Question

A charge $\sigma =\sigma_{0}cos\phi$ is glued on the cylindrical surface of a long circular cylinder of radius R. The cylinder is made of a linear dielectric material of dielectric constant $\epsilon_{r}$ .

Find the electric field inside the cylinder and show that this field is uniform.

If a small metal sphere of radius a (a<< R) gets into the center of the cylinder, find the total dipole moment of the setup by all charges: free charge, bound charge, and induced charge, given the length of the cylinder L.

Answer 1

solution » A charge o oglase is glued on the cylindrical Surface of a long circular cylinder of radios 'R! → from the symmetry, the electric field is Everywhere radial in the plane cutting normally and its magnitude only depends on the radial distance 'R! from the the area 's'. Gauss law; electric flux on the cylinder Surface Pe de Eds = a Here: E - Electric field. R Radius of cylinder of Electrix fluz. E) Relative permittivity fr> Dielectric constant. to permittivity of space. q) charge enclosed on the cylinder (-) length of the cylinder

pege. ds 4-68. ô ds. Total surface can be divided into cuts at both ends A and B, and gadial Surface 'C' Pe spate ods + g Eids pkê de f Eds cos 96 4 ff Eds corpo tf (ds Coxo = 0+0+ g tds caso E E ATRL 02 Serie A TERL1 – Now: charge q enclosed in the cylinder q=oL from the Gauss' law from Equation 0 a QERA - e 13

we know that to=of Loso / for everyone I to = tr Substituting values of o and to into Eq @ QTERL oL QTERL = oolosol E = for losol QTIRE. The value E is the electric field for Cylinders