Continuously inhomogeneous imperfect dielectric. Fig, 3.32 shows a parallel-plat

Question

Continuously inhomogeneous imperfect dielectric. Fig, 3.32 shows a parallel-plate capacitor with circular plates of radius a and a continuously inhomogeneous imperfect dielec- tric. The permittivity and conductivity of the dielectric are the following functions of the z-coordinate: (z) 2(1 + 3z/deo and (2) oo/(l +3z/d),0szs d, where oo is a constant and d is the separation between the plates. The capacitor is connected to a time-constant volt- age V. Find (a) the current distribution in the dielectric, (b) the conductance of the capacitor, (c) the power of Joule's losses in the capacitor, (d) the free charge distribution in the capac- itor, and (e) the bound charge distribution in the capacitor Figure 3.32 Parallel-plate capacitor with a continuously inhomogeneous lossy dielectric; for Problem 3.12. ed

Explanation / Answer

given parallel plate capacitor

plate radius a

continuously inhomogenous imperfect dielectric

permittivity epsilon = 2(1 + 3z/d)e [ where e is permittivity of free space]

d is plate speeration

conductivity, rho = rhoo/(1 + 3z/d)

potential difference across the plates = V

a. consider at z coordinate thickness dz of the dielectric

resistance of this plate dR = rho*dz/pia^2

dR = rhoo*dz/pi*a^2(1 + 3z/d)

integrating from z = 0 to z = d

R = rhoo*d*ln(1 + 3d/d)/3pi*a^2 = rhoo*d*ln(4)/3*pi*a^2

hence current i = V/R = V*3*pi*a^2/rhoo*d*ln(4)

b. conductance of capacitor = 1/R = 3*pi*a^2/rhoo*d*ln(4)

c. Joule's losses power = Vi = 3*pi*a^2*V^2/rhoo*d*ln(4)

d. free charge distribution at time t = i*t = V*3*pi*a^2*t/rhoo*d*ln(4)

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