I am reading Berkeley Physics Course, Volume 2 (Electricity and Magnetism by Edward M. Purcell). I am in chapter $3$, page $92$, and the book discusses conductors.
The following is from the book:
Because the surface of a conductor [in Fig $3.2$] is necessarily a surface of constant potential, the electric field, which is $-\nabla \varphi$, must be perpendicular to the surface at every point on the surface
I have omitted the picture because it is not relevant.
What is the reasoning?
I understand that the potential, $\varphi$, is a continuous function, and since $E=0$ inside the conductor and since $E=-\nabla\varphi$ I get that $\varphi=0$ inside and on the surface (from continuity) of the conductor.
However, I don't understand the reason the book gives for explaining why the field is perpendicular to every point on the surface.