(a) The amount of charge passed in time `t` through a cross-section of wire is
`q = alpha t - beta t^2`
where `alpha` and `beta` are constant.
(i) Find current in terms of time `t`.
(ii) Sketch `i` versus `t` graph.
(b) The current through a wire depends on time as
`i = 4 + 3t + 2t^2`
Find the charge crossed through a section of wire in `6 sec`.

Explanation

(a) (i) `q = alpha t - beta t^(2)`
`i = (dq)/(dt) = alpha - 2 beta t`
`t = 0, i = alpha`
`i= 0 rArr t = (alpha)/(2 beta)`
(ii) `i` versus `t` graph `y = alpha - 2 beta x` (straightline with `-ve` slope and positive intercept)
(b) `i = (dq)/(dt) rArr dq = i dt`
`q = int_0^(t) i d t = int_0^(6)(4 + 3t + 2t^2) dt`
=`|4t + (3t^(2))/(2) + 2(t^3)/(3)|_0^6`
=`4(6) + (3)/(2) (6)^2+ (2)/(3) (6)^3`
=`24 + 54 + 144 = 222 C`.
.