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Week 11 Solns

# Week 11 Solns - Electromagnetics Workshop Week 11 Solutions...

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Electromagnetics Workshop – Week 11 Solutions Dielectric Breakdown & Stored Energy 1 Problem 1: Dielectric breakdown occurs in a material whenever the magnitude of the field E exceeds the dielectric strength anywhere in the material. a) In a coaxial capacitor filled with insulating material of permittivity ε , at what value of ρ is E a maximum? b) What is the breakdown voltage if 1 a = [cm], 2 b = [cm], and the dielectric material is mica with 6 r = and [ ] 200 MV/m BR = E ? c) If 24 V = [V] for the material 6 r = , what is the energy stored per unit length in this coaxial capacitor? Problem 1 Solution: a) From Gauss’s Law, we can easily show that 2 E περ =− A Hence, E is a maximum when a = . b) () 0 ln ln 2 22 1 2 b a b Vd a ρρ πε ⎛⎞ === ⎜⎟ ⎝⎠ AA A 0 12 2 BR BR EE a a =⇒ = A A 0 62 0 12 ln 2 ln 2 200 10 1 10 ln 2 1.386 MV 12 BR BR BR Ea VE a == = × × c) 0 0 288 ln 2 24 12 ln 2 V = A A , 24 ln 2 E 2 2 2 00 6 3456 24 l n 2 l n 2 Lb e Va L Wd V d d d z π ρρφ = ∫∫∫ ∫ ∫ ∫ E [] 0 3456 / 138.7 nJ/m ln 2 e WL =≈

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Electromagnetics Workshop – Week 11 Solutions Dielectric Breakdown & Stored Energy 2 Problem 2: A typical thundercloud can be modeled as a capacitor with horizontal plates with 10 [km 2 ] area separated by a vertical distance of 5 [km]. Just before a large lightning discharge, the upper plate may have a total positive charge of up to 300 [C], with the lower plate having an equal amount of negative charge. a) What is the average electric field within the cloud? How does this compare to the dielectric breakdown field of dry air (3 [MV/m])? b) What is the potential difference between the top and bottom plates? c) Find the electrostatic energy stored in the cloud just before a discharge. Problem 2 Solution: Choosing z as our vertical axis, we can easily show using Gauss’s Law that 0 s z E ρ ε =− , [ ] 62 2 300 C 30 10 C/m 10 km s == × ⎡⎤ ⎣⎦ a) [] 6 0 30 10 3.39 MV/m avg × =≈ E , This is pretty close to the dielectric breakdown field of dry air.
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Week 11 Solns - Electromagnetics Workshop Week 11 Solutions...

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