Math Bridge: Ethernet Skin Depth and Cable Loss

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Why does 100 MHz use only a thin skin of the copper?

Connect copper frequency to skin depth, resistance scaling, and an illustrative Cat6 attenuation term.

Eddie, the electronics guideEddie guides
The one targetTurn signal frequency into a transparent copper skin-depth and loss estimate.
The chapter caseCopper at 1 to 100 MHz and the chapter’s 19.8 dB per 100 m Cat6 point.
What it buys youThe correct guided-conductor mechanism instead of a wireless multipath story.

A field team faces an unresolved physical question: Why does 100 MHz use only a thin skin of the copper? They must answer it before changing frequency on the real device. Predict the direction first.

See the relationship before changing it

The figure reads from left to right. The blue card is frequency. The middle card applies this page's relationship. The green card is skin depth. Walk the arrows once: set the input, apply the rule, then read the result with its unit.

The retained audit below checks several chapter fixtures. This added model holds every other chapter fixture fixed, so the numeric fixture does not switch without explanation.

Frequency changes skin depth An input card leads through the page relationship to the skin depth result. SET INPUT ONE CONTROL APPLY RULE predict calculate check units READ RESULT
Walk the arrows. This is guided-wave conductor physics. It does not use reflection from buildings or Doppler shift to explain the cable limit.

Derive the baseline in four named moves

  1. 1

    Name the input. The chapter baseline for frequency is 100.

  2. 2

    Name the relationship. δ=√[1.68e-8/(π·100e6·4πe-7)]=6.52 um 65.2/6.52=10.0 times thinner than at 1 MHz A10MHz=19.8√(10/100)=6.26 dB/100 m A100MHz=19.8 dB/100 m

  3. 3

    Substitute the chapter fixture. Set frequency to 100. The page ledger gives skin depth as 6.52 um.

  4. 4

    Read the result. Keep um beside the value. Use it only inside the technical boundary on this page.

Predict, then change frequency

Try Predict the direction of skin depth. Move one control, calculate, then check your prediction.

100
Chapter baseline
Skin depth

Observe This is guided-wave conductor physics. It does not use reflection from buildings or Doppler shift to explain the cable limit. Reset the control to 100 and compare skin depth.

Explain Only frequency moves here. The other chapter fixtures remain fixed.

Check yourself

What should you do before trusting a moved-control result?
Answer: Predict its direction, apply the shown relationship, keep the units, and reset to the worked baseline.
What does this small model leave out?
Answer: Only frequency moves. Field effects named in the page's technical boundary stay fixed.

Honesty note. This teaching model changes only frequency. It replays the chapter fixture; it is not a field prediction.

1. Start with the physical story

Fast-changing current crowds toward a conductor’s surface. Less cross-section carries the current, so effective resistance and cable loss rise with frequency.

Eddie: Follow one quantity at a time; every displayed result comes from the same ledger.

2. Name every algebra move

1

Convert MHz to HzMultiply by one million before using SI units.

2

Find skin depthDivide resistivity by πfμ0, then take the square root.

3

Compare with 1 MHzDivide 65.2 micrometres by the new depth.

4

Scale conductor lossUse the square root of the frequency ratio.

5

Turn dB into a voltage ratioUse 10^(−loss/20).

3. Reproduce the chapter case

δ=√[1.68e−8/(π·100e6·4πe−7)]=6.52 µm
65.2/6.52=10.0 times thinner than at 1 MHz
A10MHz=19.8√(10/100)=6.26 dB/100 m
A100MHz=19.8 dB/100 m

The arithmetic stays visible so that units and assumptions can be checked before the result is used.

4. Try one real input

TryMove frequency from 100 MHz toward 10 MHz. Predict how skin depth and the square-root loss term respond.

Frequency
Skin depth
Depth shrink vs 1 MHz
Resistance term vs 1 MHz
Loss per 100 m
Loss per metre
Voltage ratio after 100 m

ObserveA hundredfold frequency rise makes skin depth ten times smaller and the conductor-resistance term ten times larger.

ExplainThis is guided-wave conductor physics. It does not use reflection from buildings or Doppler shift to explain the cable limit.

Technical boundaries.

This transparent ledger reproduces the named chapter case.

Material
The ledger uses room-temperature bulk copper resistivity and permeability.
Cable
It scales one chapter attenuation point as a conductor-only teaching term.
Standard
It does not replace certified channel, connector, NEXT, return-loss, or PHY tests.

Correct, not complete: this ledger does not certify a Cat6 run or prove a 100 m Ethernet link.

5. Use the result in the design

Use standards-certified insertion-loss curves for design; use the square-root ledger only to explain the direction and scale of the conductor term.

6. Record the evidence state

Record cable category, conductor gauge and material, frequency, length, connectors, insertion loss, return loss, temperature, and test result.

7. Check yourself

Why does 100 times the frequency make δ ten times smaller?
Answer: Because frequency sits under a square root in the denominator: 1/√100=1/10.
Is 19.8√(f/100) the complete Cat6 loss law?
Answer: No. Dielectric loss, geometry, twist, connectors, return loss, and manufacturing limits also matter.
Why is this not a multipath calculation?
Answer: The signal is guided by the cable; the named mechanism here is current crowding in copper.
Honesty boundary.

This transparent ledger reproduces the named chapter case.

Material
The ledger uses room-temperature bulk copper resistivity and permeability.
Cable
It scales one chapter attenuation point as a conductor-only teaching term.
Standard
It does not replace certified channel, connector, NEXT, return-loss, or PHY tests.

Correct, not complete: this ledger does not certify a Cat6 run or prove a 100 m Ethernet link.

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