The work

In this chapter, the symbols \(E\) for energy (without a vector arrow) and \(W\) for work are used to distinguish between work and energy. Although the symbol \(W\) is often used colloquially for both work and energy, work is essentially a change in energy. Both quantities, work and energy, are measured in joules (J). As described in the section on energy conservation, the law of conservation of energy states that energy can neither be created nor destroyed. It merely changes its form. This association is described by the following formula:

\begin {equation} W = \Delta E [W] = \text {1 Joule = 1 J = 1 Nm} \end {equation}

This change in energy can be caused by various physical actions, including moving an object in a force field. To further illustrate these concepts, the general formula 2 for work is shown below:

\begin {equation} W = - \int _{P_\mathrm {1}}^{P_\mathrm {2}} \vec {F} \cdot \mathrm {d}\vec {s} \label {eq:WFds} \end {equation}

The equation calculates work by integrating force along a path, whereby the direction of the force relative to the direction of movement of the object is crucial. A change of sign in the formula indicates that energy is being used from an energy store for work. This sign is context-dependent and changes depending on the perspective. If energy is withdrawn, the work is considered negative as it is subtracted from the total stored energy, signalling a reduction in the system’s energy.

1 Electrical work

In electrostatics, the forces \(\vec {F}_{\mathrm {el}}\) acting between charge carriers \(Q\) and electric fields \(\vec {E}\) are central. These forces arise when electric fields act on charge carriers. The electric field \(\vec {E}\) is defined as the force per charge and describes how large the force \(\vec {F}_{\mathrm {el}}\) is in relation to the applied charge \(Q\).

\begin {equation} \vec {E} = \frac {\vec {F}_\mathrm {el}}{Q} \Rightarrow \vec {F}_\mathrm {el} = Q \cdot \vec {E} \end {equation}

If we now substitute the formula for the electric force \(\vec {F}_\mathrm {el}\) into the formula 2 for general work, we obtain the electric work \(W_\mathrm {el}\). Since the charge \(Q\) does not change over the distance \(s\), it can be moved outside the integral. Since the electric field \(\vec {E}\) is a vector and therefore has a magnitude and a direction, the product of \(\vec {E}\) and \(\mathrm {d}\vec {s}\) always depends on the chosen distance \(s\). Therefore, a line integral must be formed using the product \(\vec {E}\) and \(\mathrm {d}\vec {s}\).

\begin {equation} W_\mathrm {el} = \int _{P_\mathrm {1}}^{P_\mathrm {2}} - Q \cdot \vec {E} \cdot \mathrm {d}\vec {s} \Rightarrow W_\mathrm {el} = - Q \int _{P_\mathrm {1}}^{P_\mathrm {2}} \vec {E} \cdot \mathrm {d}\vec {s} \end {equation}

From Module 1, we know that the integral over \(\vec {E} \cdot \mathrm {d}\vec {s}\) in the constant case yields the voltage \(U\). For a consumer-independent representation, the negative sign is omitted in the following formula and simplified for the constant case by inserting the voltage \(U\).

\begin {equation} W_\mathrm {{el}} = Q \int _{P_\mathrm {1}}^{P_\mathrm {2}} \vec {E} \cdot \mathrm {d}\vec {s} \Rightarrow W_\mathrm {{el}} = Q \cdot U\label {eq:WQU} \end {equation}

Since an electric current is necessary for the transport of electrical energy, the relationship between electric current \(I\) and charge \(Q\) known from Module 1 can now be used and rearranged to replace the charge \(Q\) in formula 5. A constant current flow \(I\) results from a uniform movement of the charge carriers \(Q\) over time \(t\). In this case, the differential operator (\(\mathrm {d}\)) can be omitted. Rearranging for \(Q\) yields the following formula:

\begin {equation} I = \frac {\mathrm {d}Q}{\mathrm {d}t} \Rightarrow Q = I \cdot t \end {equation}

Applied, this now results in the following expression, which illustrates that performing or applying work requires a flow of current.

\begin {equation} W_\mathrm {{el}} = U \cdot I \cdot t [W_\mathrm {el}] = \text {1 Joule = 1 J = 1 Ws}\label {eq:WUIt} \end {equation}

2 Path integral of electrical work

The movement of a charge in an electric field depends solely on the change in position along the electric field lines. The corresponding sign is determined by the direction of movement along or against the field lines. Movements along an equipotential surface (surface with identical electric potential) have no influence on the electric work. In Figure 1, points \(P_\mathrm {1}\) and \(P_\mathrm {2}\) are shown at different potentials, which are connected by two different distances \(s_\mathrm {1}\) and \(s_\mathrm {2}\).

The magnitudes of two equidistant position changes in opposite directions along the electric field lines are identical. These properties mean that the change in position of a charge \(Q\) from point \(P_\mathrm {1}\) to point \(P_\mathrm {2}\) is equal in magnitude to any other distance from \(P_\mathrm {2}\) to \(P_\mathrm {1}\).

\begin {equation} \left | W_\mathrm {el} \right | = \left | -Q \cdot \int _{s_\mathrm {1}} \vec {E} \cdot \mathrm {d}\vec {s}\; \right | = \left | -Q \cdot \int _{s_\mathrm {2}}\vec {E} \cdot \mathrm {d}\vec {s}\; \right | \end {equation}

This behaviour reflects the fact that the work is the same, but in one case work is expended and in the other case work is done. This leads to opposite signs, which means that the sum of both works cancels out. Mathematically, this can also be expressed by saying that the ring integral along any contour back to the starting point is zero. Since the charge \(Q\) is outside the circulation integral, the result is therefore independent of the amount of charge.

\begin {equation} W_\mathrm {el} = - Q \cdot \int _{s_\mathrm {1}} \vec {E} \cdot \mathrm {d}\vec {s} -Q \cdot \int _{s_\mathrm {2}}\vec {E} \cdot \mathrm {d}\vec {s} = - Q \cdot \oint _{s}\vec {E} \cdot \mathrm {d}\vec {s} = 0 \rightarrow \oint _{s}\vec {E} \cdot \mathrm {d}\vec {s} = 0 \end {equation}

The result of the ring integral illustrates a fundamental principle of physics: in a static field, the work done over a closed path is always zero. This confirms the path independence of the work in these fields and shows that the work done depends exclusively on the starting and ending points and not on the specific path between these points.