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Step-by-Step Solution
Step 1: Identify the given parameters
• The magnetic moment of the bar magnet is given as $M = 0.4 \,\text{J T}^{-1}$.
• The uniform magnetic field is given as $B = 0.16 \,\text{T}$.
Step 2: Write down the formula for potential energy
The potential energy $U$ of a bar magnet (magnetic dipole) in a uniform magnetic field is given by
$U = - \vec{M} \cdot \vec{B} = - M B \cos \theta$,
where $\theta$ is the angle between the magnetic moment vector and the magnetic field.
Step 3: Determine the angle for stable equilibrium
For stable equilibrium, the bar magnet aligns itself with the magnetic field, implying the angle $\theta = 0^\circ.$ Hence, $\cos \theta = \cos 0^\circ = 1.$
Step 4: Substitute the values to find the potential energy
Plug $M = 0.4 \,\text{J T}^{-1}$, $B = 0.16 \,\text{T}$, and $\cos 0^\circ = 1$ into the formula:
$U = - M B = - (0.4)\times (0.16) = - 0.064 \,\text{J}.$
Step 5: State the final answer
Therefore, the potential energy of the magnet in the uniform magnetic field, in stable equilibrium, is $-\,0.064 \,\text{J}.$
