Question

The charge flowing in a conductor varies time as,
`q=at - 1/2 bt^(2) +1/6 ct^(3)`
Where a,b,c are positive constants. Then, find (i) the initial current (ii) the time after which the value of current reaches a maximum value (iii) the maximum or minimum value of current.

Answer 1

(i) Current, `i=(dq)/(dt)`
`=d/dt(at - 1/2 bt^(2) +1/6 ct^(3))`
`=a-bt +1/2 ct^(2)`
When t =0, initial current , i=a
(ii) For i to be maximum or minimum,
`(di)/(dt)=0= - b +ct` or `t =b/c`
Putting this value of t in (i), we have
`i=a - b xxb/c + 1/2 c xx b^(2)/c^(2) = a - b^(2)/c + b^(2)/(2c) = a - b^(2)/(2c)`
As this value of i is less than that at t=0, it must be minimum. So minimum value of current `a= - (b^(2))/(2c)`