Maxwell’s Equations A Student’s Guide To Maxwell’s Equations A Student’s Guide To Maxwell’s Equations by Daniel Fleisch (2008) Cambridge University Press.Daniel Fleisch Lots of videos to help Bob Eagle (UK) YouTube

Maxwell’s equations of electromagnetism All relationships between electric & magnetic fields & their sources summarized by four equations. James Clerk Maxwell

Maxwell’s equations of electromagnetism The first Maxwell equation is Gauss’s law for electric fields from Chapter 22:

Maxwell’s equations of electromagnetism Total electric flux through any closed surface equals net charge inside surface divided by  o

Gauss’ Law - for a closed surface

Gauss’ Law - for a tiny volume Divergence Scalar value of flow in/out of small volume

Gauss’ Law - for a tiny volume Divergence Scalar value of flow in/out of small volume

Gauss’ Law - for a tiny volume

Maxwell’s equations of electromagnetism The second Maxwell equation is Gauss’s law for magnetic fields from Chapter 27:

Maxwell’s equations of electromagnetism Magnetic flux through any closed surface = 0 Number of field lines that enter a closed volume = number that leave that volume Magnetic field lines cannot begin or end Isolated magnetic monopoles not observed

Gauss’ Law - for magnetism

Integrate over the closed surface – as much B flux OUT as IN; net = 0!

Gauss’ Law - for magnetism in a tiny volume

Maxwell’s equations of electromagnetism The third Maxwell equation is chapter 29’s formulation of Faraday’s law:

Maxwell’s equations of electromagnetism Describes creation of E field by a changing magnetic flux EMF = line integral of E field around closed path = rate of change of magnetic flux through surface bounded by that path Current is induced in a conducting loop placed in a time-varying B

Faraday’s Law Flux Scalar value of field flow in/out of surface area

Faraday’s Law

Divergence & Flux Divergence (from volume) & Flux (from surface area) ARE related! :)

Divergence & Flux Divergence (from volume) & Flux (from surface area) ARE related! :)

Faraday’s Law - for a tiny volume

Note ONLY electric fields induced by changing B work here! Fields from point charges don’t have a “curl”!

Maxwell’s equations of electromagnetism The fourth Maxwell equation is Ampere’s law, including displacement current: © 2016 Pearson Education Inc.

Maxwell’s equations of electromagnetism Describes creation of magnetic field by electric currents and changing electric field Line integral of magnetic field around any closed path is the given sum © 2016 Pearson Education Inc.

Ampere’s Law with Displacement Current

Ampere’s Law for a tiny volume

Maxwell’s equations in empty space

The Four Laws of Electromagnetism LawMathematical Statement Physical Meaning Gauss for EHow q produces E; E lines begin & end on q’s. Gauss for B Faraday Ampere (Steady I only) No magnetic monopole; B lines form loops. Changing  B gives emf. Moving charges give B.

Ambiguity in Ampere’s Law Consider B field in an RC circuit. Ampere’s law: Current I through any open surface S bounded by C. Current flows through surfaces 1,2,& 4. But not 3.  Ampere’s law fails ( for non-steady current ). Maxwell’s modification: Changing  E gives I, which in turn gives B.

Maxwell’s Equations LawMathematical Statement Physical Meaning Gauss for E How q produces E; E lines begin & end on q’s. Gauss for B Faraday Ampere- Maxwell No magnetic monopole; B lines form loops. Changing  B gives emf. Moving charges & changing  E give B. Maxwell’s Eqs (1864). Classical electromagnetism.

Maxwell’s Equation’s in integral form Gauss’s Law Gauss’s Law for Magnetism Faraday’s Law Ampere’s Law

Maxwell’s equations: integral form Gauss's law Gauss's law for magnetism: no magnetic monopole! Ampère's law (with Maxwell's addition) Faraday's law of induction (Maxwell–Faraday equation)

Maxwell’s Equations Differential Form

Implication A magnetic field will be produced in empty space if there is a changing electric field. (correction to Ampere) This magnetic field will be changing. (originally there was none!) The changing magnetic field will produce an electric field. (Faraday) This changes the electric field. This produces a new magnetic field. This is a change in the magnetic field.

An antenna We have changed the magnetic field near the antenna Hook up an AC source An electric field results! This is the start of a “radiation field.”

Look at the cross section E and B are perpendicular (transverse) We say that the waves are “polarized.” E and B are in phase (peaks and zeros align) Called: “Electromagnetic Waves” Accelerating electric charges give rise to electromagnetic waves