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A long straight wire carries a current, I = 2 ampere. A semi-circular conducting rod is placed beside it on two consuding parallel rails of negligible resistance. Both the rails are parallel to the wire. The wire, the rod and the rails lie in the same horizontal plane, as shown in the figure. Two ends of the semi-circular rod are at distances 1 cm and
4 cm from the wire. At time t = 0, the rod starts moving on the rails with a speed v = 3.0 m/s (see the figure).
A resistor $R = 1.4 Ω$ and a capacitor $C_{0} = 5 .0 μF$ are connected in series between the rails.
At time t = 0, C₀ is uncharged. Which of the following statement(s) is(are) correct ?
$[μ_{0} = 4 π \times 10^{- 7} SI units. Take \ln 2 = 0 .7]$

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Maximum charge on capacitor C₀ is $2.4 \times 10^{- 12}$ coulomb

Maximum charge on capacitor C₀ is $8.4 \times 10^{- 12}$ coulomb

Maximum current through R is $1.2 \times 10^{- 6}$ ampere

Maximum current through R is $3.8 \times 10^{- 6}$ ampere

answer is A, C.

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Detailed Solution

$\to$ Here V is instantaneous velocity

$Emf AB = \int (\bar{V} \times \bar{B}) \cdot dx = V \int \frac{μ_{0} i}{2 πx} dx$

$ε = \frac{μ_{0} iV}{2 π} \ln (4)$

$ε - i_{1} R = \frac{q}{C} \Rightarrow i_{1} = \frac{dq}{dt}$ ……..(i)

$ε = \frac{μ_{0} i}{2 π} (ℓn 4) (V)$

$For maximum charge \frac{dq}{dt} = i_{1} = 0$

$From (1) ε = \frac{q}{C}$

$\begin{matrix} I_{max} = \frac{μ_{0} iV}{2 π} \ln (4) \times C_{0} \\ = 8 .4 \times 10^{- 12} C \end{matrix}$

For maximum current q = 0

$\Rightarrow i_{1} = \frac{ε}{R}; \frac{μ_{0} iV}{2 π} \frac{\ln (4)}{R} = 1 .2 \times 10^{- 6} Amp$

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