91影视

$$\begin{gathered} L=2.0H \\ R=10惟 \\ E=100V \end{gathered}$$

Here we have to use the formula for energy stored by inductor鈥檚 magnetic field to calculate energy. Then calculate thermal power from the current and resistance. Then add both to find the energy delivered by the battery.

Formula:

$$\begin{gathered} U_b=0.5L{i}^2 \\ i=\frac{饾洀}{R}\left(1-e^{-\frac{t}{{蟿}_L}}\right) \\ {P}_{thermal}=i^{2}R \end{gathered}$$

The inductor鈥檚 magnetic field stores energy and is given by

$U_b=0.5L{i}^2$

The rate of change in energy stored in the inductor is

$$\begin{gathered} \frac{d{U}_b}{dt}=\frac{d\left(0.5L{i}^2\right)}{dt} \\ \frac{d{U}_b}{dt}=0.5L\frac{d\left(i^2\right)}{dt} \\ \frac{d{U}_b}{dt}=Li脳\frac{di}{dt} \end{gathered}$$

Here, current is given as

$i=\frac{E}{R}\left(1-e^{-\frac{t}{{蟿}_L}}\right)$

Plug these values in the above equation:

role="math" localid="1661856003053" $$\begin{gathered} \frac{d{U}_b}{dt}=L\frac{E}{R}\left(1-e^{-\frac{t}{{蟿}_L}}\right)脳\frac{d\left({\displaystyle \frac{饾洀}{R}}\left(1-e^{-\frac{t}{{蟿}_L}}\right)\right)}{dt} \\ \frac{d{U}_b}{dt}=L\frac{E}{R}\left(1-e^{-\frac{t}{{蟿}_L}}\right)脳\frac{E}{R}\left(\frac{e^{-\frac{t}{{蟿}_L}}}{{蟿}_l}\right) \\ \frac{d{U}_b}{dt}=\frac{E^2}{R}\left(1-e^{-\frac{t}{{蟿}_L}}\right)脳\frac{E}{R}\left(\frac{e^{-\frac{t}{{蟿}_L}}}{{蟿}_l}\right) \end{gathered}$$

Here, ${蟿}_L=\frac{L}{R}=\frac{2.0}{10}=2.0s$

$$\begin{gathered} \frac{d{U}_b}{dt}=\frac{{100}^2}{10}脳\left(1-e^{-\frac{-0.10}{0.20}}\right)脳\left(e^{-\frac{0.10}{0.20}}\right) \\ \frac{d{U}_b}{dt}=2.4脳{10}^2\mathrm{W} \end{gathered}$$

$$\begin{gathered} P_{thermal}=i^{2}R \\ {P}_{thermal}={\left(\frac{E}{R}\left(1-e^{-\frac{t}{{蟿}_L}}\right)\right)}^{2}R \\ {P}_{thermal}=\frac{E}{R}{\left(\left(1-e^{-\frac{t}{{蟿}_L}}\right)\right)}^2 \\ {P}_{thermal}=\frac{{100}^2}{10}{\left(\left(1-e^{-\frac{0.10}{0.20}}\right)\right)}^2 \\ {P}_{thermal}=1.5脳{10}^2\mathrm{W} \end{gathered}$$

Energy being delivered by battery is as follows:

$$\begin{gathered} P=P_{thermal}+\frac{d{U}_b}{dt} \\ P=2.4脳{10}^2+1.5脳{10}^2 \\ P=3.9脳{10}^2\mathrm{W} \end{gathered}$$