A Square frame with side a and a straigh conductor carrying a constant current `I` and located in the same plane. The inductance and the resistance of the frame are equal to `L` and `R` respectivaly. The frame was truned through `I80^(@)` about the axis `OO'` separated from the current carrying conductor by a distance `b` . Find the electric charge having flown through the frame.

Explanation

Magnetic field due to long wire
`B=(mu_(0)I)/(2pix),ox`
magnetic field is variable, taking a strip of length a and with `dx`
`dA=adx`
`dphi=-BdA-(mu_(0)Ia)/(2pi)(dx)/(x)`
`phi=-(mu_(0)Ia)/(2pi)int_((b-a))^(b)(dx)/(x)=-(mu_(0)Ia)/(2pi)|1nx|_((b-a))^(a)`
`=-mu_(0)(Ia)/(2pi)[1n(b)-(b-a)]=-(mu_(0)Ia)/(2pi)1n((b)/(b-a))`
`phi'=(mu_(0)Ia)/(2pi)int_(b)^(b+a)(dx)/(x)`
`=(mu_(0)Ia)/(2pi)1n((b+a)/(b))`
Change flown
`Deltaq=(phi'-phi)/(R)=(mu_(0)Ia)/(2piR)1n((b+a)/(b-a))`