\(P\) 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 (i.e. on the history of the system)
The value only depends on the coordinates (the state); differences between states are independent of the path between them:
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| \begin{equation*} P(\vec{x}_2) - P(\vec{x}_1) = \int_1^2 \vec{\nabla}P d\vec{x} \quad , \end{equation*} | (3.28) |
\(dP\) is a total differential:
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| \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*} | (3.29) |
i.e. the change of the coordinates defines completely the change of \(P\).
The ”force” follows from the gradient
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| \begin{equation*} - \vec{\nabla}P \quad . \end{equation*} | (3.30) |
We will illustrate these relations for the example of an electric potential \(W_{e}(\vec{x})\):
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 path.
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”.
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© J. Carstensen (TD Kin I)