Magnetic field at distancexxfrom long wire
B=μ0i02πx,⊗B=μ0i02πx,⊗
Force on lengthdxdxof wireABAB
dF=Bidx=μ0i0i2πdxxdF=Bidx=μ0i0i2πdxx
FAB=μ0i0i2π∫a+ladxx=μ0i0i2π|logex|a+laFAB=μ0i0i2π∫aa+ldxx=μ0i0i2π|logex|aa+l
=μ0i0i2π[loge(a+l)−loge(a)]=μ0i0i2π[loge(a+l)-loge(a)]
=μ0i0i2πloge(1+la)=μ0i0i2πloge(1+la), upward
Point of application of force acting onABAB
x'=∫L0xdF∫L0dF=μ0i0i2π∫a+ladxμ0i0i2π1n(1+la)=l1n(1+la)x′=∫0LxdF∫0LdF=μ0i0i2π∫aa+ldxμ0i0i2π1n(1+la)=l1n(1+la)
x'x′is distance from lonf wire.