question archive A rod of pure silicon (resistivity=2300 Ω*m ) is carrying a current
A rod of pure silicon (resistivity=2300 Ω*m ) is carrying a current
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A rod of pure silicon (resistivity=2300 Ω*m ) is carrying a current. The electric field varies sinusoidally with time according to E=E0 sinωt where E0=0.410 V/m, ω=2πf, f=120 Hz.
Part A:
Find the magnitude of the maximum conduction current density in the wire.
jc= (A/m2)
Part B:
Assuming , find the maximum displacement current density in the wire, and compare with the result of part A.
jD= (A/m2)
Part C:
At what frequency would the maximum conduction and displacement densities become equal if
(which is not actually the case)?
f= (Hz)
Part D:
At the frequency determined in part C, what is the relative phase of the conduction and displacement currents?
Δφ= (degrees)
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Answer:
A. Current density is given as the magnitude of the E-field over the resistivity,
∴ J = E0/ρ = 0.41/2300 = 1.78 x 10^-4 A/m^2
B. For an isotropic dielectric case the displacement current density is,
JD = ε*d/dt(E) = εd/dt(E0*sin(ωt)) = εωE0*cos(ωt)
∴ the magnitude of JD = |JD| = εωE0 = 8.85e-12*2π*120*0.41 = 2.74 x 10^-9 A/m^2
JD/J = 2.74e-9/1.78e-4 = 1.54 x 10^-5
C. We have to find f such that,
J = ε(2πf)E0
∴ f = J/(2πεE0) = 1.78e-4/(2*pi*8.85e-12*0.41) = 7.81 x 10^6 Hz = 7.81 MHz
D. J = E0/ρ*sin(ωt) and JD = εωE0*cos(ωt) = εωE0*sin(π/2-ωt)
∴ The phase different is,
Δφ = π/2 radians = 90 degrees
