Energy Stored in a Magnetic Field — Question 10

Question 10

A reversed source can recover part of the energy initially stored in an RL circuit

A series RL circuit initially carries $I_0=V/R$ in steady state. The source is reversed to -V at t=0. Until current reaches zero, $i=I_0(2e^{-t/\tau}-1)$ with $\tau=L/R$.

1. Find the energy returned to the reversed source before current first reaches zero.

2. Find the heat dissipated in that interval and verify the energy balance.

Question 10 - Solution

A reversed source can recover part of the energy initially stored in an RL circuit

(a) Method and calculation

$$i(t_0)=0\Rightarrow t_0=\tau\ln2,\quad W_{\mathrm{returned}}=\int_0^{t_0}Vi(t)dt.$$$$\boxed{W_{\mathrm{returned}}=LI_0^2(1-\ln2)}.$$

(b) Interpretation and further calculation

$$U_0=\tfrac12LI_0^2,\quad U(t_0)=0.$$$$\boxed{H=U_0-W_{\mathrm{returned}}=LI_0^2(\ln2-1/2)>0}.$$

Physical check

The reversed source absorbs energy while the current retains its original direction; both recovered energy and heat come from the initial magnetic energy.