1.6 What is a Potential

$P$ is a potential:
This sentence implies many consequences for the function $P$:

• $\overrightarrow{x}$ 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:

  $$dP=\overrightarrow{\nabla <!--FUNCTION\; APPLICATION-->}Pd\overrightarrow{x}=\frac{\partial P}{\partial x_1}d{x}_1+\frac{\partial P}{\partial x_2}d{x}_2+\frac{\partial P}{\partial x_3}d{x}_3,$$

  (1.12)

  i.e. the change of the coordinates defines completely the change of $P$.

• The ”force” follows from the gradient

  $$-\overrightarrow{\nabla <!--FUNCTION\; APPLICATION-->}P.$$

  (1.13)

This relations we will illustrate for the example of an electric potential $W_e(\overrightarrow{x})$:

• $$\overrightarrow{E}=-\overrightarrow{\nabla <!--FUNCTION\; APPLICATION-->}{W}_e(\overrightarrow{x}),$$

  (1.14)

• $$W_e({\overrightarrow{x}}_2)-W_e({\overrightarrow{x}}_1)=-{\int \; <!--nolimits-->}_1^2\overrightarrow{E}d\overrightarrow{r},$$

  (1.15)

• $$\begin{array}{ccc}\hfill d{W}_e\hfill & \hfill =\hfill & \hfill -\overrightarrow{E}d\overrightarrow{r}\hfill \\ \hfill \text{scalar}\hfill & \hfill \hfill & \hfill \text{vector}\hfill \\ \hfill \text{energy}\hfill & \hfill \hfill & \hfill \text{force}\hfill \\ \hfill \hfill \end{array}$$

  (1.16)

• Consequences for measurements:

  • Measure forces $E_i$ in all directions $x_i$ at each position $\overrightarrow{x}$

  • Calculate $d{W}_i=-E_{i}d{x}_i$ for the vector components along the path

  • The overall work is $dW={\sum <!--FUNCTION\; APPLICATION-->}_{i}d{W}_i$. 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”.