# 4.1: Four Questions

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Following the same pattern we used in Chapter 3 to develop the conservation of mass equations, we will once again begin our discussions by answering four questions:

- What is electric charge?
- How is electric charge stored in a system?
- How can it be transported across the boundary of a system?
- How can it be generated or consumed inside the system?

After we have answered the questions, we will then put it all together in the accounting framework.

## What is electric charge?

Anyone who has walked across a carpet on a dry day and then "zapped" someone or something has at least an acquaintance with electric charge. A detailed discussion of electric charge can be found in any physics textbook\(^{1}\) and is beyond the scope of the present discussion. However, certain facts are of interest as we try to understand electric charge:

- Electric charge is an attribute of matter and depends on the extent of the system, i.e. electric charge is an extensive property.
- Electric charge is granular and comes in discrete chunks.
- There are two types of electric charge. Benjamin Franklin called these positive charge and negative charge: a glass rod becomes positively charged when rubbed with a piece of silk, and a hard rubber rod becomes negatively charged when rubbed with a piece of cat's fur.
- Unlike charges attract each other, and like charges repel each other. This phenomenon is described in terms of Coulomb's Law.
- The unit of charge is the
**coulomb (**\(\mathbf{\mathrm{C}}\)**)**. The coulomb can be defined operationally using Coulomb's Law in terms of the force of attraction or repulsion between two charged particles. A more practical operational definition can be developed in terms of the force of attraction or repulsion between two parallel wires in which an electric charge is flowing. Again, an undergraduate physics text is the best resource for learning more about this. - The smallest granule of charge is given the symbol \(e\) and its magnitude in coulombs is \(e=1.602189 \times 10^{-19} \ \mathrm{C}\).

Notation for charge can sometimes be confusing. In this course we will use the following symbols and conventions:

\[\begin{align*} &q^{+} = \text{positive electric charge} = \text {n} e \text { where } \mathrm{n}=0,1,2,3, \ldots \\ &q^{-} = \text{negative electric charge} = \text {n} e \text { where } \mathrm{n}=0,1,2,3, \ldots \\ &q = q^{+} - q^{-} = \text {net electric charge} \end{align*} \nonumber \]

Although electric charge comes in discrete chunks, we will assume that the amount of charge in a system can take on any amount unless our system consists of an extremely small number of subatomic particles, atoms, or molecules.

\(^1\) For example *University Physics: Models and Applications* by W. P. Crummet and A. B. Western. Wm. C. Brown Publishers, Dubuque, IA, \(1994 .\)

## How is charge stored in a system?

Because positive charge, negative charge, and net charge are all extensive properties, the amount of charge in a system can be calculated in a manner that parallels our calculation of mass in a system. To do this we first need to define a charge density. **Charge density** \(\rho_{q}\) is the amount of charge per unit volume with typical units of \(\mathrm{C} / \mathrm{m}^{3}\) or \(\mathrm{C} / \mathrm{ft}^{3}\). As with the mass density, charge density can be a function of all three spatial coordinates and time. To calculate the net charge within a system, we once again integrate the charge density over the system volume: \[q_{sys}(t) = \int\limits_{V_{sys}} \rho_{q} (x, y, z, t) \ dV \nonumber \] A similar integration could be performed for positive and negative charge.

Under certain conditions, it may be more useful to work in terms of the charge per unit mass or unit mole rather than per unit volume. To do this, we need to define **mass specific charge** \(\widetilde{q}\) as the charge per unit mass with units of \(\mathrm{C} / \mathrm{kg}\) or \(\mathrm{C} / \mathrm{lbm}\). Similarly a molar specific charge \(\bar{q}\) could also be defined as the charge per unit mole with typical units of \( \mathrm{C} / \mathrm{kmol}\). Once the specific charge is known then the charge for the system would be calculated as \[q_{sys}(t) = \int\limits_{V_{sys}} \tilde{q}(x, y, z, t) \rho(x, y, z, t) \ dV. \nonumber \]

Similar calculations could be performed using the molar density and the molar specific charge. Once again, notice that integrating over the system volume produces a system charge that only depends upon time. Charge calculations might be useful in areas like electrochemistry or magnetohydrodynamics, where the charge characteristics of various ions are known as is the amount of matter in the system.

## How can charge be transported across the boundary of a system?

In answering this question, it is useful to consider the mechanisms for closed and open systems separately. For closed systems, experience has shown that charge can flow across the system boundary. The following symbols are used to describe the rate at which charge crosses or flows across a boundary:

\[\begin{array}{llll} &\dot{q}^{+} = \text{flow rate of positive charge, in amperes (A)} \\ &\dot{q}^{-} = \text {flow rate of negative charge, in amperes (A)} \\ &\dot{q} = \dot{q}^{+} -\dot{q}^{-} = \text{flow rate of net charge, in amperes (A)} \end{array} \nonumber \]

Recall that a flow rate by definition can only be defined with respect to a boundary. The standard unit for the flow rate of charge is the **ampere** (\(\text{A}\)), which is defined as \(1 \mathrm{~A} = 1 \mathrm{~C} / \mathrm{s}\). The "\(\mathrm{q}\)-dot" notation is consistent with our convention for describing flow rates of an extensive property; however, by long-standing convention the most commonly used symbol for the flow rate of net charge is the lower-case \(i\), e.g. \(i \equiv \dot{q}\). We will use both symbols interchangeably. Also by convention, the flow rate of net charge is referred to as the **electric current** and *the electric current is assumed to flow in the direction of the movement of positive charge*.

For an open system, there is an additional mechanism for charge to flow across the boundary — transport of charge with mass flow. An example where this mechanism is important is inside a battery where a flow of ions occurs internal to the battery to match the flow of current in the external circuit. Another application where this is important is in the design of electrostatic precipitators that remove pollutants from combustion exhaust gases. Although it is a different mechanism than the current flow into a closed system, we will not introduce a special set of transport terms. Just remember that if mass crosses the boundary of a system and the mass carries a charge, it must be included in the overall charge equation.

## How can electic charge be generated or consumed inside the system?

Experience has shown that *net charge is conserved*. In equation form this can be written as \[\begin{array}{llll} \dot{q}_{\text {gen}} =\dot{q}_{\text {gen}}^{+} -\dot{q}_{\text {gen}}^{-} \equiv 0 & & \rightarrow & & \dot{q}_{\text {gen}}^{+} = \dot{q}_{\text {gen}}^{-} \\ \dot{q}_{\text {cons}} =\dot{q}_{\text {cons}}^{+} - \dot{q}_{\text {cons}}^{-} \equiv 0 & & \rightarrow & & \dot{q}_{\text {cons}}^{+} = \dot{q}_{\text {cons}}^{-} \end{array} \nonumber \]

Based on this empirical evidence and our definition of net charge, we can see from Equation \(\PageIndex{4}\) that conservation of net charge implies that positive and negative charges can only be generated or consumed in matched pairs