Test for README.md

[summer2420]

$$ i=\begin{cases} 2\text{A},0 < t < 1 \text{s} \\ 2t^2\text{A}, t>1\text{s} \end{cases} $$

[summer2420]

$$ \begin{align}v&=\left( 10+5\int_0^t i\,dt \right)=10+5\int_0^t 5\cos 60 \pi t\\ &=10+\frac{5}{12}\sin 60 \pi t\Big|_0^t=10+\frac{5}{12\pi}\sin 60 \pi t\\ &= 10+\frac{5}{12\pi}\sin (60\pi\times0.005)\approx 10.1073 \,(\text{V})\\ i&=5 \cos (60\pi t)=5\cos(60\pi\times0.005)\\ &\approx 2.9389\,(\text{A})\\ p&=vi=10.1073\times 2.9389\approx 29.70\,(\text{W})\end{align} $$

[summer2420]

$1$ C = $6.24\times 10 ^{18}$ electrons $Q=\int_0^2i \, dt=\int_0^1 i\,dt+\int_1^2 i\,dt=2t\big|_0^1+2\frac{t^3}{3}\Big|_1^2=2+\frac{14}{3}\approx 6.667$ (C)

[summer2420]

(b) The power delivered to the element is 29.70W:

$$\begin{align} v & =\left( 10+5\int_0^t i \, dt \right)=10+5\int_0^t 5\cos 60 \pi t \\ & =10+\frac{5}{12}\sin 60 \pi t\Big|_0^t=10+\frac{5}{12\pi}\sin 60 \pi t \\ & = 10+\frac{5}{12\pi}\sin (60\pi\times0.005)\approx 10.1073 \,(\text{V}) \\ i & =5 \cos (60\pi t)=5\cos(60\pi\times0.005)\\ &\approx 2.9389 \, (\text{A}) \\ p & =vi=10.1073\times 2.9389\approx 29.70 \, (\text{W}) \end{align}$$