hw1_p5405_f07

# hw1_p5405_f07 - the gamma P γx P γy P γz/P γ 3 Given...

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HW 1 Phys5405 F07 due Th 9/6/07 1) For z > 0 there is vacuum with dielectric constant ± 0 , while for z < 0 there is material with dielectric constant ± . If a charge q is placed at ( x, y, z ) = (0 , 0 , D ), the electric ﬁeld, E , for z > 0 is the gradient of the scalar function, Φ = - 1 4 π± 0 ( q/R + q 0 /R 0 ) , (1) where R = q x 2 + y 2 + ( D - z ) 2 , R 0 = q x 2 + y 2 + ( D + z ) 2 , and q 0 = - q ± - ± 0 ± + ± 0 . Write this electric ﬁeld in cylindrical and spherical coordinates. 2) The rotations studied in class have great utility. For example, in particle physics there are theoretical calculations for the decay π + νγe + . These predict the distributions of the energy of the γ and e + and the cosine of the angle between them c . Experimenters are the ones to conﬁrm these predictions. Suppose the momentum vector of the positron is measured to have components ( P ex , P ey , P ez ), what are all the possible direction cosines of

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Unformatted text preview: the gamma, ( P γx , P γy , P γz ) /P γ ? 3) Given ﬁelds, E = E (1 , x/R, x 2 /R 2 ) e x , where E is constant. Perform the surface integral R S da E · n for each ﬁeld, where S is the surface of a sphere of radius R and n is the outward unit vector perpendicular to the surface? 1 4) A , B are arbitrary vectors that can be functions of the coordinates? Prove, ∇ ( A · B ) = ( A · ∇ ) B + ( B · ∇ ) A + A × ( ∇ × B ) + B × ( ∇ × A ) . (2) Use this result to prove, ∇ ( r · B ) = B + r ( ∇ · B ) + ( r × ∇ ) × B , (3) where r is the usual position vector. The quantity-i ( r × ∇ ) is the angular momentum operator in quantum mechanics. 2...
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## This note was uploaded on 12/24/2011 for the course PHYS 5406 taught by Professor Blecher during the Spring '10 term at Virginia Tech.

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hw1_p5405_f07 - the gamma P γx P γy P γz/P γ 3 Given...

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