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| A rod of mass $m$ and length $\ell$ and resistance $R$ starts from rest and slides on two parallel rails of zero resistance as shown in Figure. A uniform magnetic field fill the area A battery and is perpendicular and out of the plane of the paper. A battery of of voltage $V$ is connected as shown in the figure. | |
| Argue that the induced EMF in the loop is $V = Bv\ell $ when the rod has speed $v$. Write down $F = m\big(\dfrac{dv}{dt}\big)$ and integrate it so show that \begin{equation*} v(t) =\frac{V}{B\ell}\Big(1-\exp\Big(- \frac{B^2\ell^2 t}{mR}\Big)\Big). \end{equation*} |
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| Hint: Find the limiting speed and separate that out from the total $v$. | |
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