Question

A short bar magnet of magnetic moment $0.4J/T$ is placed in a uniform magnetic field of $0.16T$. The magnet is in stable equilibrium when the potential energy is

Answer 1

Step 1: Given data

1. The magnetic moment of the bar is $p_m=0.4J/T.$
2. The magnitude of the uniform magnetic field $B=0.16T.$

Step 2: The potential energy of a bar magnet

1. The potential energy stored in a bar magnet when it is placed in a magnetic field is the vector dot product of the magnetic moment and magnetic field.
2. The potential energy of a magnet is defined by the form, $U=-\overrightarrow{P_m}.\overrightarrow{B}$, where, $P_m$ is the magnetic moment and $B$ is the magnitude of a magnetic field.
3. The magnet will be in stable equilibrium when the bar magnet is aligned along the magnetic field. In this stable equilibrium, the potential energy will be minimum.

Step 3: Diagram

Step 4: Finding the potential energy

As we know, the potential energy of a magnet in a magnetic field is $U=-\overrightarrow{P_m}.\overrightarrow{B}$

So, $U=-\overrightarrow{P_m}.\overrightarrow{B}=p_m\times B\times \cos \theta$

At stable equilibrium, $\theta =0$ ( the bar magnet is aligned along the magnetic field)

Now,

$$\begin{gathered} U=p_m\times B\times \cos \theta =\left(0.4\right)\times \left(0.16\right)\times \cos 0 \\ orU=0.064\times 1 \\ orU=0.064J. \end{gathered}$$

Therefore, the potential energy when the magnet is in stable equilibrium is $0.064joule.$