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(C) $1$. The motional electromotive force (emf) induced in the sliding connector is given by $e = Bvl$. Substituting the given values: $e = 2 	imes 2

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$2$

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(C) $1$. The motional electromotive force (emf) induced in the sliding connector is given by $e = Bvl$. Substituting the given values: $e = 2 	imes 2 	imes 1 = 4 \, V$.$2$. The connector acts as a battery of emf $E = 4 \, V$ and internal resistance $r = 2 \, \Omega$.$3$. The two resistors of $6 \, \Omega$ and $3 \, \Omega$ are connected in parallel. Their equivalent resistance $R_{eq}$ is given by $\frac{1}{R_{eq}} = \frac{1}{6} + \frac{1}{3} = \frac{1+2}{6} = \frac{3}{6} = \frac{1}{2}$,so $R_{eq} = 2 \, \Omega$.$4$. The total resistance of the circuit is $R_{total} = R_{eq} + r = 2 \, \Omega + 2 \, \Omega = 4 \, \Omega$.$5$. The current $i$ flowing through the connector is $i = \frac{e}{R_{total}} = \frac{4 \, V}{4 \, \Omega} = 1 \, A$.$6$. The magnetic force acting on the connector is $F_m = Bil$. Substituting the values: $F_m = 2 \, T 	imes 1 \, A 	imes 1 \, m = 2 \, N$.$7$. To maintain a constant velocity,the external force $F_{ext}$ must be equal and opposite to the magnetic force $F_m$. Therefore,$F_{ext} = F_m = 2 \, N$.

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