In a dielectric (σ = 10−4 S/m, μr = 1, εr = 4.5), the conduction current density is given as Jc = 0.4 cos(2π x 108 t) A/m2. Determine the displacement current density.

According to the data given σ = 10−4 S/m μr = 1 εr = 4.5 Also, Jc = 0.4 cos(2π x 108 t) A/m2 Now,…

Answered: In a dielectric (σ = 10−4 S/m, μr = 1,…

Introductory Circuit Analysis (13th Edition)

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ISBN:9780133923605

Author:Robert L. Boylestad

Publisher:Robert L. Boylestad

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Question

In a dielectric (σ = 10⁻⁴ S/m, μ_r = 1, ε_r = 4.5), the conduction current density is given as J_c = 0.4 cos(2π x 10⁸ t) A/m². Determine the displacement current density.

Expert Solution

Step 1

According to the data given

σ = 10⁻⁴ S/m

μ_r = 1

ε_r = 4.5

Also,

J_c = 0.4 cos(2π x 10⁸ t) A/m²

^{Now, the conduction current density is}

^{$J_{c} = 0.4 \cos (2 π 10^{8} t) A / m^{2} σ E = 0.4 \cos (2 π 10^{8} t) (J_{c} = σ E) (10^{- 4}) E = 0.4 \cos (2 π 10^{8} t) E = (0.4 \times 10^{- 4}) \cos (2 π 10^{8} t) V / m$}

Step 2

The displacement current density will be

$J_{D} = \frac{\partial D}{\partial t} = \frac{\partial (ε_{o} ε_{r} E)}{\partial t} Since, D = ε_{o} ε_{r} E = ε_{o} ε_{r} \frac{\partial (E)}{\partial t} = \frac{10^{- 9}}{36 π} (4.5) \frac{\partial (0.4 \times 10^{4})}{\partial t} \cos (2 π \times 10^{8} t) = \frac{10^{- 9}}{36 π} (4.5) (0.4 \times 10^{4}) \cos (2 π \times 10^{8}) (- \sin (2 π \times 10^{8} t))$

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