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A small conducting loop of radius a and resistance r is pulled with velocity $υ$ perpendicular to a long straight conductor carrying a current $i_{0}$ . If a constant power P is dissipated in the loop, find the variation of velocity of the loop as a function of x. Given that $x > > a$ .

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\frac{x^{2}}{μ_{0} i_{0} a^{2}} \sqrt{\Pr}

\frac{2 x^{2}}{μ_{0} i_{0} a^{2}} \sqrt{\Pr}

\frac{2 x^{2}}{μ_{0} i_{0} a^{2}} \Pr

\frac{μ_{0} i_{0} a^{2}}{2 x^{2}} \sqrt{\Pr}

answer is B.

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Detailed Solution

Content magnetic field in the loop due to long straight current conductor

$B = \frac{μ_{0} i}{2 π x}$

$ϕ = BA = \frac{μ_{0} i_{0}}{2 π x} π a^{2}$

$but e = - \frac{d ϕ}{dt} = \frac{μ_{0} i a^{2}}{2} (\frac{1}{x^{2}}) \frac{d x}{dt} = \frac{μ_{0} i_{0} a^{2} V}{2 x^{2}}$

Power $P = \frac{e^{2}}{r} = \frac{μ_{0} r^{2} a^{4} V^{2}}{4 x^{2} r}$

$V = \frac{2 x^{2}}{μ_{0} i a^{2}} \sqrt{Pr}$

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