Work done required to turn a magnet by `90^@` from magnetic meridian is n times to that when it is rotated by `60^@` angle. What is the value of n?

Explanation

To solve the problem, we need to calculate the work done in turning a magnet from the magnetic meridian by two different angles: 90 degrees and 60 degrees. We will use the formula for potential energy of a magnetic dipole in a magnetic field, which is given by:

\[ U = -mB \cos \theta \]

Where:
- \( U \) is the potential energy,
- \( m \) is the magnetic moment,
- \( B \) is the magnetic field strength,
- \( \theta \) is the angle between the magnetic moment and the magnetic field.

### Step 1: Calculate the work done for 90 degrees rotation

1. **Initial angle (\( \theta_1 \))**: 0 degrees (magnet aligned with the field)
2. **Final angle (\( \theta_2 \))**: 90 degrees (magnet perpendicular to the field)

Using the formula for work done \( W \):
\[ W_1 = U(\theta_1) - U(\theta_2) \]
\[ W_1 = (-mB \cos 0^\circ) - (-mB \cos 90^\circ) \]
\[ W_1 = -mB(1) - (-mB(0)) \]
\[ W_1 = -mB + 0 \]
\[ W_1 = mB \]

### Step 2: Calculate the work done for 60 degrees rotation

1. **Initial angle (\( \theta_1 \))**: 0 degrees
2. **Final angle (\( \theta_2 \))**: 60 degrees

Using the same work done formula:
\[ W_2 = U(\theta_1) - U(\theta_2) \]
\[ W_2 = (-mB \cos 0^\circ) - (-mB \cos 60^\circ) \]
\[ W_2 = -mB(1) - (-mB(1/2)) \]
\[ W_2 = -mB + \frac{mB}{2} \]
\[ W_2 = -\frac{mB}{2} \]

### Step 3: Relate the two work done expressions

According to the problem, the work done for the 90 degrees rotation is \( n \) times the work done for the 60 degrees rotation:
\[ W_1 = n \cdot W_2 \]

Substituting the values we calculated:
\[ mB = n \left(-\frac{mB}{2}\right) \]

### Step 4: Solve for \( n \)

Rearranging the equation:
\[ mB = -\frac{n \cdot mB}{2} \]
Dividing both sides by \( mB \) (assuming \( mB \neq 0 \)):
\[ 1 = -\frac{n}{2} \]
Multiplying both sides by -2:
\[ n = -2 \]

However, since we are looking for the absolute value of \( n \) in the context of work done, we take:
\[ n = 2 \]

### Final Answer
The value of \( n \) is \( 2 \).

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