We see in the Laplace equation (del^2)V=0 only if volume charge density is zero. But electric field intensity(E) is not equal to zero. What can be the physical situations to satisfy this equation.?

We see in the Laplace equation (del^2)V=0 only if volume charge density is zero. But electric field intensity(E) is not equal to zero. What can be the physical situations to satisfy this equation.?

We see in the Laplace equation (del^2)V=0 only if volume charge density is zero. But electric field intensity(E) is not equal to zero. What can be the physical situations to satisfy this equation.?

It's really rather simple. Consider a sphere having a uniform charge density having a total charge of q Then inside the sphere the electric field is non-zero (find what the field is. It's a nice exercise to work that out) but outside the sphere the charge density is zero. However the electric field outside the sphere is zero. In fact the field outside the sphere is exactly the same thing that you'd find if the charge q as a point charge.