The current in a coil of self inductance $$L = 2.0$$ H is increasing according to the law $$i = 2\sin t^2$$. Find the amount of energy spent (in J) during the period when the current changes from $$0$$ to $$2$$ A.

The self-inductance L = 2.0 H and the current is given by i = 2 sin(t²). We need to find the energy spent during the period when the current changes from 0 to 2 A.

The energy stored in an inductor carrying current $$i$$ is given by $$U = \frac{1}{2}Li^2.$$

For $$i = 0$$, the energy stored is $$U_1 = \frac{1}{2} \times 2 \times 0^2 = 0 \text{ J},$$ and for $$i = 2$$ A, $$U_2 = \frac{1}{2} \times 2 \times 2^2 = 4 \text{ J}.$$

The energy spent equals the change in energy stored, $$\Delta U = U_2 - U_1 = 4 - 0 = 4 \text{ J}.$$ Thus the amount of energy spent is 4 J.