An inductor coil of resistance `10 Omega` and inductance 120 mH is connected across a battery of emf 6 V and internal resistance `2 Omega`. Find the charge which flows through the inductor in (a) 10 ms, (b) 20ms and (c ) 100ms after the connections are made.

Explanation

Here `L=120mH=0.120H`
`R=10W, Emf=6, r=2`
`i=i_0(1-e^(1-e^(-t)//(tau))`
Now `dQ=idt=i_0(1-e^(-t//tau))dt`
`Q=int dQ`
`=int_0^(t) i_(0)=int_(0)^(t) i_(0)(1-e^(-t//tau))dt`
`i_0[int_(0)^(t) (e^(-t//tau))dt]`
`i_0[t-(-(tau))int_(0)^(1) (e^(-t//tau))(dt)]`
`i_0[t+t{e^(-t//tau)}-t]`
`=i_0(1-`tau(e))[(1-(tau)e)int_(0)^(1)(e^(-t//tau))(dt)]`
Now, `i_(0)=(6)/(10+2)=(6/12)=0.5A.`
`(tau)=(L/R)=(120/12)=0.01`
(a) `t=0.01s`
so, `Q=0.5[0.01xx+0.01e^(-0.1//0.01)-0.01]`
=0.00108=1.8xx10^(-3)=1.8mQ`
(b) `t=20ms=2xx10^(-2)s=0.02s`
`Q=0.5[0.02+0.01xxe^(-0.02//0.01)`
`-(0.01)]`
`=0.005676=5.6xx10^(-3)C`
`=5.6mC`
(c ) `t=100ms=0.1s`
`Q=0.5[0.1+0.1xxe-^(0.1//0.01)-0.01]`
`=0.045C=45mc.`