Two long rectilineal conductors, parallel between them, have the distance between them of r=4 cm.An electrical current passes through both of them.
They are attracting each other with the force of F'=mN/m (on unity of length). Which is the value of mechanical work, on unity of length, for reduplication of distance between them?
The magnetic field from wire one is: B = uI1/2PIr. For electrons moving in wire two, the differential force along dL of the wire felt at distance r is F = I2*dL*B = u*I1*I2*dL/2PIr
Work is Int(F dr), so W = Int(u*I1*I2*dL/2PIr dr) = (u*I1*I2*dL/2PI)*Int(1/r dr) = K*Int(1/r dr)
Recalling that Int(1/r dr) = ln(r), the work done moving the wires from 4 cm separation to 8 cm separation is K*ln(8cm/4cm).
So W/dL = (u*I1*I2/2PI)*ln(2)