EMI

ByOm Sharma

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 cm and 4 cm from the wire. At time t=0, the rod starts moving on the rails with a speed v=3.0 \mathrm{~m} / \mathrm{s} (see the figure).
A resistor \mathrm{R}=1.4 \Omega and a capacitor \mathrm{C}<em>{0}=5.0 \mu \mathrm{~F} are connected in series between the rails. At time t=0, C</em>{0} is uncharged. Which of the following statement(s) is(are) correct?
[ \mu_{0}=4 \pi \times 10^{-7} SI units. Take \ln 2=0.7 ]


(a) Maximum current through R is 1.2 \times 10^{-6} ampere
(b) Maximum current through R is 3.8 \times 10^{-6} ampere
(c) Maximum charge on capacitor \mathrm{C}<em>{0} is 8.4 \times 10^{-12} coulomb (d) Maximum charge on capacitor \mathrm{C}</em>{0} is 2.4 \times 10^{-12} coulomb

Answer: a c

Solution:
At first we will estimate the emf develop in the semi circular rod

d \varepsilon=B v d l \sin (90-\theta)
d \varepsilon=B v d \cos \theta (Towards left in segment)
\int d \varepsilon=\int_{1}^{4} \frac{\mu_{0}}{2 \pi x} v d x{d 1 / \cos \theta=d x}
\varepsilon=\frac{\mu_{0} I V}{2 \pi}[\ln x]<em>{1}^{4} \varepsilon=\frac{\mu</em>{0} I V}{2 \pi} \ln (4)

\varepsilon=\frac{\mu_{0} I v}{\pi} \ln 2 (This emf is in anticlockwise sense)
In this question further it is assumed that the velocity of rod is constant through out.
Now the given circuit is as like RC charging circuit
\varepsilon=\frac{\mu_{0} I v}{\pi} \ln 2
Given \mathrm{I}=2 \mathrm{~A}, \mathrm{v}=3 \mathrm{~m} / \mathrm{s}, \ln 2=0.7
\varepsilon=\frac{4 \pi \times 10^{-7} \times 2 \times 3 \times 0.7}{\pi}=168 \times 10^{-8} \mathrm{volt}
Maximum current at \mathrm{t}=0(\mathrm{R}=1.4 \Omega)
I_{\text {max }}=\frac{\varepsilon}{R}=\frac{168 \times 10^{-8}}{1.4}=1.2 \times 10^{-6} \mathrm{~A} \quad (A correct)
Maximum charge at steady state ( \mathrm{C}<em>{0}=5.0 \mu \mathrm{~F} ) Q</em>{\text {max }}=C_{0} \varepsilon=168 \times 10^{-8} \times 5 \times 10^{-6}=8.4 \times 10^{-12} \mathrm{C}
Ans AC

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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 cm and 4 cm from the wire. At time t=0, the rod starts moving on the rails with a speed v=3.0 \mathrm{~m} / \mathrm{s} (see the figure).A resistor \mathrm{R}=1.4 \Omega and a capacitor \mathrm{C}_0=5.0 \mu \mathrm{~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} SI units. Take \ln 2=0.7 ](2021)

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