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HR32_post(1) - Chapter 32 Maxwell’s equations Magnetism...

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Unformatted text preview: Chapter 32 Maxwell’s equations Magnetism in matter Will cover the following topics • Gauss’ law for magnetic field • The missing term from Ampere’s law added by Maxwell 1 • The magnetic field of the Earth • Orbital and spin magnetic moment of the electron • Diamagnetic materials • Paramagnetic materials • Ferromagnetic materials – Let’s say we have a permanent bar magnet with well defined north and south poles. If we attempt to cut the magnet into pieces, we do not get isolated orth and south poles separately. Gauss’ Law for Magnetic Fields • In electrostatics the positive and negative charges can be separated • This is not the case with magnetic poles: 2 north and south poles separately. Instead, we get again magnetic dipoles. Not even if we break the magnet down to its individual atoms and then to its electrons and nuclei, will we be able to produce magnetic monopole – Thus: The simplest magnetic structure that can exist is a magnetic dipole. Magnetic monopoles do not exist (as far as we know) • Gauss’ law for magnetic field is a formal way of saying that magnetic monopoles do not exist: the net magnetic flux through any closed surface is zero: ∫ = ⋅ = Φ A d B B r r • We can compare this equation to the Gauss’ law for electric field: ∫ = ⋅ = Φ q A d E enc E r r Gauss’ Law for Magnetic Fields (cont’d) 3 ε –Gauss’ law for the electric field says that the integral is proportional to the net enclosed charge q enc –Gauss’ law for the magnetic field says that there is no net magnetic flux through the surface, because there can be no “magnetic charge” enclosed. The simplest magnetic structure that can exist is a dipole, which consists of both, a north and south poles. Thus there must always be as much magnetic flux into the surface as out of it, producing zero net magnetic flux • In both equations, the integral is taken over closed Gaussian surface • According to Faraday’s law of induction, changing magnetic flux induces an electric field: ∫- = ⋅ dt d s d E B Φ r r • Using a symmetry argument, J.C. Maxwell guessed that, the similar statement can be also made about changing electric flux: Induced Magnetic Fields 4 ∫ = ⋅ dt d ε μ s d B E Φ r r • In both equations, the integral is taken along a closed loop Changing electric flux induces a magnetic field • This is mathematically given by Maxwell’s law of induction: • Thus, according to Maxwell’s law of induction, changing electric flux induces a magnetic field: • Example demonstrating Maxwell’s law of induction: h Consider a parallel-plate capacitor with circular plates as shown in the figure Induced Magnetic Fields (cont’d) ∫ = ⋅ dt d ε μ s d B E Φ r r 5 h The charge on the capacitor is being increased at a steady rate by a constant current i . As a result, the electric field between the plates also increases at a steady rate h The bottom figure is a view of right-hand plate....
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HR32_post(1) - Chapter 32 Maxwell’s equations Magnetism...

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