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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. |
Answer» Solution :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)` |
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