# HR32 - Chapter 32 Maxwells equations Magnetism in matter In...

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Chapter 32 Maxwell’s equations; Magnetism in matter In this chapter we will discuss the following topics: -Gauss’ law for magnetism -The missing term from Ampere’s law added by Maxwell -The magnetic field of the earth -Orbital and spin magnetic moment of the electron -Diamagnetic materials -Paramagnetic materials -Ferromagnetic materials (32 – 1)

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Fig.a Fig.b In electrostatics we saw that positive and negative charges can be separated. This is not the case with magnetic poles, as is shown in the figure. In fig.a we have a p Gauss' Law for the magnetic field ermanent bar magnet with well defined north and south poles. If we attempt to cut the magnet into pieces as is shown in fig.b we do not get isolated north and south poles. Instead new pole faces appear on the newly cut faces of the pieces and the net result is that we end up with three smaller magnets, each of which is a i.e. it has a north and a south pole. This result can be expr magnetic dipole essed as follows: The simplest magnetic structure that can exist is a magnetic dipole. Magnetic monopoles do not exists as far as we know. (32 – 2)
i B G ˆ i n i φ Δ A i 123 The magnetic flux through a closed surface is determined as follows: First we divide the surface into area element with areas , ,..., n n AA A A ΔΔ Δ Δ B Magnetic Flux Φ For each element we calculate the magnetic flux through it: cos ˆ Here is the angle between the normal and the magnetic field vectors at the position of the i-th element. The inde ii i i i BA nB Δ Φ= Δ G 11 x runs from 1 to n We then form the sum cos Finally, we take the limit of the sum as The limit of the sum becomes the integral: cos nn i i B i n BdA B dA == ΔΦ = Δ →∞ Φ= = ∑∑ ∫∫ SI magnetic flux un G G vv 2 T m known as the "Weber" (Wb) it : (32 – 3) B B dA Φ =⋅ G G v

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Gauss' law for the magnetic field can be expressed mathematically as follows: For any closed surface Contrast this with Gauss' law for cos the electric field: 0 B enc E o BdA B dA q EdA ε φ Φ= ⋅= = = ∫∫ G G G G v v v Gauss' law for the magnetic field expresses the fact that there is no such a thing as a " ". The flux of either the electric or the magnetic field through a surface is proportional Φ magnetic charge to the net number of electric or magnetic field lines that either enter or exit the surface. Gauss' law for the magnetic field expresses the fact that the magnetic field lines are closed. The number of magnetic field lines that enter any closed surface is exactly equal to the number of lines that exit the surface. Thus 0. B (32 – 4) 0 B BdA = G G v
Faraday's law states that: This law describes how a changing magnetic field generates (induces) an electric field. Ampere's law in its original form reads: B d Ed S dt Φ ⋅= Induced magnetic fields G G G v . Maxwell using an elegant symmetry argument guessed that a similar term exists in Ampere's law.

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HR32 - Chapter 32 Maxwells equations Magnetism in matter In...

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