An LR circuit with emf `epsilon` is connected at t = 0. (a) find the charge Q which flows through the battery during O to t. (b) Calculate the work done by the battery during this period. (d) find the magnetic field energy stored in the circuit at time t. (e) Verify that the results in the three parts above are consistent with enegy conservation.

Explanation

(a)`(dq)=idt`
`=i_(0)(1-e^(-t//(tau)))dt`
`=i_0(1-e^(-t//(tau)))dt[ :. (tau)=(L/R)]`
Q=int_(0)^(t) (dq)`
`=i_(0)[int_(0)^(t)(dt)-int_(0)^(t)(e^(-(tR)//L)(dt)]`
`i_(0)[t-(L/R)(1-(e^(-(tR)//L))]`
`i_(0)[t-(-(L)/(R))(1-(e^(-(tR)//L))]_(0)^(t)`
`Q=(E/R)[t-((L)/(R))(1-(e^(-(tR)//L))]`
(b) similarly as we know workdone
=Vl=Ei`
`Ei_0[t-((L)/(R))(1-(e^(-(tR)//L)))]`
`=(E^2)/(R)[t-((L)/(R))(1-(e^(-(tR)//L))]`
(c) `H=int_(0)^(t) (i_2)(R).(dt)`
`=(E^2)/(R^2).R.int_(0)^(t)(1-(e^(-(tR)//L))^2.(dt)`
`(E^2)/(R)int_(0)^(t)(1-(e^(-(2tR)//L))-2e^(-(tR)//L))(dt)]`
`(E^2)/(R)(t-(L)/(2R)e^(-2tR//L)+(L)/(R)2.(e^(-(tR)//L)))_(0)^(t)`
`(E^2)/(R)(t-(L)/(2R)e^(-2tR//L)+(L)/(R)2.(e^(-(tR)//L)))`
`-(-(L)/(2R)+(2L)/(R))`
`(E^2)/(R){t-(L)/(2R)(x^(2)-4x+3)}`
`[where x=e^(-tR//L)]`
(d) `E=(1/2)Li^(2)=1/2 (L)((E^(2))/(R^2)(1-(e^(-(tR)//L)))^2`
`((LE)^2)/(2 R^(2)) (1-x)^2`
`[where x=(e^(-(tR)//L))]`
(e) Total energy stored in magnetic field
`(E^2)/(R) t-(E^(2)L)/(R^2) (e^(-(tR)//L))`
`(E^2)/(R){t-(L)/(2R)(1-x)}`
=Energy drawn from battery.
(Hence conservation of energy hold good).