Given , `epsilon-L(di)/(dt)=iR`, find the value of `i` at any time t in terms of constant ` epsilon`, L and R. At t=0 , `i=0` .

Explanation

We have , `epsilon-L (di)/(dt)=iR`
or `(epsilon -iR)=L (di)/(dt)`
or `(di)/((epsilon-iR))=(dt)/(L) " " ` ...(i)
Integrating both sides of the equation (i), we get
`int (di)/((epsilon-iR))= int (dt)/(L)`
Here limit of time varies from 0 to t and corresponding limits of i varies from 0 to i.
` :. "" int_(0)^(i)(di)/((epsilon-iR))=int _(0)^(t) (dt)/(L)`
For integration of LHS, substitute ` epsilon-iR=z`
Also `"" (d)/(di)(epsilon-iR)=(dz)/(di)`
or `(0-R)=(dz)/(di)`
`:. "" di=(dz)/((-R))`
and ` int _(0)^(t)(di)/((epsilon-iR)) = int _(0)^(i) (dzl(-R))/(z)`
` =((1)/(-R)) int_(0)^(i) (dz)/(z)`
`=((1)/(-R))|ln z|_(0)^(i)`
`=((1)/(-R))|ln (epsilon-iR)|_(0)^(i)`
`=((1)/(-R)){ln(epsilon-iR)-ln(epsilon-0)}`
`=((1)/(-R)) ln ((epsilon-iR))/(epsilon)" " ` ...(ii)
and RHS ` int_(0)^(t) (dt)/(L)=((1)/L))|t|_(0)^(t)=(1)/(L)(t-0)`
`=(t)/(L)`
From equation (i) and (ii) , we have
`(-(1)/(R)) ln ((epsilon-iR)/(epsilon))=(t)/(L)`
or ` " " ln((epsilon-iR)/(epsilon))=-(R)/(L)t`
or ` ((epsilon-iR)/(epsilon))=e^(-(Rt)/(L))`
or `" " i=(epsilon)/(R)(1-e^(-(tR)/(L)))`