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Consider a metal can placed co-axially inside a long solenoid. The metal can be modeled as a thin cylindrical shell of radius $r$ . The magnetic field due to the solenoid varies with time $t$ as Kt where $K$ is a positive constant. The resistivity of the can is $ρ$ and its breaking stress is $σ$ . Ignore the self-inductance of the can and any currents induced in the flat surfaces of the can. Assume that the magnetic field produced by the induced currents is negligible. Compute the time at which the can breaks

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$t = \frac{σ ρ}{r^{2} K^{2}}$

$t = \frac{2 σ ρ}{r^{2} K^{2}}$

$t = \frac{σ ρ}{2 r^{2} K^{2}}$

$t = \frac{σ ρ}{3 r^{2} K^{2}}$

answer is B.

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Detailed Solution

$B = K t \Rightarrow \frac{d B}{d t} = K$

Emf is induced in the can (due to induced electric field)

$\begin{array}{l} E m f, e = \frac{d B}{d t} π r^{2} = K π r^{2} \\ R = \frac{ρ \times 2 π r}{A} \\ i = \frac{ε}{R} = \frac{K π r^{2} A}{2 ρ π r} \end{array}$

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From the diagram,

$\begin{array}{l} i \times 2 r B = 2 σ A \\ \frac{K π r^{2} A}{2 ρ π r} (2 r K t) = 2 σ A \\ t = \frac{2 σ ρ}{r^{2} K^{2}} \end{array}$

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