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,

\[q =  \int_0^t idt\]

\[ \Rightarrow q =  \int_0^t \left( i_0 + \alpha t \right)  dt\]

\[ \Rightarrow q =  i_0 t + \frac{\alpha t^2}{2}\]

\[ \therefore q = 10 \times 10 + 4 \times \frac{\left( 10 \right)^2}{2}\]

\[ = 100 + 200\]

\[ = 300  C\]