A long charged cylinder of linear charged density `lambda` is surrounded by a hollow co-axial conducting cylinder. What is the electric field in the space between the two cylinders?

Explanation

Inner radius two co-axial conducting cylinder is a and outer radius b. The length of cylinder is L. `gt gt` a and b. `m lambda Cm^(-1)` is the uniform linear charged density.

Let electric field produced at point P in the space between the two cylinders and let the - Gaussian swface having radius r. According to Gauss.s law Hux passing through Gaussian.s surface,
`vecE.vecA=(Q)/(in_(0))`
`:. Exx2pirL = (lambdaL)/(in_(0)) [ because cos 0^(@) =1]`
`:. E= (lambda)/(2pi in_(0)r)` where r is the distance of point from axis of cylinder. Electric field is radial and normal to axis.
Potential difference between two cylinder
`V=-int_(a)^(b)vecE.dvecr=intEdr`
`[because vecE "and" vecr` are opposite in direction]
`=int_(a)^(b)(lambda)/(2piin_(0))dr=(lamda)/(2piin_0)int_(a)^(b)(1)/(r)dr`
`:. V = (lambda)/(2piin_(0))[lnr]_(a)^(b)`
`:. V= (lambda)/(2pi in_(0))` ( lnb-lna)
`:. V = (lambda)/(2pi in_(0)) "in"(b)/(a)`
Total charge on each cylinder `Q= L lambda`
Capacitance of cylindrical capacitor
`C =(Q)/(V)`
`C=(Llambda)/((lambda)/(2piin_(0))"in"(b)/(a))or C = (2piin_(0)L)/("In"(b)/(a))`