A magnet is cut in three equal parts by cutting it perpendicular to its length. The time period of original magnet is `T_0` in a uniform magnetic field B. Then, the time period of each part in the same magnetic field is

Options

A. `T_(0)`
B. `(T_(0))/(3)`
C. `(T_(0))/(2)`
D. `4T_(0)`

Answer

To solve the problem, we need to analyze how cutting the magnet into three equal parts affects its time period in a uniform magnetic field.

### Step-by-step Solution:

1. **Understand the Original Time Period Formula**:
The time period \( T_0 \) of the original magnet in a uniform magnetic field \( B \) is given by:
\[ T_0 = 2\pi \sqrt{\frac{I}{M B}} \]
where \( I \) is the moment of inertia, \( M \) is the magnetic moment, and \( B \) is the magnetic field.

2. **Cutting the Magnet**:
When the magnet is cut into three equal parts, the length of each part becomes \( \frac{L}{3} \). Since the magnetic moment \( M \) is proportional to the length of the magnet, the magnetic moment of each part \( M' \) will be:
\[ M' = \frac{M}{3} \]

3. **Calculating the Moment of Inertia for Each Part**:
The moment of inertia \( I \) of a uniform rod about its center is given by:
\[ I = \frac{ML^2}{12} \]
For each part, the mass \( m' \) will be \( \frac{M}{3} \) and the length will be \( \frac{L}{3} \). Therefore, the moment of inertia \( I' \) for each part is:
\[ I' = \frac{m' \left(\frac{L}{3}\right)^2}{12} = \frac{\left(\frac{M}{3}\right) \left(\frac{L}{3}\right)^2}{12} \]
Simplifying this:
\[ I' = \frac{M \cdot \frac{L^2}{9}}{36} = \frac{ML^2}{324} = \frac{I}{27} \]

4. **Finding the New Time Period**:
Now, substituting \( I' \) and \( M' \) into the time period formula for each part:
\[ T' = 2\pi \sqrt{\frac{I'}{M' B}} = 2\pi \sqrt{\frac{\frac{I}{27}}{\frac{M}{3} B}} \]
This simplifies to:
\[ T' = 2\pi \sqrt{\frac{I}{27} \cdot \frac{3}{MB}} = 2\pi \sqrt{\frac{3I}{27MB}} = 2\pi \sqrt{\frac{I}{9MB}} = 2\pi \sqrt{\frac{I}{MB}} \cdot \frac{1}{3} \]
Thus, we can express the new time period \( T' \) in terms of the original time period \( T_0 \):
\[ T' = \frac{T_0}{3} \]

5. **Conclusion**:
The time period of each part of the magnet in the same magnetic field is:
\[ T' = \frac{T_0}{3} \]