Gradient Use

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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!