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ISRO Scientist Electrical 2020 Paper

Option 3 : \(\frac{\mu }{{8\pi }}H/m\)

Building Material & Concrete Technology

15652

20 Questions
20 Marks
25 Mins

Consider a straight wire of circular cross-section.

\(MMF = \oint H.ds = I\)

\(\oint Hdx = {I_x}\)

\( \Rightarrow H = \frac{{{I_x}}}{{2\pi x}}\)

As the current density is uniform,

\({I_x} = \frac{{{x^2}}}{{{r^2}}}{\rm{I}}\)

\(H = \frac{I}{{2\pi {r^2}}}x\)

\( \Rightarrow B = {\mu _0}H = \frac{{{\mu _0}I}}{{2\pi {r^2}}}x\)

\(d\phi = Bdx = \frac{{{\mu _0}I}}{{2\pi {r^2}}}xdx\)

\( \Rightarrow d\lambda = \frac{{{\mu _0}I{x^3}}}{{2\pi {r^4}}}dx\)

\( \Rightarrow \lambda = \mathop \smallint \limits_0^r \frac{{{\mu _0}I{x^3}}}{{2\pi {r^4}}}dx = \frac{{{\mu _0}I}}{{8\pi }}\)

\( \Rightarrow L = \frac{{{\mu _0}}}{{8\pi }}\;H/m\)