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Question 7.55: In the discussion of motional emf (Sect. 7.1.3) I assumed th...

In the discussion of motional emf (Sect. 7.1.3)I assumed that the wire loop (Fig. 7.10)has a resistance R;the current generated is then I=υBh/R.But what if the wire is made out of perfectly conducting material, so that R is zero?In that case, the current is limited only by the back emf associated with the self inductance L of the loop (which would ordinarily be negligible in comparison with I R). Show that in this régime the loop (mass m) executes simple harmonic motion, and find its frequency ^{33} . [Answer: \omega=B h / \sqrt{m L}]

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ε =v B h=-L \frac{d I}{d t} ; F=I h B=m \frac{d v}{d t} ; \quad \frac{d^{2} v}{d t^{2}}=\frac{h B}{m} \frac{d I}{d t}=-\frac{h B}{m}\left(\frac{h B}{L}\right) v, \frac{d^{2} v}{d t^{2}}=-\omega^{2} v, \text { with } \omega=\frac{h B}{\sqrt{m L}} .

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