Prove that the electric field is always perpendicular to equipotential surface.
Solution.
(The solution below is based on the manuscript of 2016-2017 PHYS3450 Electromagnetism Homework 2 Solution.)
is the electric field vector; 
is a line element vector on the equipotential surface.
For any two arbitrary points 
and 
on the equipotential surface, we have the same potential there (i.e., 
). From 
. Thus 
, or, 
.