Question

A long straight wire carries a current, $i=2$ampere. A semi-circular conducting rod is placed beside it on two conducting 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 $1\text{cm}$ and $5\text{cm}$ from the wire. At time $t=0$, the rod starts moving on the rails with a speed $v=3.0\text{m/s}$ (see the figure). A resistor $R=1.4\Omega$ and a capacitor $C_0=5.0\mu F$ are connected in series between the rails. At time $t=0,C_0$ is uncharged. Which of the following statement(s) is(are) correct ?
[${\mu }_0=4\pi \times {10}^{–7}\text{SI units. Take}\ln 2=0.7$]

• A. The maximum current through resistor is $1.2\times {10}^{-6}$ ampere.
• B. Tha maximum current through resistor is $3.8\times {10}^{-6}\text{A}$
• C. The maximum charge on the capacitor is $8.4\times {10}^{-12}C$
• D. The maximum charge on the capacitor is $2.4\times {10}^{-12}C$

Answer 1

Answer: (A), (C)

The maximum charge on the capacitor is $8.4\times {10}^{-12}C$


EMF developed across the semi-circular rod $={\int }_1^4\frac{{\mu }_{0}i}{2\pi r}vdr=\frac{{\mu }_{0}iv}{2\pi }\ln 4=\frac{{\mu }_{0}iv}{\pi }\ln 2$
From the given value we have
$\epsilon =\frac{4\pi \times {10}^{-7}\times 2\times 3\times 0.7}{\pi }=24\times 7\times {10}^{-8}$
$\Rightarrow i_{max}=\frac{E}{R}=\frac{24\times 7\times {10}^{-8}}{1.4}=1.2\times {10}^{-6}\text{A}$
Also, maximum charge on capacitor is given by
$Q_{max}=C_0\epsilon =(5\times {10}^{-6})\times (24\times 7\times {10}^{-8})=8.4\times {10}^{-12}\text{C}$