Question

A short bar magnet of magnetic moment \[0.4JT\]is placed in a uniform magnetic field of\[0.16{\text{ }}T\]. The magnet is in stable equilibrium when the potential energy is
 $ \left( A \right)0.064J \\$
 $ \left( B \right) - 0.064J \\$
 $\left( C \right)zero \\$
 $\left( D \right)0.082J \\$

Answer 1

Hint: In order to solve this question, we are going to first write the formula for the potential energy of the magnet that depends on the magnetic moment and the magnetic field, then finding the minimum potential energy corresponding to the stable equilibrium, i.e. found for the angle equal to zero.

Formula used:
The potential energy of a magnet with the magnetic moment\[\vec M\]that is placed in magnetic field \[\vec B\]is given by
\[U = - \vec M \cdot \vec B\]

Complete step-by-step answer:
As we know that the potential energy of a magnet with the magnetic moment\[\vec M\]that is placed in magnetic field \[\vec B\]is equal to the negative of the dot product of the two quantities
i.e, it is given by
\[U = - \vec M \cdot \vec B\]
Now solving this further, we get
\[U = - MB\cos \theta \]
The dot product is the smallest for\[\theta = 0\] as it is given negative,
The stable equilibrium means that the potential energy of the magnet is the minimum at that value. Hence, we get minimum potential energy as
\[{U_{\min }} = - MB = - 0.4 \times 0.16J = - 0.064J\]
Hence, the option \[\left( B \right) - 0.064J\]is the correct answer to this question.

So, the correct answer is “Option B”.

Note: It is important to note that the magnet approaches the equilibrium with a decrease in the potential energy. The stable equilibrium corresponds to the minimum energy which is thus found in the question for the angle \[\theta = 0\]. The potential energy directly depends on vectors' magnetic moment and magnetic field and the angle between them.