Question

A steel pipe is constructed of a material for which μ_r = 200 and σ = 5 × 10⁶ mho/m. The outer and inner radii are 8 and 6 mm respectively and the length is 80 m. If the total current carried by the pipe is 2cos10⁴ πt A, Then the skin depth will be
1. 0.225 × 10^-3 m
2. 0.300 × 10^-3 m
3. 0.352 × 10^-3 m
4. 0.125 × 10^-3 m 

Answer 1

Correct Answer - Option 1 : 0.225 × 10^-3 m

Concept:

σ = 5 × 10 mho/m

Frequency f = 5000 Hz

Permittivity,

\(\varepsilon = {\varepsilon _0}{\varepsilon _r} = \frac{1}{{36\pi }} \times {10^{ - 9}}\;F/m\) assume {ε_r = 1}

\(\frac{\sigma }{{\omega \varepsilon }} = 1.8 \times {10^{13}} > \; > \; > 1\)

Hence, given steel pipe is a Good conductor.

Formula: To obtain skin depth in a Good conductor:

\(\alpha = \beta = \frac{1}{\delta } = \sqrt {\pi f\mu \sigma } \)     

 \(\left\{ {\begin{array}{*{20}{c}} {\alpha \to attenuation\;constant\;\left( {\frac{{Neper}}{m}} \right)}\\ {\beta \to Phase~{\rm{constant}}\left( {\frac{{rad}}{m}} \right)}\\ {\delta \to skin\;depth\;\left( m \right)} \end{array}} \right.\) 

Calculation:

∴ \(\delta = \frac{1}{{\sqrt {\pi f\mu \sigma } }} = 0.225 \times {10^{ - 3}}m\)