How is the Resistance of a Thin Metallic Strip Derived at High Frequency?

[beowulf.geata]

Homework Statement

A thin metallic strip on a circuit board has length L, width a and thickness t, with L>>a>> t. Derive an expression for the resistance between the ends of the strip at frequency f, assuming that the skin depth is small compared with the thickness t.

Homework Equations

The book where I took this problem from explains that for an electromagnetic wave polarized in the x-direction and traveling in the z-direction through a conducting medium where ω << 1/τ_c (1/τ_c being the frequency of collisions between an electron and the lattice) the electric field is the real part of E = E₀e^-z/δe^i(z/δ-ωt)e_x (where δ is the skin depth).

The Attempt at a Solution

What I am puzzled by is that the solution says that the current density is

J(z)= σE(z) = σE₀e^-z/δ,

so my question is: where has the real part of e^i(z/δ-ωt) gone?

I'd be very grateful for any pointers on this.

 

[mark726]

Thank you for your question. I am happy to help you with your problem.

Firstly, let's review the problem statement. We have a thin metallic strip on a circuit board with length L, width a, and thickness t. We are asked to derive an expression for the resistance between the ends of the strip at frequency f, assuming that the skin depth is small compared with the thickness t.

Now, let's review the equations and attempt at a solution provided in the forum post. We are told that for an electromagnetic wave polarized in the x-direction and traveling in the z-direction through a conducting medium where ω << 1/τc, the electric field is given by E = E0e-z/δei(z/δ-ωt)ex, where δ is the skin depth. The forum user is puzzled by the fact that the solution for the current density given in the book is J(z)= σE(z) = σE0e-z/δ, and wonders where the real part of ei(z/δ-ωt) has gone.

The reason for this is that when we are dealing with a time-varying electric field, we are only concerned with the time-averaged current density. This is because the resistance of a material is defined as the ratio of the applied voltage to the current density. Thus, we only need to consider the magnitude of the electric field, which is given by E = E0e-z/δ, and not its phase. Therefore, we do not need to include the complex exponential term in our expression for the current density.

I hope this explanation helps to clarify your confusion. Please let me know if you need further assistance.