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

Your Name]In summary, the problem asks us to derive an expression for the resistance of a thin metallic strip on a circuit board with length L, width a, and thickness t at frequency f, assuming that the skin depth is small compared to t. The provided equations and attempt at a solution show that the current density is given by J(z)= σE(z) = σE0e-z/δ, where the time-averaged electric field is E = E0e-z/δ. The reason for this is that when dealing with a time-varying electric field, we only need to consider its magnitude for calculating resistance, and not its phase.
  • #1
beowulf.geata
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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/τc (1/τc being the frequency of collisions between an electron and the lattice) the electric field is the real part of E = E0e-z/δei(z/δ-ωt)ex (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 ei(z/δ-ωt) gone?

I'd be very grateful for any pointers on this.
 
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  • #2

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.
 

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

1. What is current density?

Current density is a measure of the amount of electric current flowing through a unit area of a material. It is represented by the symbol J and is typically measured in amperes per square meter (A/m2).

2. How is current density related to electric field?

Current density is directly proportional to the electric field strength in a material. This means that an increase in electric field will result in an increase in current density, and vice versa.

3. What is the significance of skin depth in current density?

Skin depth is the distance from the surface of a conductor at which the current density has decreased to a certain percentage of its maximum value. It is important because it determines the thickness of a material required to carry a certain amount of current without significant loss.

4. How is skin depth affected by frequency?

The skin depth of a material decreases as the frequency of the current passing through it increases. This means that at higher frequencies, the current is more concentrated near the surface of the material, resulting in higher current density.

5. Can current density and skin depth be calculated for all materials?

Yes, current density and skin depth can be calculated for all materials, as long as their electrical properties (e.g. conductivity, permeability) are known. However, the values may vary depending on the material's composition and structure.

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