1. Magnetic field due to the long straight wire

At a distance $r$ from a long straight wire carrying current $I=2\,\text{A}$,

$B(r)=\frac{\mu_0 I}{2\pi r}$

The semicircular rod lies in the same plane as the wire, so the magnetic field is perpendicular to the plane.

2. Motional emf across the moving rod

The rod moves with speed $v=3\,\text{m/s}$ along rails parallel to the wire. The motional emf across a moving conductor is

$\mathcal E = \int (\vec v\times \vec B)\cdot d\vec l$

Since $\vec v$ is same for all elements and $B$ depends only on distance $r$ from the wire, we can integrate from $r_1=1\,\text{cm}=0.01\,\text{m}$ to $r_2=4\,\text{cm}=0.04\,\text{m}$:

$\mathcal E = v\int_{r_1}^{r_2} B(r)\,dr$

$\mathcal E = v\int_{0.01}^{0.04} \frac{\mu_0 I}{2\pi r}\,dr$

$\mathcal E = \frac{\mu_0 I v}{2\pi}\ln\left(\frac{0.04}{0.01}\right)$

$\mathcal E = \frac{(4\pi\times10^{-7})(2)(3)}{2\pi}\ln 4$

Now,

$\frac{(4\pi\times10^{-7})(2)(3)}{2\pi}=12\times10^{-7}=1.2\times10^{-6}$

and

$\ln 4 = 2\ln 2 = 2(0.7)=1.4$

So,

$\mathcal E = 1.2\times10^{-6}\times1.4 = 1.68\times10^{-6}\,\text{V}$

This emf is constant because the distances of the rod ends from the wire remain $1\,$cm and $4\,$cm while it slides.

3. Circuit behavior

The rails, rod, resistor, and capacitor form a series $RC$ circuit driven by constant emf $\mathcal E$.

At $t=0$, capacitor is uncharged, so current is maximum:

$I_{\max}=\frac{\mathcal E}{R}$

Given $R=1.4\,\Omega$,

$I_{\max}=\frac{1.68\times10^{-6}}{1.4}=1.2\times10^{-6}\,\text{A}$

So Option A is correct and Option B is incorrect.

4. Maximum charge on capacitor

In steady state, current becomes zero and capacitor gets charged to voltage $\mathcal E$.

Thus maximum charge,

$Q_{\max}=C\mathcal E$

Given $C=5.0\,\mu\text{F}=5.0\times10^{-6}\,\text{F}$,

$Q_{\max}=5.0\times10^{-6}\times1.68\times10^{-6}$

$Q_{\max}=8.4\times10^{-12}\,\text{C}$

So Option C is correct and Option D is incorrect.

5. Final evaluation of options

• A: Correct
• B: Incorrect
• C: Correct
• D: Incorrect

Therefore the correct options are A, C.

6. Comparison with stored correct answer

Stored correct answer: A, C

My derived answer: A, C

They match.