Question

The current through a wire depends on time as i = i₀ + αt,
where i₀ = 10 A and α = 4 As^–1. Find the charge that crosses through a section of the wire in 10 seconds.

Answer 1

Current through the wire varies with time as
i = i₀ + α t
Time interval, t = 10 s
i₀ = 10 A
α = 4 A/sec
We know that for variable current,

$$\begin{gathered} q={\int }_0^{t}i\mathrm{d}t \\ \Rightarrow q={\int }_0^t\left(i_0+\alpha t\right)\mathrm{d}t \\ \Rightarrow q=i_{0}t+\frac{\alpha t^2}{2} \\ \therefore q=10\times 10+4\times \frac{{\left(10\right)}^2}{2} \\ =100+200 \\ =300\mathrm{C} \end{gathered}$$