Time period of cillations of a magnet of magnetic moment M and moment of inertia l in a vertical plane perpendicular to the magneitc meridian at a place where earth's horizontal and vertical component of magnetic field are `B _(H) and B_(v)` respectively is

Options

A. `T = 2pi sqrt ((1)/(M (B_(v) ^(2) + B_(H) ^(2))))`
B. `T = 2pi sqrt ((l )/(MB _(v)))`
C. `T = 2pi sqrt ((l )/(MB _(H)))`
D. Infinite

Answer

To find the time period of oscillations of a magnet with magnetic moment \( M \) and moment of inertia \( I \) in a vertical plane perpendicular to the magnetic meridian, we can follow these steps:

### Step 1: Understand the System
We have a magnet with a magnetic moment \( M \) and a moment of inertia \( I \). The magnet is oscillating in a vertical plane that is perpendicular to the magnetic meridian. The Earth's magnetic field has two components: the horizontal component \( B_H \) and the vertical component \( B_V \).

### Step 2: Identify the Relevant Formula
The time period \( T \) of oscillations of a magnetic dipole in a magnetic field is given by the formula:
\[ T = 2\pi \sqrt{\frac{I}{M B}} \]
where \( B \) is the magnetic field strength acting on the magnet.

### Step 3: Determine the Effective Magnetic Field
Since the magnet is oscillating in a vertical plane perpendicular to the magnetic meridian, the effective magnetic field that influences the oscillation is the vertical component of the Earth's magnetic field \( B_V \). The horizontal component \( B_H \) does not contribute to the oscillation in this orientation.

### Step 4: Substitute the Values
Now, substituting \( B = B_V \) into the formula, we get:
\[ T = 2\pi \sqrt{\frac{I}{M B_V}} \]

### Step 5: Final Expression
Thus, the time period of the oscillations of the magnet is:
\[ T = 2\pi \sqrt{\frac{I}{M B_V}} \]

### Conclusion
The time period of oscillations of the magnet is determined by its moment of inertia, magnetic moment, and the vertical component of the Earth's magnetic field.

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