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3 The axis of an infinitely long circular dielectric tube of inner radius ri and outer radius r. coincides with the z-axis. A33

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(a) The relation between cartesian and cylindrical coordinates are, x=r cos 0 y=r sinº ZEZ = a, ą = a, cos 0-0, sin 0 ay sinºSubstitute P. (2,3x + a, 4y) for P in equation (1). Pps= P. (2_3x +a, 4y).a, (2) Write the relation between unit vectors a,,aFor outer surface, r=r an=-a, Write the formula for polarization surface charge density on the inner surface of the dielectriWrite the formula for polarization volume charge density of the dielectric tube. (5) Ppv=-D.P a a a Substitute P. (2,3x + a,Substitute Por(3+sin’ ø) for Pps. and r dødz for ds in equation (6) Q;, = S5pps, ds 21 1 0 0 2x 1 ESS(Por. (3+ sin? 2°) )(r,dSubstitute - Por; (3+ sin? ö) for Pps: and rdødz for ds in equation (6) lei = S Spyds 21 1 00 21 1 ESS(-Por: (3+ sin? +sin’ø)Substitute -7P, for P pv and rdrdº dz for dv in equation (8) Q, = [pydv 1 21 = S SS(-7P.)(rdrdodz) OOT 1 21 =-7P.S | dødz 2 0Therefore, the total charge Q on the dielectric tube is 0. (d) E and D inside the dielectric is, E Ē A = Ēx(2rH) = Q: +Q Ēx(2

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