A parallel plate capacitor with circular plates of radius 16mm and gap width 5 m

Question

A parallel plate capacitor with circular plates of radius 16mm and gap width 5 mm has a uniform electric field between the two plates. Starting at time zero, the potential difference between the plates is V=V0e^-t/T where T equals 12ms and V0=100 volts.

A) at a radial distance of .8R from the central axis, what is the magnetic field magnitude as a function of time, for times greater than zero?

B) What is that magnetic field if the time is 3T?

C) At a radial distance of 1.2 R from the central axis, what is the magnetic field magnitude as a function of time, for times greater than zero?

D) What is that magnetic field if the time is 3T?

E) At what radial distance, r, will the magnetic field be maximum?

F) At what two radial distances, r, will the magnetic field be minimum?

Explanation / Answer

Jd = e0 dE/dt

E = V/d = (100 / 5 x 10^-3) e(-t/T)

= 20,000 e^(-t/T)

Jd = Id / A

Id = e0 A dE/dt

(A) B.L = u0 Id

  
B ( 2 pi r ) = u0 e0 (pi r^2) d(20000 e^(-t/T))/dt

B = - (20,000 u0 e0 r / 2 T) e^(-t/T)

r = 0.8R

B = (8000 u0 e0 R / T) e(-t/T)

(B) t = 3T

B= (8000 x 4pi x 10^-7 x 8.854 x 10^-12 x 16 x 10^-3 / 12 x 10^-3) e^-3

B = 5.91 x 10^-15 T

(C)

B ( 2 pi r ) = u0 e0 (pi R^2) d(20000 e^(-t/T))/dt

B = - (20,000 u0 e0 R^2 / 2 r T) e^(-t/T)

r = 1.2R

B = (8333.3 u0 e0 R / T) e(-t/T)

(D) B = (8333.33 x 4pi x 10^-7 x 8.854 x 10^-12 x 16 x 10^-3 / 12 x 10^-3) e^-3

B = 6.15 x 10^-15 T

(E) Bmax at r = R

(D) Bmin at r = 0 and infinity

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