Two identical bar magnets are placed on above the other such that they are mutually perpendicular and bisect each other. The time period of this combination in a horizontal magnetic field is T. The time period of esch magnet in the same field is

Options

A. `sqrt(2)T`
B. `2^(1/4)T`
C. `2^(-1/4)T`
D. `2^(-1/2)T`

Explanation

To solve the problem, we need to analyze the time period of two identical bar magnets placed perpendicularly to each other and how it relates to the time period of each magnet in a horizontal magnetic field.

### Step-by-Step Solution:

1. **Understanding the Setup**:
We have two identical bar magnets placed one above the other, and they are mutually perpendicular. This means that if one magnet is aligned along the x-axis, the other is aligned along the y-axis.

2. **Time Period of the Combination**:
The time period \( T \) of the combination of these two magnets in a horizontal magnetic field \( B_H \) is given by the formula:
\[ T = 2\pi \sqrt{\frac{I}{B_H}} \]
where \( I \) is the moment of inertia of the system.

3. **Moment of Inertia**:
Since the two magnets are identical and placed perpendicularly, the moment of inertia of the combination can be expressed as:
\[ I = I_1 + I_2 \]
where \( I_1 \) and \( I_2 \) are the moments of inertia of each individual magnet.

4. **Time Period of Each Magnet**:
The time period \( T_H \) of each individual magnet in the same horizontal magnetic field \( B_H \) is given by:
\[ T_H = 2\pi \sqrt{\frac{I_1}{B_H}} \]
Since both magnets are identical, \( I_1 = I_2 \).

5. **Relating the Time Periods**:
Since the two magnets are identical, we can express the moment of inertia of the combination in terms of the moment of inertia of one magnet:
\[ I = 2I_1 \]
Substituting this back into the equation for the time period \( T \):
\[ T = 2\pi \sqrt{\frac{2I_1}{B_H}} \]

6. **Expressing \( T \) in Terms of \( T_H \)**:
Now, we can relate \( T \) to \( T_H \):
\[ T_H = 2\pi \sqrt{\frac{I_1}{B_H}} \]
Therefore, we can write:
\[ T = 2\pi \sqrt{2} \sqrt{\frac{I_1}{B_H}} = \sqrt{2} T_H \]

7. **Final Expression**:
Thus, the time period of each magnet in the same field is:
\[ T_H = \frac{T}{\sqrt{2}} \]

### Final Answer:
The time period of each magnet in the same field is \( \frac{T}{\sqrt{2}} \).