A current through a wire depends on time as $$i = \alpha_0 t + \beta t^2$$, where $$\alpha_0 = 20$$ A s$$^{-1}$$ and $$\beta = 8$$ A s$$^{-2}$$. Find the charge crossed through a section of the wire in 15 s.

We are given the current as a function of time: $$i = \alpha_0 t + \beta t^2$$, where $$\alpha_0 = 20$$ A s$$^{-1}$$ and $$\beta = 8$$ A s$$^{-2}$$.

The charge flowing through a section of wire in time $$t$$ is obtained by integrating the current: $$q = \int_0^{t} i \, dt$$.

Substituting, $$q = \int_0^{15} (\alpha_0 t + \beta t^2) \, dt = \left[\frac{\alpha_0 t^2}{2} + \frac{\beta t^3}{3}\right]_0^{15}$$.

Now we compute each term. The first term is $$\frac{\alpha_0 \times 15^2}{2} = \frac{20 \times 225}{2} = \frac{4500}{2} = 2250$$.

The second term is $$\frac{\beta \times 15^3}{3} = \frac{8 \times 3375}{3} = \frac{27000}{3} = 9000$$.

So the total charge is $$q = 2250 + 9000 = 11250$$ C.

Hence, the correct answer is Option B.