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© Copyright Dean P. Neikirk 2004-2009 Mikhail Belkin, EE 325, ECE Dept., UT Austin 1 13. Time-varying fields, Faraday Law, and Maxwell’s equations EE325 Mikhail Belkin

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© Copyright Dean P. Neikirk 2004-2009 Mikhail Belkin, EE 325, ECE Dept., UT Austin 2 Summary of electrostatics and magnetostatics summarizing everything we have so far in the static case but what happens if something changes in time??? H J ∇× = r r 0 B = r r g { r o B H μ μ μ = r r v D ρ ∇• = r r { r o D E ε ε ε = r r J E σ = r r 0 E ∇× = r
© Copyright Dean P. Neikirk 2004-2009 Mikhail Belkin, EE 325, ECE Dept., UT Austin 3 Faraday experiment Faraday knew that current produces magnetic field He wondered if magnetic field somehow produce current? -- yes, if it changes in time! Different versions of Faraday experiments. He was trying to see if/how magnetic field may produce currents. He produced magnetic field by sending current through the coil. Faraday observed that a change in magnetic field (produced by changing current through the coil that generates the field) results in a current through the second coil. The current was detected by the deflection of the magnetic needle as shown.

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© Copyright Dean P. Neikirk 2004-2009 Mikhail Belkin, EE 325, ECE Dept., UT Austin 4 Faraday’s law Faraday’s law: an electromotive force (emf) is induced in a loop if the magnetic flux through the loop changes in time this emf is an induced voltage recall we get a voltage by doing a path integral of an electric field magnitude of induced voltage is proportional to how fast the flux through the loop changes in time the sign of the induced voltage is such that it would induce a current in the loop that would produce a flux that opposes the change in the flux through the loop (Lenz’s law) i.e., the “induced current” tries to keep the flux from changing in integral form Faraday’s law is the time variation could be due to either movement of the path or time variations in the magnetic field B note that we can take any ‘surface bounded by P’ and the flux of B through that surface is the same because div(B)=0 surface closed bounded path P by P d emf E dl B dS dt = = - r r r r g g Ñ
© Copyright Dean P. Neikirk 2004-2009 Mikhail Belkin, EE 325, ECE Dept., UT Austin 5 Direction of the induced emf and current Lenz’s law: the sign of the induced voltage is such that it would induce a current in the loop that opposes the change in the flux through the loop (Lenz’s law) the sign convention for the Faraday Law above works in such a way that if you go around the path in a given direction, the surface normal direction is set by the right hand rule surface closed bounded path P by P d emf E dl B dS dt = = - r r r r g g Ñ dl r dS r

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© Copyright Dean P. Neikirk 2004-2009 Mikhail Belkin, EE 325, ECE Dept., UT Austin 6 Faraday law for a moving circuit
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