Struggling with Electromagnetics Problem in Cheng's Book?

[ECE302]

Hello. I am an electrical engineer who is trying to improve my skills in several areas, one of which is electromagnetics.

I am using the book by David Cheng "Fundamentals in Engineering Electromagnetics" and doing some problems in that book.

I am having trouble with a problem in chapter 6.8 anyway here is the problem...

6.8) Calculations concerning the electromagnetic effect of currents in a good conductor usually neglect the displacement current even at microwave frequencies.

a) Assuming Er=1 and tau=5.70 x 10^7 (S/m) for copper, compare the magnitude of the displacement current density with that of the conduction current density at 100(GHZ)
b) Write the governing differential equation for magnetic field intensity H in a source-free good conductor.

can someone help me with a worked solution, i am just stuck and don't know exactly what to do. thx!

 

[Nate810]

a) The displacement current density is given by jd = (1/tau)*E. The conduction current density is given by jc = Er*E. At 100GHz, the magnitude of the displacement current density is jd = 5.7 x 10^7 A/m^2 and the magnitude of the conduction current density is jc = 1 x 10^9 A/m^2. Thus, the conduction current density is much greater than the displacement current density at this frequency. b) The governing differential equation for magnetic field intensity H in a source-free good conductor is given by curl(H) = -(1/tau)*E.

 

[thepatientmental]

Hello there,

Electromagnetics can definitely be a challenging subject, but it's great that you're working on improving your skills. Let me try to help you with this problem.

For part a), we can use the equation for displacement current density, which is given by:

Jd = epsilon * (dE/dt)

Where Jd is the displacement current density, epsilon is the permittivity of the medium (in this case, air with Er=1), and dE/dt is the time derivative of the electric field. We can also use Ohm's law to calculate the conduction current density, which is given by:

Jc = sigma * E

Where Jc is the conduction current density, sigma is the conductivity of the material (in this case, copper with tau=5.70 x 10^7 S/m), and E is the electric field.

At 100 GHz, the time derivative of the electric field can be approximated as the frequency multiplied by the electric field, so we can rewrite the displacement current density as:

Jd = epsilon * (2*pi*f*E)

Plugging in the values for epsilon, f, and E, we get:

Jd = (8.85 x 10^-12) * (2*pi*100 x 10^9) * 1 = 5.55 x 10^-2 A/m^2

For the conduction current density, we can use the given values for sigma and E to get:

Jc = (5.70 x 10^7) * 1 = 5.70 x 10^7 A/m^2

As we can see, the magnitude of the conduction current density is much larger than that of the displacement current density. This is because at high frequencies, the displacement current is negligible compared to the conduction current.

For part b), the governing differential equation for magnetic field intensity H in a source-free good conductor is given by:

∇^2H = -sigma * (dE/dt)

Where ∇^2 is the Laplacian operator. This equation relates the magnetic field intensity H to the time derivative of the electric field, with the conductivity of the material as a proportionality constant.

I hope this helps you in solving the problem. If you're still stuck, I would suggest seeking help from your professor or a tutor. Best of luck!