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Unformatted text preview: 1 Physics 1C Winter Quarter 2009 W. Gekelman – Notes-5© Let us write down Maxwell’s equations in both integral and differential form. Differential Form Integral Form ! • ! E = " # ! E i ˆ ndA = q ! " ! • ! B = ! B i ˆ ndA = ! ! " ! E = # $ ! B $ t ! E i d ! l = ! d dt " " ! B i ˆ ndA " ! " ! B = μ ! j + μ # $ ! E $ t ! B i d ! l = μ " ¡ i + μ ¢ d dt ! E i ˆ ndA ¡ What are the curl the gradient and the divergence? In rectangular coordinates: ! " ! E = ˆ i ˆ j ˆ k # # x # # y # # z E x E y E z $ % & & & & & ’ ( ) ) ) ) ) = # E z # y * # E y # z + ,- . / ˆ i + # E x # z * # E z # x + ,- . / ˆ j + # E y # x * # E x # y + ,- . / ˆ k CURL ¡ i ! E = ¢ E x ¢ x + ¢ E y ¢ y + ¢ E z ¢ z DIVERGENCE ! 2 ! E = " 2 E x " x 2 + " 2 E y " y 2 + " 2 E z " z 2 These equations are all familiar except for the term in red. This is Maxwell’s great contribution. How do you go from integral to differential form? Take Faraday’s law: ! E i d ! l = ! d dt " " ! B i ˆ ndA " This is integral form. Then use the mathematical theorem: ! A i d ! l = ! " ! A ( ) # " # i ˆ ndA This is true for any vector A ! " ! E ( ) i ˆ ndA = # d dt " $ ! B i ˆ ndA $ and equating the integrands: ! " ! E = # $ ! B $ t For Coulombs law: 2 ! E i ˆ ndA = q ! " Now use the vector divergence theorem: ! i ! E ( ) " dV = ! E i ˆ ndA " Since the charge is the integral over the volume charge density: q = ! dV " we arrive at: ! i ! E ( ) " dV = ! E i ˆ ndA " = 1 # $ dV " . Again equating the integrands: ! • ! E = " # Consider a capacitor when it is charging Inside the capacitor no charge flows and Ampere’s law with Maxwell’s term becomes: (1) !...
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This note was uploaded on 01/21/2010 for the course PHYS 1C taught by Professor Whitten during the Winter '07 term at UCLA.
- Winter '07