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ECE-305-Lecture4

# ECE-305-Lecture4 - Engineering Electro-Magnetics...

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Engineering Electro-Magnetics Engineering Electro-Magnetics ECE-305, Lecture 4 Dennis McCaughey, Ph.D. 26 January 2010

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01/26/2010 Dennis McCaughey, ECE305,Spring 2010 2 Homework Homework Problems 2.3 and 2.7 Due Next Tues February 2
01/26/2010 Dennis McCaughey, ECE305,Spring 2010 3 2.3 Field Due to a Continuous Volume 2.3 Field Due to a Continuous Volume Charge Distribution Charge Distribution Region filled with charges separated by minute distances Define a charge density, charge per unit volume The integral is a triple Difficult to evaluate in all but the most symmetrical problems 3 0 Charge Density: ( ) Small charge in a small volume: lim Total charge within some finite volume v v v V v vol C m Q V Q V Q dv ρ ρ ρ ρ = = =

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01/26/2010 Dennis McCaughey, ECE305,Spring 2010 4 Example 2.3: Charge in an Electron Example 2.3: Charge in an Electron Beam Beam ( 29 5 5 5 0.04 2 0.01 6 10 0.02 0 0 0.04 0.01 5 10 0.02 0 .04 0.01 5 10 5 0.0 .02 0.01 10 2000 4000 0 2000 10 5 10 first 10 next 10 = 10 = 10 10 2000 v vol z z z z z Q dv e d d dz e d dz z e d e e d e e π ρ ρ ρ ρ ρ ρ ρ ρ ρ φ φ π ρ ρ π ρ ρ ρ π ρ π - - - - = - - = - - - - - - = = - × - × - - - - = - - - ∫ ∫ ∫ ∫ ∫ .01 4000 0 10 4000 1 1 10 0.0785 4000 2000 pC ρ π - - ≈ - - = Use cylindrical coordinates
01/26/2010 Dennis McCaughey, ECE305,Spring 2010 5 Example 2.3: Setup in Rectangular Example 2.3: Setup in Rectangular Coordinates Coordinates { } ( 29 2 2 5 , , 0.01,0.02 0.04 0.04 2 0.01 6 10 0.02 0 0 : 5 10 v vol v x y z x y z z Q dv dxdydz Jacobian dxdydz d d dz dz d d e π ρ ρ ρ ρ ρ φ φ ρ ρ + ≤ ≤ - - = = = = - × ∫∫∫

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01/26/2010 Dennis McCaughey, ECE305,Spring 2010 6 Electron Beam Current Estimate Electron Beam Current Estimate Assume the electrons are moving at 10% of the speed of light Can estimate the current= coulombs/sec ( 29 ( 29 2 9 8 12 9 6 2 10 2 10 sec 0.1 3 10 3 40 10
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ECE-305-Lecture4 - Engineering Electro-Magnetics...

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