Question

The intensity of magnetic field is $H$ and moment of magnet is $M$ . The maximum potential energy is
A. $MH$
B. $2MH$
C. $3MH$
D. $4MH$

Answer 1

Hint:
This problem is based on a Magnetic Field and we know that potential energy directly varies with the uniform magnetic field. Also, the formula for magnetic potential energy is $U = - MB\cos \theta $ hence, use this relation to predict the correct value of $\theta $ to find out the maximum potential energy in the given problem.





Complete step by step solution:
We know that the magnetic potential energy is defined as: -
$U = - \overrightarrow M .\overrightarrow B $
or, $U = - MB\cos \theta $
where M = magnetic dipole moment
B = uniform magnetic Field
and, U = Magnetic Potential Energy
For maximum potential energy, $\cos \theta $ should be maximum and we know that the maximum value of $\cos \theta $ is $1$ when $\theta = {0^ \circ }$.
Since the potential energy is negative, therefore, $\cos \theta $ must be $ - 1$ to maximize the magnetic potential energy, which is possible only when $\theta = {180^ \circ }$.
Thus, Maximum Potential Energy in a given magnetic field $H$ and magnetic moment $M$will be: -
$ \Rightarrow U = - MB\cos \left( {{{180}^ \circ }} \right)$
$ \Rightarrow U = - M(H)( - 1) = MH$ $\left( {\therefore \cos {{180}^ \circ } = - 1{\text{ }}and{\text{ B = H}}} \right)$
Hence, the correct option is (A) $MH$ .


Therefore, the correct option is A.




Note:
Since this is a problem related to a uniform magnetic field and potential energy hence, quantities that are required to calculate the Maximum Potential Energy such as magnetic moment and angle $\theta $ must be identified on a prior basis as it gives a better understanding of the problem and helps to solve the question further.