📂Mathematical Physics

Potential, A General Definition of Potential Energy

Definition¹

Scalar Potential

Let’s assume that the vector field $\mathbf{V}$ is a conservative field. In other words, let’s say $\nabla \times \mathbf{V} = \mathbf{0}$. Then, there exists a scalar field $W$ that satisfies $\mathbf{V} = -\nabla W$, and this is called the scalar potential of $\mathbf{V}$.

Vector Potential

Assume the vector field $\mathbf{V}$ satisfies $\nabla \cdot \mathbf{V} = 0$. Then, there exists a vector field $\mathbf{A}$ that satisfies $\mathbf{V} = \nabla \times \mathbf{A}$, and this is called the vector potential of $\mathbf{V}$.

Description

The scalar potential is simply referred to as potential.

If the unit of vector field $\mathbf{V}$ is force, it is specifically called potential energy.

Mechanics

Potential Energy

In one dimension, the potential energy of the force $F$ is defined as $V$ that satisfies:

$$F(x) = -\dfrac{d V(x)}{d x}$$

This is just a simple reduction of the above definition from three to one dimension. Also, since $F$ is a force, $V$ is referred to as potential energy.

Electromagnetism

Electric Potential

For an electric field $\mathbf{E}$, the scalar field $V$ that satisfies the following is called electric potential:

$$\mathbf{E} = - \nabla V$$

The electric force experienced by a test charge with a charge $Q$ is given by Coulomb’s law as $\mathbf{F} = Q \mathbf{E}$. Therefore, since the unit of the electric field is not force, the electric potential is not referred to as potential energy.

Magnetic Potential

For a magnetic field $\mathbf{B}$, the vector field $\mathbf{A}$ that satisfies the following is called magnetic potential:

$$\mathbf{B} = \nabla \times \mathbf{A}$$

Similarly, since the magnetic force is given as $\mathbf{F}_{m} = Q (\mathbf{v} \times \mathbf{B})$ by the Lorentz force law, the unit of $\mathbf{B}$ is not force, therefore magnetic potential is not referred to as potential energy.