The magnetic moments of two bar magnets of same size are in the ratio `1:2.` When they are placed one over the other with their similar poles together, then their perlod of oscillation in a magnetic field is 3s. If one of the magnets is reversed, then the period of oscillation in the same field will be q

Options

A. `sqrt3s`
B. `3sqrt3s`
C. `3 s`
D. `6 s`

Answer

To solve the problem, we will follow these steps:

### Step 1: Define the Magnetic Moments
Let the magnetic moments of the two bar magnets be:
- \( m_1 = x \) (for the first magnet)
- \( m_2 = 2x \) (for the second magnet)

### Step 2: Calculate the Resultant Magnetic Moment When Similar Poles Are Together
When both magnets are placed with their similar poles together (i.e., north-north or south-south), the resultant magnetic moment \( m_3 \) is the sum of the individual magnetic moments:
\[ m_3 = m_1 + m_2 = x + 2x = 3x \]

### Step 3: Write the Expression for the Period of Oscillation
The period of oscillation \( T \) for a magnetic moment in a magnetic field is given by the formula:
\[ T = 2\pi \sqrt{\frac{I}{mB}} \]
where:
- \( I \) is the moment of inertia,
- \( m \) is the magnetic moment,
- \( B \) is the magnetic field strength.

Given that the period of oscillation when similar poles are together is \( T = 3 \, \text{s} \), we can write:
\[ 3 = 2\pi \sqrt{\frac{I}{3xB}} \tag{1} \]

### Step 4: Calculate the Resultant Magnetic Moment When One Magnet Is Reversed
When one of the magnets is reversed, the resultant magnetic moment \( m_4 \) is the difference of the two magnetic moments:
\[ m_4 = m_2 - m_1 = 2x - x = x \]

### Step 5: Write the Expression for the New Period of Oscillation
Now, we can write the expression for the period of oscillation when one magnet is reversed:
\[ T' = 2\pi \sqrt{\frac{I}{xB}} \tag{2} \]

### Step 6: Take the Ratio of the Two Periods
To find the new period \( T' \), we can take the ratio of equations (2) and (1):
\[ \frac{T'}{T} = \sqrt{\frac{3xB}{3xB}} = \sqrt{3} \]
Thus,
\[ T' = T \cdot \sqrt{3} = 3 \cdot \sqrt{3} \, \text{s} \]

### Final Answer
The period of oscillation when one of the magnets is reversed is:
\[ T' = 3\sqrt{3} \, \text{s} \]

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