imagine two identical dielectric cylinders imagine its axis

imagine two identical dielectric cylinders (imagine it\'s axis is placed on the k hat axis). these two dielectric cylinders have a radius of R and a length of L, with a specific dieelectric constant of K =2 sits in a vacuum.

one cylinder \"a)\" (look at pic above) has a uniform electric field inside the cylinder with a field strength E = (Eo)( i hat).

the other one \"b)\" has a uniform field E= (Eo)(i hat) incident on it from the vaccum (in this situation the E field is outside the cylinder going in the positive i hat direction).

what would the surface bound charge density at theta = 45 degrees be for both of these cylinders?

can refer to pic above for a) and b)

Solution

cylinder 1: using gauss law,

E₀RLcos45=Q/2e₀

=>q= 1.414e₀E₀R

cylinder 2:

Field inside=(2/(2+1))E₀=(2/3)E0

Field due to surface charge= -E₀(1-2/3)=-E₀/3

applying gauss law, -E₀RLcos45/3=Q/2e₀

=>q= -0.471e₀E₀R