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Physics Lesson 16.5.3 - Energy of a Magnetic Dipole

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Welcome to our Physics lesson on Energy of a Magnetic Dipole, this is the third lesson of our suite of physics lessons covering the topic of Magnetic Dipole Moment, you can find links to the other lessons within this tutorial and access additional physics learning resources below this lesson.

Energy of a Magnetic Dipole

As we know from Section 5, objects and systems tend to occupy a state in which they have the lowest energy possible. For example, when you roll a ball inside a hole, it swings several times until it stops at bottom of the hole in which the ball is closer to the ground and therefore it has the lowest gravitational energy. Likewise, when you make a pendulum swing, it will stop at the vertical position, in which the bob is closer to the ground. In this case, the bob has the lowest gravitational potential energy possible. Another example: people lie on the bed when sleeping so that the centre of gravity of the body be as close as possible to the ground. In this case, they have the lowest energy possible and therefore, they can take a rest.

This rule is also true when a magnetic dipole is inserted inside an external magnetic field. When the vectors of magnetic field and magnetic dipole moment are collinear, the system has the lowest energy as the magnetic moment vector attempts to align with the magnetic field. Following this reasoning, we can say that when magnetic moment vector is antiparallel to the magnetic field lines (as in the previous solved example), the system has the highest energy possible.

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Therefore, we can say that a magnetic dipole in an external field has an energy that depends on the orientation of dipole moment in respect to the magnetic field direction. For electric dipoles, we have for the energy U in terms of the angle θ:

U(θ) = -p⃗ ∙ E⃗

where p⃗ is the electric dipole moment and E⃗ is the electric field. Similarly, we obtain for the energy U of magnetic dipoles in terms of the angle θ:

U(θ) = -μ⃗ ∙ B⃗

Note that here we have a dot product of two vectors because energy is a scalar (the dot product of two vectors gives a scalar). The sign minus is because the maximum value of energy is obtained for θ = 180° for which cos θ = -1. In this way, we obtain a positive value for the maximum energy.

Example 3

Calculate the energy of the magnetic dipole shown in the figure below if the radius of loop is r = 3cm, the current flowing through the loop is I = 5A and the magnetic field lines punch the area of loop at α = 60°. The magnitude of magnetic field is B = 40mT.

Physics Tutorials: This image provides visual information for the physics tutorial Magnetic Dipole Moment

Solution 3

Clues:

r = 3 cm = 3 × 10-2 m
I = 5A
α = 600
B = 40 mT = 4 × 10-2 T
N = 1
U = ?

The vector of magnetic dipole moment is directed upwards. This is found using the right hand rule mentioned earlier.

The angle formed by the magnetic dipole moment vector to the magnetic field lines is θ = 90° + α = 90° + 60° = 150°.

The magnitude of magnetic dipole moment μ is

μ = N ∙ I ∙ A
= N ∙ I ∙ π ∙ r2
= 1 ∙ (5A) ∙ 3.14 ∙ (3 × 10-2 m)2
= 1.413 × 10-2 A ∙ m2

The energy of the magnetic dipole therefore is

U(θ) = -μ⃗ ∙ B⃗
= -μ ∙ B ∙ cosθ
= -(1.413 × 10-2 A ∙ m2 ) ∙ (4 × 10-2 T) ∙ (cos1500 )
= -(1.413 × 10-2 A ∙ m2 ) ∙ (4 × 10-2 T) ∙ (-0.866)
= 4.89 × 10-4 J

You have reached the end of Physics lesson 16.5.3 Energy of a Magnetic Dipole. There are 4 lessons in this physics tutorial covering Magnetic Dipole Moment, you can access all the lessons from this tutorial below.

More Magnetic Dipole Moment Lessons and Learning Resources

Magnetism Learning Material
Tutorial IDPhysics Tutorial TitleTutorialVideo
Tutorial		Revision
Notes		Revision
Questions
16.5Magnetic Dipole Moment
Lesson IDPhysics Lesson TitleLessonVideo
Lesson
16.5.1What is Magnetic Moment? Magnetic Dipole Moment
16.5.2Torque in Terms of Magnetic Dipole Moment
16.5.3Energy of a Magnetic Dipole
16.5.4Work Done on a Magnetic Dipole

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