Gradient Use

So where does gradient come into electromagnetism?

The infinitesimal change in electric potential $ dV $ is defined as the work done per unit positive charge moving an infinitesimal displacement $ d\vec{l} $ within an electric field $ \vec{E} $. It is given by the familiar force times displacement relationship, and the minus sign is simply a convention to make the work done positive when moving in a direction anti-parallel to the electric field $$ dV = - \vec{E} \circ d\vec{l} \nonumber $$ $$ dV = -(E_x \hat{i} + E_y \hat{j} + E_z \hat{k}) \circ (dx \hat{i} + dy \hat {j} + dz \hat {k} ) \nonumber $$ $$ dV = -E_x dx -E_y dy -E_z dz $$ $ dV $ can also be written as $$ dV = \frac{\partial V}{\partial x} dx + \frac{\partial V}{\partial y} dy + \frac{\partial V}{\partial z} dz $$ So comparing equations (1) and (2) for $ dV $ we find, $$ E_x = -\frac{\partial V}{\partial x}, E_y = -\frac{\partial V}{\partial y}, E_z = -\frac{\partial V}{\partial z} \nonumber $$ So now we can write $ \vec{E} $ in terms of $ V $ as, $$ \vec{E} = - \Bigg(\frac{\partial V}{\partial x} \hat{i} + \frac{\partial V}{\partial y} \hat{j} + \frac{\partial V}{\partial y} \hat{k} \Bigg) $$ $$ \vec{E} = -\vec{\nabla}V $$ The electric potential, a scalar field, determines the direction and magnitude of the electric field, a vector field!