TST2-351-F03 - 3 3) The charge density inside a spherical...

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Dr. Gangopadhyaya Test II Physics 351 May 8, 2005 As always, to receive full credit you must show your work in a neat, organized and a legible form. Some Relevant Formuae are given below: V sph ( r, θ, φ ) = X l =0 ± A l r l + B l r l +1 ² P l (cos θ ) (1) V cyl ( ρ, φ ) = ( a 0 + b 0 log ρ ) ( c 0 + d 0 φ ) + X n =1 ( a n ρ n + b n ρ - n ) ( c n cos( ) + d n sin( )) (2) ( E 2 - E 1 ) · ˆ n 1 2 = σ ± 0 ; ( E 2 - E 1 ) × ˆ n 1 2 = 0 ; cos ( a + b ) = cos a cos b - sin a sin b (3) P 0 ( x ) = 1 , P 1 ( x ) = x , P 2 ( x ) = 1 2 (3 x 2 - 1) , P 3 ( x ) = 1 2 (5 x 3 - 3 x ) ; (4) Z 1 - 1 dx P m ( x ) P n ( x ) = 2 2 m + 1 δ mn (1 + x ) n = 1 + n x + n ( n - 1) 2! x 2 + n ( n - 1)( n - 2) 3! x 3 + ··· (5) 1
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1) A sheet of charge density σ = σ 0 sin x cos y δ ( z ) separates two regions. Electric field under the interface ( z = 0 - ) is given by ~ E = ax ˆ i + by 2 ˆ j + c ˆ k , where a, b, c are constants. Find the electric field just over the interface ( z = 0 + ). 2
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2) The potential on a sphere of radius R = . 1 m is given by V 0 cos (2 θ ) where V 0 = 75 . 0 V olts . a) Determine the potential in the region r > R . In particular at r = 1 . 0 m . b) Determine the charge density on the surface of the sphere (assume no charges inside or outside the sphere.)
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Unformatted text preview: 3 3) The charge density inside a spherical balloon of radius R varies as ρ ( r ) = ρ e-r R r 2 . The density outside the sphere is zero. Determine, 1. the total charge Q ; 2. the electric field inside and outside the sphere using Gauss’ law; 4 4) The electric potential due to a charged ring ( q ) of radius R at a distance z from its center and on its axis is given by V ( q, R, z ) = 1 4 π± q √ R 2 + z 2 = q 4 π± z ± 1 + R 2 z 2 ²-1 2 . (6) However, this problem can also be viewed as an azimuthally symmetric problem and hence the potential for r > R is also given by V sph ( r, θ, φ ) = ∞ X l =0 B l r l +1 P l (cos θ ) . Using, P l (1) = 1, determine the first two non-vanishing B l . 5...
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This note was uploaded on 01/28/2010 for the course PHYS 351 taught by Professor Gangopashyaya during the Spring '09 term at Loyola Chicago.

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TST2-351-F03 - 3 3) The charge density inside a spherical...

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