intensity of the Earth's magnetic field at that place is:
\[ B = \sqrt{2} B_0 \]

### Final Answer:
The total intensity at the place will be \( \sqrt{2} B_0 \).

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