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[QUE/EM-09010] --- EM-PROBLEM

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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*}

 Rod connected to a battery and sliding on a wire in magnetic field
Hint: Find the limiting speed and separate that out from the total $v$.

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