A short bar magnet of magnetic moment $0.4 ext{ J T}^{-1}$ is placed in a uniform magnetic field of $0.16

Question

A short bar magnet of magnetic moment $0.4 	ext{ J T}^{-1}$ is placed in a uniform magnetic field of $0.16 	ext{ T}$. The magnet is in stable equilibrium when the potential energy is:

Accepted Answer

1. **Formula:** The potential energy ($U$) of a magnetic dipole with magnetic moment $\mathbf{m}$ in a uniform magnetic field $\mathbf{B}$ is given by the dot product:
   $$ U = -\mathbf{m} \cdot \mathbf{B} = -mB \cos 	heta $$
   where $	heta$ is the angle between the magnetic moment vector and the magnetic field vector .
2. **Stable Equilibrium:** The magnet is in stable equilibrium when the potential energy is at its minimum. This occurs when the magnetic moment is aligned parallel to the magnetic field, i.e., $	heta = 0^\circ$ .
3. **Calculation:**
   - Magnetic Moment ($m$) = $0.4 	ext{ J T}^{-1}$
   - Magnetic Field ($B$) = $0.16 	ext{ T}$
   - Angle ($	heta$) = $0^\circ$
   $$ U = -(0.4) 	imes (0.16) 	imes \cos 0^\circ $$
   $$ U = -0.064 	imes 1 = -0.064 	ext{ J} $$
4. **Conclusion:** The potential energy in the stable equilibrium position is $-0.064 	ext{ J}$.

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