Gradient

Definition

A gradient is a vector that represents the rate and direction of change of a scalar field. In physics, it indicates how the electric potential changes with respect to position.

5 Must Know Facts For Your Next Test

1. The gradient of the electric potential gives the electric field: $\vec{E} = -\nabla V$.
2. A gradient points in the direction of the steepest ascent of a function.
3. The magnitude of the gradient represents how rapid the change is at a point.
4. In Cartesian coordinates, the gradient operator is $\nabla = \left( \frac{\partial}{\partial x}, \frac{\partial}{\partial y}, \frac{\partial}{\partial z} \right)$.
5. For an electric potential $V(x,y,z)$, the electric field components are $E_x = -\frac{\partial V}{\partial x}$, $E_y = -\frac{\partial V}{\partial y}$, and $E_z = -\frac{\partial V}{\partial z}$.

Review Questions

• What does the gradient of an electric potential indicate about an electric field?
• How do you mathematically express the gradient in Cartesian coordinates?
• Why does the electric field point in the direction opposite to that of the gradient of electric potential?

Related terms

Electric Field: A vector field representing the force per unit charge exerted on a positive test charge at every point in space.

Potential Gradient: The spatial rate of change of electric potential; essentially another term for 'gradient' when discussing potentials.

$\nabla$ (Nabla): A symbol used to denote vector differential operators such as gradient, divergence, and curl.