An infineltly long cylindrical surface density `sigma = sigma_(0) cos varphi`. Where `varphi` is the polar angle of the cylindrical coordinate system whose `z` axis coincides with the axis of the given surface. Find the magnitude and direction of the electric fiels strength vector on the `z` axis.

When an uncharged conducting ball of radius R is placed in an external uniform electric field, a surface charge densityh sigma = sigma_(0) cos theta is induced on the ball's surface charge (here sigma_(0) is a constant, theta is a polar angle). Find the magnitude of the resultant electric force acting on an induced charge of the same sign.

Two infinite planes sepated by a distance l carrying a uniform surface charge of densities sigma and -sigma . The planes have round coaxial holes of radius R , with l lt lt R . Taking the origin O and the x cordinates axis as shown in the figure, find the potentail of the electric field and the projection of its strength vector E_(x) on the area of the system as functions of the x cordiante. Draw the appoximate plot varphi (x) .

A thin straight infinitely long conducting wire having charge density lambda enclosed bya cylindrical surface of radius rand length l_(r) , its axis coinciding with the length of the wire. Find the expression for the electric flux through the surface of the cylinder.

A thin straight infinitely long conducting wire having charge density lambda enclosed bya cylindrical surface of radius r and length l_(r) , its axis coinciding with the length of the wire. Find the expression for the electric flux through the surface of the cylinder.

A thin straight infinitely long conducting wire having charge density lambda is enclosed by a cylindrical surface of radius r and lengh l, its axis coinciding with the length of the wire. Find the expression for the electric flux through the surface of the cylinder.

A long thin flat sheet has a uniform surface charge density sigma . The magnitude of the electric field at a distance .r. from it is given by

The direction angles of a vector with x-axis is (pi)/(4) with y -axis is (pi)/(3) .Find the direction angle of the vector with z-axis.