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MIT6_641s09_sol_quiz2009

# MIT6_641s09_sol_quiz2009 - MIT OpenCourseWare...

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MIT OpenCourseWare http://ocw.mit.edu 6.641 Electromagnetic Fields, Forces, and Motion Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .

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6.641 Electromagnetic Fields, Forces, and Motion Spring 2009 Quiz 1 - Solutions - Spring 2009 Prof. Markus Zahn MIT OpenCourseWare Problem 1 Figure 1: A diagram of a perfectly conducting hollow sphere filled with a perfectly insulating dielectric with a uniform distribution of volume charge (Image by MIT OpenCourseWare). A perfectly conducting hollow sphere of radius R is filled with a perfectly insulating dielectric ( σ = 0) with a uniform distribution of volume charge: ρ = ρ 0 0 < r < R within a medium with permittivity . The sphere is grounded at r = R so that the scalar electric potential at r = R is zero, Φ( r = R ) = 0. There is no point charge at r = 0 so that E r ( r = 0) must be finite. A Question: What is the EQS electric field E ( r ) for 0 < r < R? Solution: 1 ρ o d ρ o r 2 � • E = r 2 ∂r ( r 2 E r ) = dr ( r 2 E r ) = r 2 E r = ρ 0 r 3 + C 1 E r = ρ o r + C 1 3 3 r 2 E r ( r = 0) is finite C 1 = 0 ρ o r E r ( r ) = 0 < r < R 3 1
C Quiz 1: 2009 - Solutions 6.641, Spring 2009 B Question: What is the scalar electric potential Φ( r ) where E ( r ) = −� Φ( r ) ? Solution: d Φ ρ o r ρ o r 2 E r = dr = 3 Φ = 6 + C 2 ρ o R 2 ρ o R 2 Φ( r = R ) = 0 = + C 2 C 2 = 6 6 Φ( r ) = ρ o ( r 2 R 2 ) 6 Question: What is the free surface charge density σ s ( r = R ) on the inside surface of the sphere at r = R ? Solution: �ρ o R ρ 0 R σ s ( r = R ) = �E r ( r = R ) = 6 = 3 Problem 2 Figure 2: A diagram of a uniform line charge of height L standing perpendicular to the ground plane (Image by MIT OpenCourseWare). A uniform line charge λ coulombs/meter of height L stands perpendicularly on a perfectly conducting ground plane of infinite extent in free space with dielectric permittivity 0 . 2

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Quiz 1: 2009 - Solutions 6.641, Spring 2009 A Question: Find the electric field at the ground plane surface E ( r, z = 0 + ) where r is the cylindrical radial coordinate shown above. See integrals in hint below.
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MIT6_641s09_sol_quiz2009 - MIT OpenCourseWare...

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