Question

(3.) (40 pts) An insulating sphere of radius a has an embedded nonuniform charge den- sity: p(P) = (1+) where pe is a positive constant (this charge density is only valid for r <a). Sur- rounding the sphere, and concentric with it, is a conducting shell of inner radius b> a and outer radius c which is charged (not neutral). I do not know what the charge on the conductor is, but I am told that a measurement of the radial electric field right on the outer surface of the conducting shell (at r = c+e where we take the limit e + 0), shows that it is positive and equal to E. a b p(r) (a) What is the total charge contained in the insulating sphere of radius a? (b) Find the electric field everywhere in space: for r <a, a <r<b,b<r< cand r> c. Plot the electric field versus radius. (c) What is the total charge on the conducting shell in terms of the given informa- tion? What is the charge density on the inner and outer surfaces? (a) Find the electric potential everywhere in space for r <a,a<r<b,b<r<c and r > c. Plot the electric potential versus radius. Define the electric potential to be zero at infinity (e) What is the work that I do to assemble charges in this configuration, i.e., the to- tal electric potential energy of this configuration? Use your answer to part (d), and the relationship between the electric potential energy and electric potential for a collection of charges. (1) What is energy stored in the electric field? Use your answer to part (b), and the relationship between energy density and electric field. Compare the electric field energy in this part with the total electric potential energy in part (e). Are they the same or not?

Please help. Thank you.

Answer 1

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constant c) Since outer produs suyau the the charge present there be electric field to &: {r-c Thus induud charge at inner suyau of shell, r= b is shell on conducting Thu soral charge 7. Раз 6 -O- in innur suyau charge density 4 ñ 6² 3 Charge densing on outo suyou - 4 n c3 3 Eo - Q = 408 (² E 428 C² in inne swyou G - Charge density 4 KEC²E - TA 4 t b3 386 FO 7 2 3 a3 6 le

ch 4480 22 E on outer density harge o sulau 4 K c² 3 = 3€ c. Eo