In a deflection magnetometer which is adjusted in the usual way. When a magnet is introduced, the deflection observed is `theta` and the period of oscillation of the needle in the magnetometer is `T`. When the magnet is removed, the period of oscillation is `T_(0)`. The relation between `T` and `T_(0)` is

Options

A. `T^(2)=T_(0)^(2)cos theta`
B. `T=T_(0) cos theta`
C. `T=(T_(0))/(cos theta)`
D. `T^(2)=(T_(0)^(2))/(cos theta)`

Answer

To solve the problem, we need to establish the relationship between the periods of oscillation of the magnetometer with and without the magnet. Let's denote the following:

- \( T \): Period of oscillation when the magnet is introduced.
- \( T_0 \): Period of oscillation when the magnet is removed.
- \( \theta \): Deflection observed when the magnet is introduced.

### Step 1: Understand the Magnetic Fields
When the magnet is introduced, the total magnetic field \( B \) acting on the needle is a combination of the horizontal component \( B_H \) and the magnetic field due to the magnet \( B \). The relationship can be expressed as:
\[ B_{\text{total}} = \sqrt{B_H^2 + B^2} \]

### Step 2: Write the Expression for the Period of Oscillation
The period of oscillation \( T \) when the magnet is present can be expressed as:
\[ T = 2\pi \sqrt{\frac{I}{m}} \cdot \frac{1}{\sqrt{B_H^2 + B^2}} \]
where \( I \) is the moment of inertia of the needle and \( m \) is the magnetic moment.

When the magnet is removed, the period of oscillation \( T_0 \) is given by:
\[ T_0 = 2\pi \sqrt{\frac{I}{m}} \cdot \frac{1}{B_H} \]

### Step 3: Relate the Two Periods
Now, we can relate \( T \) and \( T_0 \) by dividing the two equations:
\[ \frac{T}{T_0} = \frac{\sqrt{B_H^2}}{\sqrt{B_H^2 + B^2}} \]

### Step 4: Substitute the Deflection Angle
From the deflection angle \( \theta \), we know that:
\[ B = B_H \tan(\theta) \]
Substituting this into our previous equation gives:
\[ \frac{T}{T_0} = \frac{B_H}{\sqrt{B_H^2 + (B_H \tan(\theta))^2}} = \frac{B_H}{\sqrt{B_H^2(1 + \tan^2(\theta))}} = \frac{B_H}{B_H \sec(\theta)} = \cos(\theta) \]

### Step 5: Final Relationship
Thus, we find that:
\[ T = T_0 \cos(\theta) \]

### Conclusion
The relationship between the periods of oscillation when the magnet is introduced and when it is removed is:
\[ T = T_0 \cos(\theta) \]