The electric current through a wire varies with time as $$I = I_0 + \beta t$$, where $$I_0 = 20$$ A and $$\beta = 3$$ A s$$^{-1}$$. The amount of electric charge crossed through a section of the wire in $$20$$ s is:

We need to find the total electric charge that crosses a section of wire in 20 seconds, given that the current varies as $$I = I_0 + \beta t$$.

Current is the rate of flow of charge: $$I = \frac{dQ}{dt}$$. Therefore, the total charge is the integral of current over time:

$$Q = \int_0^{t} I \, dt$$

Given $$I = I_0 + \beta t$$ with $$I_0 = 20$$ A and $$\beta = 3$$ A/s:

$$Q = \int_0^{20} (I_0 + \beta t) \, dt = \int_0^{20} (20 + 3t) \, dt$$

$$Q = \left[20t + \frac{3t^2}{2}\right]_0^{20}$$

$$= \left(20 \times 20 + \frac{3 \times 20^2}{2}\right) - (0)$$

$$= 400 + \frac{3 \times 400}{2}$$

$$= 400 + 600 = 1000 \text{ C}$$

The correct answer is Option (2): 1000 C.