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A current I flows in a long single -layer solenoid with cross-sectional radius R. The number of turns per unit length of the solenoid equals n. Find the limiting current at which the winding may rupture if the tensile strength of the wire is equal to $F_{lim}$

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I_{\lim} = \sqrt{{4F}_{lim} / μ_{0} nR}

I_{\lim} = \sqrt{{3F}_{lim} / μ_{0} nR}

I_{\lim} = \sqrt{{2F}_{lim} / μ_{0} nR}

I_{\lim} = \sqrt{F_{lim} / μ_{0} nR}

answer is D.

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Detailed Solution

Force on an element $Δ l$ is

$dF = \frac{1}{2} μ_{0} n I^{2} Δ l$

$T Δ θ = dF$

$T Δ θ = \frac{1}{2} μ_{0} n I^{2} d l$

$T Δ θ = \frac{1}{2} μ_{0} n I^{2} R Δ θ$

$I^{2} = \frac{2 T}{μ_{0} nR}$

$I = \sqrt{\frac{2 F_{l i n}}{μ_{0} nR}}$

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