In an LR circuit connected to a battery,the r | vedclass

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(A) The energy stored in an inductor is given by $U = \frac{1}{2} LI^2$.The rate at which energy is stored is the power $P = \frac{dU}{dt} = LI \left(

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(A) The energy stored in an inductor is given by $U = \frac{1}{2} LI^2$.The rate at which energy is stored is the power $P = \frac{dU}{dt} = LI \left( \frac{dI}{dt} ight)$.For an $LR$ circuit,the current at time $t$ is $I(t) = I_0 (1 - e^{-Rt/L})$,where $I_0 = \frac{E}{R}$ is the maximum current.The rate of change of current is $\frac{dI}{dt} = \frac{I_0 R}{L} e^{-Rt/L}$.Substituting these into the power equation:$P = L \left[ I_0 (1 - e^{-Rt/L}) ight] \left[ \frac{I_0 R}{L} e^{-Rt/L} ight] = I_0^2 R (e^{-Rt/L} - e^{-2Rt/L})$.At $t = 0$,$I = 0$,so $P = 0$.As $t 	o \infty$,$\frac{dI}{dt} 	o 0$,so $P 	o 0$.Since the rate is zero at both $t = 0$ and $t = \infty$ and positive in between,the curve must start from zero,rise to a maximum,and return to zero.

In an $LR$ circuit connected to a battery,the rate at which energy is stored in the inductor is plotted against time during the growth of current in the circuit. Which of the following best represents the resulting curve?

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