The electric field is mathematically related to the electric potential through the gradient operator. The formula is E equals negative gradient of V. This means the electric field points in the direction of decreasing potential. In this visualization, the blue surface represents the electric potential due to a point charge, while the red arrows show the direction of the electric field, which points away from the charge.
In Cartesian coordinates, the gradient operator applied to a scalar potential gives a vector with components equal to the partial derivatives in each direction. For the electric field, we take the negative of these components. The gradient of the potential, shown in green, points in the direction of steepest increase of the potential. The electric field, shown in red, points in the opposite direction - the direction of steepest decrease. This is why we have the negative sign in the relationship E equals negative gradient of V.
Electric field lines and equipotential surfaces have a special geometric relationship: they are always perpendicular to each other. This is a direct consequence of the gradient relationship. Since the electric field is the negative gradient of the potential, it points in the direction of steepest decrease of the potential. Equipotential surfaces, by definition, have constant potential values. The direction of maximum change must be perpendicular to surfaces of constant value. Mathematically, this means the dot product of the electric field and the gradient of the potential equals zero, indicating they are perpendicular vectors.
Let's apply the gradient relationship to a specific example: the electric field of a point charge. The electric potential due to a point charge q is given by V equals q divided by 4 pi epsilon-zero times r, where r is the distance from the charge. To find the electric field, we take the negative gradient of this potential. The gradient of 1 over r points radially inward with magnitude 1 over r-squared. When we apply the negative sign, we get the electric field pointing radially outward for a positive charge, with magnitude proportional to 1 over r-squared. This is exactly Coulomb's law for the electric field of a point charge, derived from the gradient relationship.
To summarize what we've learned about the gradient in electric fields: First, the electric field is mathematically defined as the negative gradient of the electric potential, expressed as E equals negative gradient of V. In Cartesian coordinates, this means the electric field components are the negative partial derivatives of the potential with respect to each coordinate. A key geometric insight is that electric field lines are always perpendicular to equipotential surfaces, which follows directly from the gradient relationship. When applied to a point charge, this relationship explains why the electric field follows the inverse square law, decreasing as one over r-squared. This fundamental relationship between electric field and potential gradient is central to electrostatics and electromagnetic theory, providing a powerful mathematical framework for understanding electric phenomena.