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E_M_Chap2 - σ on it You will need the Heavyside...

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Chapter 2 How would you write down the volume charge densities of following objects: A ring of radius R carries a total charge Q (uniformly distributed) placed on the xy -plane with its center at the origin. ans. ρ ( ~ r ) = Q 2 πR 2 δ ( r - R ) δ ( θ - π 2 ) . A ring of radius R carries a total charge Q (uniformly distributed) placed on the z = R 2 -plane and has its center on the z -axis. ans. ρ ( ~ r ) = Q 2 πR 2 δ ( r - R ) δ ( θ - θ 0 ) . A disk of radius R has a uniform charge density
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Unformatted text preview: σ on it. You will need the Heavyside Θ-function, defined below, for this problem. Θ( x ) = x < 1 x > 1 2 x = 0 It can be represented by 1 2 ‡ 1 + x | x | · . Another nice representation is Θ( x ) = lim ² → 1 2 ‡ 1 + tanh x ² · . Note that lim ² → 1 2 ² sec h 2 x ² = d Θ( x ) dx = δ ( x )....
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