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Question: A magnetic dipole is placed in a uniform magnetic field, \(B\), the potential energy is minimum when...

A magnetic dipole is placed in a uniform magnetic field, BB, the potential energy is minimum when dipole is
A. Parallel to BB
B. Perpendicular to BB
C. Inclined at an angle of 450 to BB
D. Anti- parallel to BB

Explanation

Solution

Using the relation between the potential energy and the angle between the magnetic moment of the dipole and the uniform magnetic field, try to analyze for different cases and find the minimum potential energy.

Complete step by step answer:
We know that
The potential energy of a magnetic dipole placed in a uniform magnetic field BBis
P=−M⋅Bcos⁡θP=-M\cdot B\cos \theta
Where M=magnetic moment of the magnetic dipoleM=\text{magnetic moment of the magnetic dipole}
B=uniform magnetic fieldB=\text{uniform magnetic field}
θ=angle between them\theta =\text{angle between them}
We know that the
The potential energy is varying with θ\theta
i.e,
Case (I):
When θ=0o\theta ={{0}^{o}} that is the dipole is aligned parallel to BB
⇒P=−M⋅Bcos⁡θ\Rightarrow P=-M\cdot B\cos \theta
⇒P=−M⋅B\Rightarrow P=-M\cdot B
Therefore, the potential energy of a magnetic dipole placed in a uniform magnetic field BBis
P=−M⋅BP=-M\cdot B
Case (ii):
When θ=90o\theta ={{90}^{o}} that is the dipole is aligned perpendicular to BB
⇒P=−M⋅Bcos⁡θ\Rightarrow P=-M\cdot B\cos \theta
⇒P=0\Rightarrow P=0
Therefore, the potential energy of a magnetic dipole placed in a uniform magnetic field BBis
P=0P=0
Case (iii):
When θ=180o\theta ={{180}^{o}} that is the dipole is aligned anti-parallel to BB
⇒P=−M⋅Bcos⁡θ\Rightarrow P=-M\cdot B\cos \theta
⇒P=M⋅B\Rightarrow P=M\cdot B
Therefore, the potential energy of a magnetic dipole placed in a uniform magnetic field BBis
P=M⋅BP=M\cdot B
From the above three cases we can observe that the potential energy of a magnetic dipole placed in a uniform magnetic field BBis
Maximum when θ=0o\theta ={{0}^{o}}
Minimum when θ=180o\theta ={{180}^{o}}
Therefore we can conclude that the potential energy of a magnetic dipole placed in a uniform magnetic field BB is minimum when θ=0o\theta ={{0}^{o}} that is the dipole is aligned parallel to BB
Hence option (A) is correct.

Note:
Care should be taken to avoid calculation errors while dealing with the different angles. Also note that there is a negative sign in the formula we are using which should be considered in the calculation. So you should take care while dealing with the above formula in order to avoid conceptual errors.