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Physics PHYS 354 Electricity and Magnetism II Problem Set #4 1. Vector Transformations Consider two right-handed, orthogonal coordinate systems, their axes being defined by the unit vectors ê1 , ê2 , ê3 and ê1 , ê2 , ê3 respectively. The two systems have a common origin O. The position vector of point P may then be written as r xi eˆi xieˆi . It is clear that every ê can be written as a linear combination of the ê 's: eˆi Rij eˆ j . a) Establish the relations Rij eˆi eˆ j , which imply that eˆi eˆi eˆ j eˆ j . b) Show that it follows at once that eˆi eˆj R ji . c) Use the results of (a) and (b) to establish the transformation rules for the vector components x and x': xi Rij x j and xi xj R ji . d) Use the orthogonality relations for the basis vectors ê and ê : eˆi ˆe j ij and eˆi eˆj ij to establish the orthogonality relations for the tensors R ij : Rin R jn ij and Rni Rnj ij . e) The three unit vectors êi (i=1, 2, 3), as well as the êi , satisfy the relations eˆ eˆ eˆ i j k ijk where ijk is the totally antisymmetric tensor. Use this property to show that the determinant of the transformation matrix Rij has the value 1: det Rij 1 . It may be helpful to show that det Rij ijk R1i R2 j R3k first. This can be done by brute force. 2. Binomial Expansion of P x-y The electric potential x for the charge distribution x is given by x 1 x 3 x d y 4 0 xy y where the vectors are defined according to the diagram at right. a) o Show, using the binomial expansion, that xˆ 1 1 1 ˆˆ j yj 3 xi x j 3 yi y j ij yk yk . x y xi xi xi xi 2xk xk 2 when x y . b) Show then that the electric potential can be written in the form: xˆ xˆ xˆ q x j p j i j 3 Qij . xi xi xi xi 2xk xk 2 c) Write down expressions for the charge q , the dipole moment p j , and the quadrupole moment tensor Qij . d) e) Now let the axes be translated by Y , that is, y y Y . Find the expressions for q , pi , and Qij . Show that the quadrupole moment of a spherically symmetric charge distribution is zero, assuming that the origin is located at the center of the distribution. Is the quadrupole moment still zero if the origin is located elsewhere? 2 f) We showed in part (b) that the electric potential due to a quadrupole is xˆ xˆ x i j 3 Qij . 2xk xk 2 This is obtained when a charge distribution has a net charge of zero, and a dipole moment of zero. An object consisting of two positive charges +e flanking a negative charge -2e, each at a distance L has this property. Using the relationship E x x , show that the electric field is given by: Ek Qkm xˆm 5 xˆi xˆ j Qij 4 xk . 2 x5 x 3. Triple Cross Product Identity Derive the vector identity a a 2a using index notation. Hint: use the fact that ijk lmk il jm im jl . 4. Magnetic Dipole The magnetic vector potential of a magnetic dipole at x 0 is given by x A x m 3 , x where m is the magnetic moment. Show that the magnetic field is given by 3nm n m Bx 3 x where n is a unit vector in the same direction as x . 5. Electric Multipole Expansion A charge density x is confined to a small region near x 0 . It is acted on by an electric field due to distant charges which therefore varies slowly in space over the region of . Describe this external electric field by a potential x with a Taylor series expansion near x 0 : 3 x 0 i xi 12 ij xi x j where ij ji . a) ii 0 . Why? b) Show that the total force F acting on can be expanded as a series whose first two terms are F qE p E x x 0 where q and p are the charge and dipole moment of . c) Show that the first term in the expansion of the torque on is N p E 0 . 4