Electric Field outside a Charged Cylinder
Electric field due to a charged cylinder can be calculated using Gauss's law. The charge density of the cylinder is given as \( \rho = ks \), where \( k \) is a constant and \( s \) is the surface charge density.

Gauss's Law:
According to Gauss's law, the electric field \( E \) at a distance \( r \) from the axis of the cylinder can be calculated by considering a Gaussian surface in the form of a cylindrical shell of radius \( r \) and length \( L \).

Electric Field Calculation:
- The electric field \( E \) at a distance \( r \) from the axis of the charged cylinder is given by:
\[ E = \frac{\rho r}{2\epsilon_0} \]
- Using cylindrical coordinates, the electric field can be expressed as:
\[ \vec{E} = \frac{\rho}{2\epsilon_0} \hat{r} \]
- where \( \hat{r} \) is the unit vector in the radial direction.

Explanation:
The electric field outside the charged cylinder is radial and its magnitude is directly proportional to the distance from the axis of the cylinder. This means that the electric field is stronger closer to the cylinder and decreases as the distance from the cylinder increases.
In conclusion, the electric field outside a charged cylinder with charge density \( \rho = ks \) can be calculated using Gauss's law and expressed in cylindrical coordinates as \( \vec{E} = \frac{\rho}{2\epsilon_0} \hat{r} \).