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: -
$\Rightarrow U=-MB\cos \left({180}^{\circ }\right)$
$\Rightarrow U=-M(H)(-1)=MH$ $\left(\therefore \cos {180}^{\circ }=-1and\text{ B = H}\right)$
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.