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Notes 5 Spring 2005

# Notes 5 Spring 2005 - Notes 5 Spring 2005 Development of...

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Notes 5 Spring 2005.doc Development of Maxwell’s equations : Faraday’s law of induction Recall Stokes’ theorem: ( ) = × l d A S d A s G G G G G Since Stokes’ theorem is valid for any vector field, we can substitute the electric field vector, E G , for A G to get ( ) = × l d S d s G G G G G E E and note that the expression for Faraday induction, dt d l d Φ = G E , describes how a changing magnetic flux induces a voltage (emf) about a closed path. Substituting this result into the boxed equation above yields () dt d S d s Φ = × G G G E Provided we are working with an area that is constant in time, we can associate the time-derivative operator with only the magnetic field while using the definition of magnetic flux, = Φ s S d G G B , to obtain = = × s s s S d dt d S d dt d S d G G G G G G G B B E then we equate the integrands to obtain dt d B E G G G = × This is what we will refer to as Maxwell’s first equation . Since it was derived directly from Faraday’s law of induction it is often also referred to as Faraday’s law of induction . It shows that whenever a time-varying magnetic field is present there is also (necessarily present) a non-zero circulation or curl of the electric field, i.e., an electric field is present. 1

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Notes 5 Spring 2005.doc Ampere’s law Similar to what we did with the electric field, we can substitute H G into Stokes’ Theorem to obtain () ∫∫ × = s S d l d G G G G G H H We see from unit analysis that the left side of this equation has units of Amperes and so the right hand side must also have units of Amperes. This implies that H G G × has units of 2 m A which are the units of a current density. In fact it turns out that H G G × is the sum of two current densities, and c J G d J G , the conduction and displacement current densities, respectively. So we write d c J J G G G G + = × H The displacement current density, , results from a time-varying electric flux density, so it is common to write d J G t J c + = × D H G G G G This is Maxwell’s second equation , Ampere’s law , which shows that a current density results in a circulation or curl of a magnetic field around that current density.
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Notes 5 Spring 2005 - Notes 5 Spring 2005 Development of...

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