PHY2061 Enriched Physics 2 Lecture Notes Magnetic Dipoles Magnetic Dipoles Disclaimer These lecture notes are not meant to replace the course textbook The content may be incomplete Some topics may be unclear These notes are only meant to be a study aid and a supplement to your own notes Please report any inaccuracies to the professor Magnetic Field of Current Loop z B r I R y x For distances R r the loop radius the calculation of the magnetic field does not depend on the shape of the current loop It only depends on the current and the area as well as R and 0 cos Br 2 4 R 3 B B 0 sin 4 R 3 where iA is the magnetic dipole moment of the loop Here i is the current in the loop A is the loop area R is the radial distance from the center of the loop and is the polar angle from the Z axis The field is equivalent to that from a tiny bar magnet a magnetic dipole We define the magnetic dipole moment to be a vector pointing out of the plane of the current loop and with a magnitude equal to the product of the current and loop area K K i A i The area vector and thus the direction of the magnetic dipole moment is given by a right hand rule using the direction of the currents D Acosta Page 1 10 24 2006 PHY2061 Enriched Physics 2 Lecture Notes Magnetic Dipoles Interaction of Magnetic Dipoles in External Fields Torque By the F iL B ext force law we know that a current loop and thus a magnetic dipole feels a torque when placed in an external magnetic field B ext The direction of the torque is to line up the dipole moment with the magnetic field F Bext F i Potential Energy Since the magnetic dipole wants to line up with the magnetic field it must have higher potential energy when it is aligned opposite to the magnetic field direction and lower potential energy when it is aligned with the field To see this let us calculate the work done by the magnetic field when aligning the dipole Let be the angle between the magnetic dipole direction and the external field direction W F ds F sin ds r F sin d where ds rd r F d W d Now the potential energy of the dipole is the negative of the work done by the field U W d The zero point of the potential energy is arbitrary so let s take it to be zero when 90 Then 2 2 U d D Acosta B sin d Page 2 10 24 2006 PHY2061 Enriched Physics 2 Lecture Notes Magnetic Dipoles The positive sign arises because d d and are oppositely aligned Thus U B cos 2 B cos Or simply U B The lowest energy configuration is for and B parallel Work energy is required to re align the magnetic dipole in an external B field B B Lowest energy Highest energy The change in energy required to flip a dipole from one alignment to the other is U 2 B D Acosta Page 3 10 24 2006 PHY2061 Enriched Physics 2 Lecture Notes Magnetic Dipoles Force on a Magnetic Dipole in a Non uniform Field or why magnets stick Two bar magnets stick together when opposite poles are brought together north south and repel when the same poles are brought together north north south south The magnetic field of a small bar magnet is equivalent to a small current loop so two magnets stacked end to end vertically are equivalent to two current loops stacked z N i2 1 S N i1 S The potential energy on one dipole from the magnetic field from the other is U 1 B 2 z1 Bz 2 choosing the z axis for the magnetic dipole moment Now force is derived from the rate of change of the potential energy F U U z for this particular case z For example the gravitational potential energy of a mass a distance z above the surface of the Earth can be expressed by U mgz Thus the force is F mg z i e down For the case of the stacked dipoles Fz B U 1z 2 z z z or in general any magnetic dipole placed in a non uniform B field Fz z B z Thus there is a force acting on a dipole when placed in a non uniform magnetic field D Acosta Page 4 10 24 2006 PHY2061 Enriched Physics 2 Lecture Notes Magnetic Dipoles For this example the field from loop 2 increases with z as loop 1 is brought toward it B2 from below 0 z Thus the force on loop 1 from the non uniform field of loop 2 is directed up and we see that there is an attractive force between them North South attract Another way to see this attraction is to consider the F iL B ext force acting on the current in loop 1 in the presence of the non uniform field of loop 2 i2 F F i1 D Acosta Page 5 10 24 2006 PHY2061 Enriched Physics 2 Lecture Notes Magnetic Dipoles Atomic Dipoles Now why do some materials have magnetic fields in the absence of electric currents Consider the atom An electron orbiting an atom is like a small current loop Let s consider first a classical model of the atom with the electron in a circular orbit L r e i We can determine the magnetic dipole moment as follows i A e T r2 2 r period v e v r 2 erv 2 r 2 T But angular momentum is defined as L r p L L rp mvr for circular orbits This implies e L 2m The amazing thing is that this relation which was defined classically also holds in quantum physics The details of the orbit are not important only that there is some net angular momentum An atom with an electron in an orbit with angular momentum is a small current loop which implies that it is also a magnetic dipole Now consider an atom immersed in an external magnetic field applied along the z axis The potential energy is U B z Bz and the magnetic dipole moment is e Lz z 2m D Acosta Page 6 10 24 2006 PHY2061 Enriched Physics 2 Lecture Notes Magnetic Dipoles From quantum physics we have that the average angular momentum about the z axis is given by mh Lz A 2 where mA 0 1 2 is an integer the quantum number and h 6 626 10 34 J s is a new constant in nature known as Planck s Constant Thus for the potential energy of an atom in an external magnetic field we have U B B mA B eh 5 79 10 5 eV T Bohr magneton 4 me The Stern Gerlach Experiment and the Spin of the Electron Let s consider the Stern Gerlach experiment of 1922 In that experiment a neutral atomic beam is passed through …
View Full Document
Unlocking...