A bar magnet is hung by a thin cotton thread in a uniform horizontal magnetic field and is in equilibrium state. The energy required to rotate it by 60^(@) is W. Now the torrue required to keep the magnet in this new position is

Options

A. `(sqrt(3)W)/2`
B. `(2W)/sqrt(3)`
C. `W/sqrt(3)`
D. `sqrt(3)W`

Explanation

To solve the problem step by step, we will analyze the situation of the bar magnet in a magnetic field and derive the required torque.

### Step 1: Understand the Initial Condition
The bar magnet is in equilibrium in a uniform horizontal magnetic field. In this position, the magnetic moment (M) of the bar magnet is aligned with the magnetic field (B), which means the angle (θ) between M and B is 0 degrees.

**Hint:** Remember that the torque (τ) acting on a magnetic dipole in a magnetic field is given by τ = M × B × sin(θ).

### Step 2: Energy Required to Rotate the Magnet
The energy (W) required to rotate the magnet by 60 degrees from the equilibrium position can be expressed in terms of the potential energy change. The work done (W) is given by:
\[ W = PE \cdot (1 - \cos(θ)) \]
where PE is the potential energy associated with the magnetic moment in the magnetic field.

For θ = 60 degrees:
\[ W = PE \cdot (1 - \cos(60^\circ)) \]
Since \(\cos(60^\circ) = \frac{1}{2}\):
\[ W = PE \cdot (1 - \frac{1}{2}) = PE \cdot \frac{1}{2} \]

### Step 3: Torque at the New Position
When the magnet is rotated to 60 degrees, the torque (τ) required to keep it in this position can be calculated using:
\[ τ = PE \cdot \sin(θ) \]
For θ = 60 degrees:
\[ τ = PE \cdot \sin(60^\circ) \]
Since \(\sin(60^\circ) = \frac{\sqrt{3}}{2}\):
\[ τ = PE \cdot \frac{\sqrt{3}}{2} \]

### Step 4: Relate Torque and Work Done
Now we have expressions for both W and τ:
1. \( W = PE \cdot \frac{1}{2} \)
2. \( τ = PE \cdot \frac{\sqrt{3}}{2} \)

To find the ratio of τ to W:
\[ \frac{τ}{W} = \frac{PE \cdot \frac{\sqrt{3}}{2}}{PE \cdot \frac{1}{2}} \]
This simplifies to:
\[ \frac{τ}{W} = \frac{\sqrt{3}}{1} = \sqrt{3} \]

### Step 5: Calculate the Torque
From the ratio, we can express τ in terms of W:
\[ τ = \sqrt{3} \cdot W \]

### Conclusion
Thus, the torque required to keep the magnet in the new position of 60 degrees is:
\[ τ = \sqrt{3} \cdot W \]

**Final Answer:** The torque required to keep the magnet in this new position is \( \sqrt{3} W \).

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