Torques `tau_(1)` and `tau_(2)` are required for a magnetic needle to remain perpendicular to the magnetic fields at two different places. The magnetic field at those places are B1 and B2 respectively, then `(B_(1))/(B_(2))` is

Explanation

To solve the problem, we need to analyze the relationship between the torques acting on a magnetic needle in two different magnetic fields. Here's a step-by-step solution:

### Step 1: Understanding Torque
The torque (\( \tau \)) on a magnetic needle in a magnetic field is given by the formula:
\[ \tau = \mathbf{m} \times \mathbf{B} \]
Where:
- \( \tau \) is the torque,
- \( \mathbf{m} \) is the magnetic moment of the needle,
- \( \mathbf{B} \) is the magnetic field.

### Step 2: Torque When Needle is Perpendicular
When the magnetic needle is perpendicular to the magnetic field, the angle \( \theta \) between the magnetic moment and the magnetic field is \( 90^\circ \). Therefore, the sine of \( 90^\circ \) is 1. Thus, the torque can be simplified to:
\[ \tau = mB \sin(90^\circ) = mB \]

### Step 3: Expressing Torques for Two Different Fields
Let \( \tau_1 \) be the torque required for the magnetic needle to remain perpendicular in the magnetic field \( B_1 \), and \( \tau_2 \) be the torque in the magnetic field \( B_2 \). We can express these as:
\[ \tau_1 = mB_1 \]
\[ \tau_2 = mB_2 \]

### Step 4: Finding the Ratio of the Magnetic Fields
To find the ratio of the magnetic fields \( \frac{B_1}{B_2} \), we can set up the following equation using the expressions for \( \tau_1 \) and \( \tau_2 \):
\[ \frac{\tau_1}{\tau_2} = \frac{mB_1}{mB_2} \]
Since the magnetic moment \( m \) is constant for the same needle, it cancels out:
\[ \frac{\tau_1}{\tau_2} = \frac{B_1}{B_2} \]

### Step 5: Conclusion
Thus, we can conclude that:
\[ \frac{B_1}{B_2} = \frac{\tau_1}{\tau_2} \]

### Final Answer
The ratio of the magnetic fields \( \frac{B_1}{B_2} \) is equal to the ratio of the torques \( \frac{\tau_1}{\tau_2} \).

---