If `tau_1,tau_2,tau_3 and tau_4` are the magnetic torques acting on the bar magnet when it is kept at angles of `30^@ , 60^@ , 90^@ and 135^@` respectively with the direction of the magnetic field , then which among then following is correct ?

Options

A. `tau_1gttau_2gttau_3 gt tau_4`
B. `tau_3gttau_1gttau_2 gt tau_4`
C. `tau_4gttau_3gttau_2 gt tau_1`
D. `tau_3gttau_2gttau_4 gt tau_1`

Answer

To solve the problem of determining the order of magnetic torques acting on a bar magnet at different angles with respect to the magnetic field, we will follow these steps:

### Step-by-Step Solution:

1. **Understanding Magnetic Torque**:
The magnetic torque (\( \tau \)) acting on a bar magnet in a magnetic field is given by the formula:
\[ \tau = m \cdot B \cdot \sin(\theta) \]
where:
- \( m \) is the magnetic moment of the magnet,
- \( B \) is the magnetic field strength,
- \( \theta \) is the angle between the magnetic moment and the magnetic field.

2. **Calculating Torque for Each Angle**:
We will calculate the torque for each angle given in the problem.

- **For \( \tau_1 \) at \( 30^\circ \)**:
\[ \tau_1 = mB \sin(30^\circ) = mB \cdot \frac{1}{2} = \frac{mB}{2} \]

- **For \( \tau_2 \) at \( 60^\circ \)**:
\[ \tau_2 = mB \sin(60^\circ) = mB \cdot \frac{\sqrt{3}}{2} = \frac{\sqrt{3}mB}{2} \]

- **For \( \tau_3 \) at \( 90^\circ \)**:
\[ \tau_3 = mB \sin(90^\circ) = mB \cdot 1 = mB \]

- **For \( \tau_4 \) at \( 135^\circ \)**:
\[ \tau_4 = mB \sin(135^\circ) = mB \cdot \sin(45^\circ) = mB \cdot \frac{\sqrt{2}}{2} = \frac{\sqrt{2}mB}{2} \]

3. **Comparing the Magnitudes of the Torques**:
Now we will compare the calculated torques:
- \( \tau_1 = \frac{mB}{2} \)
- \( \tau_2 = \frac{\sqrt{3}mB}{2} \)
- \( \tau_3 = mB \)
- \( \tau_4 = \frac{\sqrt{2}mB}{2} \)

To compare these values, we can express them in terms of \( mB \):
- \( \tau_1 = \frac{1}{2} mB \)
- \( \tau_2 = \frac{\sqrt{3}}{2} mB \)
- \( \tau_3 = 1 \cdot mB \)
- \( \tau_4 = \frac{\sqrt{2}}{2} mB \)

Now, we can order them:
- \( \tau_3 \) is the largest (\( mB \))
- \( \tau_2 \) is next (\( \frac{\sqrt{3}}{2} mB \))
- \( \tau_4 \) follows (\( \frac{\sqrt{2}}{2} mB \))
- \( \tau_1 \) is the smallest (\( \frac{1}{2} mB \))

4. **Final Order of Torques**:
The order of the torques from largest to smallest is:
\[ \tau_3 > \tau_2 > \tau_4 > \tau_1 \]
Thus, the correct option is:
\[ \text{D: } 3, 2, 4, 1 \]