A metal bar AB can slide on two parallel thick metallic rails separated by a distance l . A resistance R and an inductance L are connected to the rails as shown in the figure. A long straight wire, ca

Question

A metal bar AB can slide on two parallel thick metallic rails separated by a distance l . A resistance R and an inductance L are connected to the rails as shown in the figure. A long straight wire, carrying a constant current I 0 is placed in the plane of the rails as shown. The bar AB is held at rest at a distance x 0 from the long wire. At t = 0 , it is made to slide on the rails away from the wire. It is observed that at time t = T , the metal bar AB is at a distance of 2 x 0 from the long wire and the resistance R carries a current i 1 . The net charge that has flown through resistance R from t = 0 to t = T is

TestHub · Accepted Answer

Let v be velocity of AB when AB is at a distance x from the wire,

$e_{AB} = B . ℓ . v = \frac{μ_{0} I_{0}}{2 π x} I . v = \frac{d ϕ}{d t}$

For circuit Loop,

$\frac{d ϕ}{d t} - i R - L \frac{d i}{d t} = 0$

$\frac{d ϕ}{d t} - \frac{d q}{d t} R - L \frac{d i}{d t} = 0$

$\Rightarrow d q = \frac{d ϕ - L d i}{R}$

$Δ q = \frac{Δ ϕ - L Δ I}{R}$

$= \frac{1}{R} [(ϕ_{1} - ϕ_{2}) - L i_{1}]$

$Δ q = \frac{1}{R} [\frac{μ_{0} ℓ I_{0}}{2 π} \ln (2) - L i_{1}]$

Physics - EMI/AC Question with Solution | TestHub

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