The magnetic moment of a bar magnet is \( 0.5 \, \text{Am}^2 \). It is suspended in a uniform magnetic field of \( 8 \times 10^{-2} \, \text{T} \). The work done in rotating it from its most stable to most unstable position is:
- \( 16 \times 10^{-2} \, \text{J} \)
- \( 8 \times 10^{-2} \, \text{J} \)
- \( 4 \times 10^{-2} \, \text{J} \)
- \( \text{Zero} \)
The Correct Option is B
Approach Solution - 1
To solve this problem, we need to calculate the work done in rotating a bar magnet from its most stable position (parallel to the magnetic field) to its most unstable position (antiparallel to the magnetic field). The formula for the work done \( W \) in rotating a magnetic dipole in a magnetic field is given by:
where PE (Potential Energy) is calculated using:
Here, \( M \) is the magnetic moment, \( B \) is the magnetic field, and \( \theta \) is the angle with the magnetic field.
1. **Most Stable Position**: The magnetic dipole is aligned with the magnetic field. Therefore, \( \theta = 0^\circ \).
2. **Most Unstable Position**: The magnetic dipole is antiparallel to the magnetic field. Thus, \( \theta = 180^\circ \).
Substituting these into the formula for work done:
3. **Substitute Given Values**: \( M = 0.5 \, \text{Am}^2 \) and \( B = 8 \times 10^{-2} \, \text{T} \).
Therefore, the work done in rotating the magnet is \(\mathbf{8 \times 10^{-2} \, \text{J}}\).
The correct option is: \( \mathbf{8 \times 10^{-2} \, \text{J}} \).
Approach Solution -2
Magnetic Potential Energy:
The potential energy U of a magnetic moment m in a uniform magnetic field B is given by:
U = -mB cos θ.
Stable and Unstable Equilibrium Positions:
1. At stable equilibrium (θ = 0°):
Ustable = -mB cos 0° = -mB.
2. At unstable equilibrium (θ = 180°):
Uunstable = -mB cos 180° = +mB.
Work Done:
The work done in rotating the magnet from the stable to the unstable position is the change in potential energy:
W = ΔU = Uunstable - Ustable = mB - (-mB) = 2mB.
Substitute Values:
Given:
m = 0.5 Am2, B = 8 × 10-2 T,
W = 2 × 0.5 × 8 × 10-2 = 8 × 10-2 J.
Answer: 8 × 10-2 J
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