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 magnetic torques acting on a bar magnet at different angles with respect to a magnetic field, we can follow these steps:

### Step 1: Understand the Torque Formula
The torque (\( \tau \)) acting on a magnetic dipole in a magnetic field is given by the formula:
\[ \tau = M \cdot B \cdot \sin(\theta) \]
where:
- \( M \) is the magnetic moment of the bar magnet (constant for a given magnet),
- \( B \) is the magnetic field strength (constant in this case),
- \( \theta \) is the angle between the magnetic moment and the magnetic field.

### Step 2: Identify the Angles and Calculate Sine Values
We need to calculate the sine values for the angles given in the problem:
- For \( \theta_1 = 30^\circ \):
\[ \sin(30^\circ) = \frac{1}{2} \]
- For \( \theta_2 = 60^\circ \):
\[ \sin(60^\circ) = \frac{\sqrt{3}}{2} \]
- For \( \theta_3 = 90^\circ \):
\[ \sin(90^\circ) = 1 \]
- For \( \theta_4 = 135^\circ \):
\[ \sin(135^\circ) = \sin(180^\circ - 45^\circ) = \sin(45^\circ) = \frac{1}{\sqrt{2}} \]

### Step 3: Compare the Sine Values
Now we can compare the sine values to determine the order of the torques:
- \( \sin(30^\circ) = \frac{1}{2} \)
- \( \sin(60^\circ) = \frac{\sqrt{3}}{2} \)
- \( \sin(90^\circ) = 1 \)
- \( \sin(135^\circ) = \frac{1}{\sqrt{2}} \approx 0.707 \)

### Step 4: Determine the Order of Torques
Since torque is directly proportional to \( \sin(\theta) \), we can rank the torques:
- \( \tau_1 \) (at \( 30^\circ \)) is the smallest.
- \( \tau_4 \) (at \( 135^\circ \)) is next.
- \( \tau_2 \) (at \( 60^\circ \)) is greater than \( \tau_4 \).
- \( \tau_3 \) (at \( 90^\circ \)) is the largest.

Thus, the order of torques from smallest to largest is:
\[ \tau_1 < \tau_4 < \tau_2 < \tau_3 \]

### Step 5: Conclusion
The maximum torque occurs when the angle is \( 90^\circ \) between the magnetic moment and the magnetic field, which corresponds to \( \tau_3 \).

### Final Answer
The correct statement is that \( \tau_3 \) (the torque at \( 90^\circ \)) is the largest.

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