2013S_721_HW4_sol - Homework 4 Solutions Physics 721 Spring...

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Homework 4 Solutions Physics 721 Spring 2013 1 Jackson 4.2 A point dipole with dipole moment v p is located at the point vx 0 . From the properties of the derivative of a Dirac delta function, show that for calculation of the potential Φ or the energy of a dipole in an external ±eld, the dipole can be described by an e²ective charge density ρ eff ( ) = v p · ∇ δ ( 0 ) Assuming charge density of the form given, Φ( ) = 1 4 πǫ 0 i d 3 x ρ ( ) | | = 1 4 πǫ 0 i d 3 x v p · ∇ δ ( 0 ) | | = 1 4 πǫ 0 i d 3 x δ ( 0 ) v p · ∇ 1 | | = 1 4 πǫ 0 i d 3 x δ ( 0 ) v p · ∇ 1 | | = 1 4 πǫ 0 v p · ( 0 ) | 0 | 3 which is the expected potential for a dipole. Calculating energy, W = i d 3 ( )Φ( ) = i d 3 xv p · ∇ δ ( 0 )Φ( ) = i d 3 ( 0 ) v p · ∇ Φ( ) = v p · v E ( 0 ) As expected for a dipole 2 Jackson 4.5 A localized charge density ρ ( x,y,z ) is placed in an external electrostatic ±eld described by a potential Φ (0) ( ). The external potential varies slowly in space over the region where the charge density is di²erent from zero. 1
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(a) From frst principles calculate the total ±orce acting on the charge distribution as an expansion in multipole moments times derivatives o± the electric feld, up to and including the quadrupole moments. Show that the ±orce is v F = q v E (0) + p b v p · v E (0) BP 0 + 1 6 s j,k Q jk v E (0) j ∂x k ( vx )
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2013S_721_HW4_sol - Homework 4 Solutions Physics 721 Spring...

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