1.6 What is a Potential

\(P\) is a potential:
This sentence implies many consequences for the function \(P\):

• \(\vec{x}\) describes the location (state)

• The value of \(P\) does not depend on the path (on the history of the system)

• The value only depends on the coordinates (the state):

• \(dP\) is a total differential:

  \begin{equation*} dP = \vec{\nabla}P d\vec{x} = \frac{\partial P}{\partial x_1} dx_1 +\frac{\partial P}{\partial x_2} dx_2 + \frac{\partial P}{\partial x_3} dx_3 \quad , \end{equation*}

  (1.12)

  i.e. the change of the coordinates defines completely the change of \(P\).

• The ”force” follows from the gradient

  \begin{equation*} - \vec{\nabla}P \quad . \end{equation*}

  (1.13)

This relations we will illustrate for the example of an electric potential \(W_{e}(\vec{x})\):

• \begin{equation*} \vec{E} = - \vec{\nabla}W_e(\vec{x}) \quad , \end{equation*}

  (1.14)

• \begin{equation*} W_e(\vec{x}_2) - W_e(\vec{x}_1) = -\int_1^2 \vec{E} d\vec{r} \quad , \end{equation*}

  (1.15)

• \begin{equation*} \begin{array}{ccc} dW_e & = &-\vec{E} d\vec{r}\\ \mbox{scalar} & & \mbox{vector}\\ \mbox{energy} & & \mbox{force}\\ \end{array} \end{equation*}

  (1.16)

• Consequences for measurements:

  • Measure forces \(E_i\) in all directions \(x_i\) at each position \(\vec{x}\)

  • Calculate \(dW_i = - E_i dx_i\) for the vector components along the path

  • The overall work is \(dW = \sum_i dW_i\). This does not depend on the sequence of the measurements of the electric fields (forces).

  • This procedure needs a lot of single measurements for the three directions at each position along the way

• Direct measurement of the potential difference

  • just one measurement

• Both procedures are equivalent and you may switch between them, depending on which quantity is easier to measure.

• Just knowing that a function is a potential makes ”life” much easier, even if you want to measure forces:

  • ”Search for the easiest path from A to B on which you can sum up the forces to calculate the potential difference”.

© J. Carstensen (Stat. Meth.)