A coil of wire having finite inductance and resistance has a conducting ring placed coaxially within it. The coil is connected to a battery at time t $$=$$ $$0$$, so that a $$time-dependent$$ current $$I1(t)$$ starts flowing through the coil. If $$I2(t)$$ is the current induced in the ring and $$B(t)$$ is the magnetic field at the axis of the coil due to $$I1(t)$$ then as a function of time (t$$ $$0), the product $$I2$$ (t) $$B(t)$$

Question

A coil of wire having finite inductance and resistance has a conducting ring placed coaxially within it. The coil is connected to a battery at time t $$=$$ $$0$$, so that a $$time-dependent$$ current $$I1(t)$$ starts flowing through the coil. If $$I2(t)$$ is the current induced in the ring and $$B(t)$$ is the magnetic field at the axis of the coil due to $$I1(t)$$ then as a function of time (t$$ > $$0), the product $$I2$$ (t) $$B(t)$$

Infinity Learn · Accepted Answer

Using $$k1$$, $$k2$$ etc. as different constants, we $$haveI1(t)=k11$$−e−$$t/$$τ,$$B(t)=k2I1(t)I2(t)=k3dB(t)dt=k4e$$−$$t/$$τ$$I2(t)B(t)=kb1$$−e−$$t/$$τe−$$t/$$τThis quantity is zero for t $$=$$ $$0$$ and $$t=$$∝ and positive for other value of t. It must, therefore, pass through a maximum.

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