Please solve parts a,b,c,d and e Homework Problem 6.17 We build a spherical capa

Question

Please solve parts a,b,c,d and e Homework Problem 6.17 We build a spherical capacitor out of two concentric spherical conductors. The inner conductor and the inner surface of the outer conductor are coated with a thin layer (exaggerated in the figure) of the dielectric K-Resin, a popular commercial dielectric, KKR 80. The space between the two layers of K-Resin is filled with the super dielectric PZLT with dielectric constant KPZLT-1500. Introduce +Q on the inner conductor and -Q on the outer conductor in your calculations Kee The radii are a = 1cm, b = 1.25cm, c 3cm, and d = 3.25cm conducior (a)Symbolically compute the electric field in regions 1, 11, (b)Is the potential difference 1nd between the inner and (c)Compute AVad symbolically in terms of Q. a, b, c, d and III outer conductors positive or negative? Why? KKR, KpzLT, and constants. (d)Compute the capacitance symbolically (e)Compute the capacitance numerically

Explanation / Answer

given,

radius of inner sphere = a

radius of the inner sphere + layer of dielectric K resin = b

dielectrric constant of kresin = k

radius of outer sphere - dielectric kpzlt = c

radius of outrer sphere = d

dielectric constant of kpzlt = K

a) charge on inner conductor = +Q, and that on outer conductor = -Q

consider gaussean surface at radius r form the center

then for region I

from gauss law

E*4*pi*r^2 = Q/k*epsilon

E = Q/4*pi*epsilon*k*r^2 [ where epsilon is permittivity of free space]

similiarly

for region 2

E = Q/4*pi*epsilon*r^2

for region 3

E = Q/4*pi*epsilon*K*r^2

b) Vi - Vo > 0 ( +ve) because inner conductor is +vely charged and hence at a higher potential

c) we know that

dV = -E*dr

now, DV(ad) = integrate(-EI*dr in region I) + integrate(-EII*dr in region II) + integrate(-EIII*dr in region III)

hence DV(ad) = Q(1/a - 1/b)/4*pi*epsilon*k + Q(1/b - 1/c)/4*pi*epsilon + Q(1/c - 1/d)/4*pi*epsilon*K

DV(ad) = Q(1/ka - 1/kb + 1/b - 1/c + 1/Kc - 1/Kd)/4*pi*epsilon

d) capacitance = Q/D(Vad) = Q/Q(1/ka - 1/kb + 1/b - 1/c + 1/Kc - 1/Kd)/4*pi*epsilon = 4*pi*epsilon / (1/ka - 1/kb + 1/b - 1/c + 1/Kc - 1/Kd)

e) k = 80

K = 1500

a = 0.01 m

b = 0.0125 m

c = 0.03 m

d = 0.0325 m

1/4*pi*epsil;on = 8.98*10^9

hence

C = 2.373*10^-12 F

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