An infinitely long line, with uniform positive charge density, lies along the z-axis. In cylindrical coordinates (r, ϕ, z), at any point $\vec{P}$ not on the z-axis, the direction of the electric field is

Understanding the Electric Field of a Line Charge

The problem asks for the direction of the electric field $\vec{E}$ at a point $\vec{P}$ located away from an infinitely long, straight line carrying a uniform positive charge density $\lambda$. The line lies along the z-axis, and we are using cylindrical coordinates $(r,\phi ,z)$.

Applying Symmetry Arguments

Due to the infinite length and uniform charge distribution, the electric field setup possesses significant symmetry:

- Axial Symmetry: The charge distribution is symmetric around the z-axis. This means the electric field pattern looks the same if we rotate the system around the z-axis. Therefore, the electric field cannot depend on the azimuthal angle $\phi$. Also, there can be no component of the electric field in the $\phi$ direction (i.e., $\vec{E}\cdot \hat{\phi }=0$).
- Translational Symmetry: The charge distribution is uniform along the z-axis. This implies that the electric field pattern should be the same at any height $z$. Consequently, there can be no component of the electric field along the z-axis (i.e., $\vec{E}\cdot \hat{z}=0$). Consider any point $\vec{P}$ at a distance $r$ from the z-axis. There is another point ${\vec{P}}^{\prime }$ at the same distance $r$ but at the opposite z-coordinate relative to the midpoint. The z-components of the electric field produced by the charge elements at these points will cancel each other out.

Direction in Cylindrical Coordinates

Based on the symmetry arguments, the electric field $\vec{E}$ can only have a component in the radial direction. In cylindrical coordinates, the radial unit vector $\hat{r}$ points directly away from the z-axis. Since the charge density ($\lambda$) is positive, the electric field must point radially outwards, away from the line charge.

Therefore, at any point $\vec{P}$ not on the z-axis, the direction of the electric field is purely radial and points away from the z-axis.

Conclusion

In cylindrical coordinates, the direction pointing away from the z-axis is represented by the unit vector $\hat{r}$. Thus, the direction of the electric field is $\hat{r}$.