week5 - 1 Physics 1C Winter Quarter 2009 W. Gekelman –...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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) !...
View Full Document

This note was uploaded on 01/21/2010 for the course PHYS 1C taught by Professor Whitten during the Winter '07 term at UCLA.

Page1 / 6

week5 - 1 Physics 1C Winter Quarter 2009 W. Gekelman –...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online