Question

Problem 3 A spherical shell of dielectric material with inner radius a and outer radius b has a polarisation P(r) kr which is frozen into the material, and where k is a constant. As usual, r is the distance from the centre. There is no free charge 1) Calculate all the bound charges 2) Calculate the electric field inside the dielectric by first calculating the electric displacement D 3) Cross-check your result by using Gauss's law (i.e. for E without first calculating the electric displacement). [Hint: To find the electric field you should find Qenclosed by finding the total surface and volume charges in the usual way. But you should exclude the surface charge at b since you only care about the electric field inside the dielectric material 15 marks

Answer 1

Anso- Polanzalton Pio) = K ( 7 2 2 + 8) 8 K → constant * No foce charge figure volume charge density. Po=-D.P = -1 2 (22[0 + 1 1/2 2 JK) = a (2 +1) 32 ar -k 22 or = -K (372+0) = -3K and surface charge density So=p. Whise ñ nomal unit vetor. tpos = K (1/2 + 8) bora K (1 32 + b) 31 1 - p.Ñ --K(z++)834= -* ( gazte eb Qrenc=0 we have PD.da = Qtene where De elatie displacement- &..da=0 - D=0 everywhere now, D = EOE + P = 0 E = - 11/26 E=0 for sea E= 1 kl 1/2+2) and E= o for a E= -1/0 1 1 2 + r) & I for a crcb

we have Coauss law & Eda = Qeni E.uar2 Dons = to now for Er war Qema for rca Qeni zo so Eso sb Qene so sol E=0 for acrcb Qence s.dr t 1 Soda - Jisada + lebat = -Klázta) aña 2t -3k añõz dr - A = -47K (1793) - Lak zdr = HRK (1+03) - 128K 3 19 =-40K(1+93) - 48K (73_93) =-unk-urka3 - 45K 83 turka3 = -4AK - 8 unkra Qonx. = - 4KK (1+r3) + E= I - 40k (1+r3) CREO E = -x [ / r2 + 17