A rectangular loop of wire ABCD is kept close to an infinitely long wire carrying a current `I(t)=I_0 (1-t/T)` for `0 le t le T` and I(0)=0 for t `gt` T as shown in figure. Find the total charge passing through a given point in the loop, in time T. The resistance of the loop is R.

Explanation

Considering strip of thickness dr and length I at distance r from very long current carrying wire.

Magnetic flux linked with this strip,
`dphi=BA=(mu_0I)/(2pir) (L_1dr)`
`dphi=(mu_0IL_1)/(2pir)dr`
Net flux linked with whole loop ABCD,
`phi=int_x^(L_2+x) (mu_0IL_1)/(2pir)dr`
`=(mu_0IL_1)/(2pi)[ln r]_x^(L_2+x)`
`phi=(mu_0IL_1)/(2pi)ln ((L_2+x)/x)`
`I(t)=I_0 (1-t/T)`
`therefore phi=(mu_0L_1)/(2pi)I_0(1-t/T)ln ((L_2+x)/x)`
Now induced charge,
`Q=(Deltaphi)/R`
`Q=(phi_1-phi_2)/R`
For t=0 flux
`phi_1=(mu_0L_1)/(2pi) I_0(1-0)ln ((L_2+x)/x)=(mu_0L_1I)/(2pi) ln ((L_2+x)/x)`
For t=T time flux ,
`phi_2=(mu_0L_1)/(2pi) I_0(0)ln ((L_2+x)/x)=0`
`rArr Q=(Deltaphi)/R =(mu_0L_1I_0)/(2piR)ln ((L_2+x)/x)`