Derive the expression for the skin depth (delta) of a good conductor and calculate (delta) for silver (conductivity (sigma = 6.30 $imes10^7$,\text{S m}^{-1}), relative permeability (${\mu }_r$ = 1)) at a frequency of (100,\text{kHz}). Express (delta) in metres and micrometres.

Step-by-step solution

1. Step 1: Recall that for a good conductor the complex wave number is (k=sqrt{iomegamusigma}).
2. Step 2: The skin depth is defined as the distance over which the field amplitude falls by a factor (1/e), i.e. (delta = 1/operatorname{Re}(k)).
3. Step 3: For (omegallsigma/varepsilon) the real part of (k) simplifies to (k approx (1+i)sqrt{\frac{omegamusigma}{2}}).
4. Step 4: Hence (delta = sqrt{\frac{2}{omegamusigma}}).
5. Step 5: Substitute the values: (omega = 2pi f = 2pi(1.00 $imes10^5$) = 6.283 $imes10^5$,\text{rad s}^{-1}).
6. Step 6: Compute the magnetic permeability: (mu = ${\mu }_0{\mu }_r$ = 4pi $imes10^{-7}$\times1 = 1.2566 $imes10^{-6}$,\text{H m}^{-1}).
7. Step 7: Plug into the skin depth formula: (delta = sqrt{\frac{2}{(6.283 $imes10^5$)(1.2566 $imes10^{-6}$)(6.30 $imes10^7$)}}).
8. Step 8: Compute the denominator: ((6.283 $imes10^5$)(1.2566 $imes10^{-6}$)=0.789); multiply by (sigma) to get (4.97 $imes10^7$).
9. Step 9: Take the reciprocal and square root: (delta = sqrt{\frac{2}{4.97 $imes10^7$}} = sqrt{4.03 $imes10^{-8}$} = 6.35 $imes10^{-4}$,\text{m}).
10. Step 10: Convert to micrometres: (6.35 $imes10^{-4}$,\text{m}=6.35 $imes10^2$,mu\text{m}=635,mu\text{m}).

Concept tested: Skin depth of a good conductor

Published 14 September 2026 · Last updated 16 September 2026 · How this solution was prepared and checked