A long straight wire is located parallel to an infinite conducting plate. The wire is located parallel to an infinite conducting plate. The wire cross-sectional radius is equal to `a`, the distance between the axis of the wire and the plane equals `b`. FInd the mutual capacitance of this system per unit length of the wire under the condition `a lt lt b`.

Explanation

The field in the region between teh conducting plane and the wire can be obtained by using an oppositely charged wire as an image on the other side.
Then the poltential difference between the wire and the plane,
`Delta varphi = int_(b) vec(E). vec(dr)`
`= int_(a) = [(lambda)/(2pi epsilon_(0) r) + (lambda)/(2pi epsilon_(0) (2b - r))] dr`
`= (lambda)/(2pi epsilon_(0)) In (b)/(a) - (lambda)/(2pi epsilon_(0)) In (b)/(2b - a)`
`= (lambda)/(2pi epsilon_(0)) In (2b - a)/(a)`
`= (lambda)/(2pi epsilon_(0)) In (2b)/(a)`, as `b gt gt a`
Hence, the sought mutal capacitance of the system per unit length of the wire
`= (lambda)/(Delta varphi) = (2pi epsilon_(0))/(In 2b//a)`